3.3 Malaysian Rnnit
37
O Y = 2f(x)
a
a = 2 vertical stretch with scale factor 2.
Y f(x)
2 Each point on the graph of y = 2f(x) is
b 2 times the distance to the x-axis as the
c corresponding point on the graph of
x
Reflection of y in the y-axis
x
q = 2 horizontal compression with scale
factor —1
2
Each point on the graph of y =f(2x) is
half the distance to the y-axis as the
corresponding point on the graph of
x
d
h = I horizontal translation right I unit.
k = —3 and vertical translation down 3
units.
x
oeoooooo•
139
MODELLING RELATIONSHIPS: LINEAR AND OUADRATIC FUNCTIONS
Example 19
Functions g, r and s are transformation of the graph of f. Find the functions g, r and s in terms of f.
x
r(x) = —f(x + 4) + 5 The graph of g is a horizontal stretch of the
s(x) = 2f(x— graph of f with scale factor 4.
11
q4
The graph of r can be obtained by translating
the graph of f left 4 units, reflecting in the
x-axis, and then translating up 5 units.
h = —4 and k = 5
The graph of g can be obtained by translating
the graph of fright 5 units, vertical stretching
with scale factor 2, and translating up 4 units.
a-2 11=5
Reflect What is the relationship between the graphs of y =f(x) and
The graph ofy =f(qx) is a horizontal stretch or compression ofthe graph ofy = f(x).
For which values ofq is the transfomation a stretch ratherthan a compression?
Exercise 3K a g(x) = f(—x) b g(x) = -f(x)
c g(x) = f(2x) d g(x) = 3f(x)
1 The graph of y=f(x), where —3 S 6, is f g(x) =f(x) — 3
shown. Copy the graph of fand draw these g(x) =
functions on the same axes.
axaaaaoo
x
Malaysian Rnnit
37
2 The graphs of functions r and s are 3 The diagram shows the graph of y for
transformation of the graph off. Find the
functions r and s in terms of f. 2 sxS8.
a Write down the range of f.
Let g(x) =f(—x).
b Sketch the graph of g.
c Write down the domain of g.
The graph of h can be obtained by a vertical
translation of the graph of g. The range of h
d Find the equation for h in terms of g.
e Find the equation for h in terms off.
Noauaonoo
Noanaoo•o
woavaaooon
Developing inquiry skills
In the opening scenario for this chapter you looked at how crates of
emergency supplies were dropped from a plane.
=The function h(t) —4.9t2 + 720 gives the height ofthe crate during free fall.
How could you transform the parent graph h (t) = t2 to give the function
h (t) = —4.9t2 + 720? What do these transformations tell you about the
motion of the crate in this context?
Npp
Rf1Æf
141
MODELLING RELATIONSHIPS: LINEAR AND OUADRATIC FUNCTIONS
3.4 Graphing quadratic functions
In section 3.3, you studied transformations of quadratic graphs. In this
section, you will further study the graphs of quadratic functions and
different forms of the equations for quadratics. First you will graph
with a GDC and then without.
Transferring a graph from GDC to paper
You may be asked to sketch, on paper, the graph of a quadratic
function. You could do this by plotting the graph on your GDC and
then carefully re-creating a sketch of the graph on paper.
When you "sketch" a parabola, your sketch should show the general
shape of the graph accurately, and label the key features including:
the coordinates of x- and y-intercepts
the coordinates of the vertex
the equation of the axis of symmetry
Example 20
The quadratic function f(x) = —0.5x2 + 7.5x— 18 is said to be in general form. Use
technology to plot the graph of f(x) = —0.5x2 + 7.5x— 18 and then sketch this on
paper.
Your sketch should show the general correct shape of the graph, with key features
labelled.
Also state the domain and range of this function.
•Ungaved Graph the function on your GDC.
flCr)-o x +7sæ18
20
y-intercept is (0, —18) Read the y-intercept from your GDC.
The zeros are at x = 3 and x = 12. Use your GDC to find the zeros of the function.
Use your GDC to find the coordinates of the
So, (3, 0) and (12, 0) are also points on
vertex.
the graph.
Vertex is (7.5, 10.125)
142
3.4 Malaysian Rnnit
O
(7.5, 10.125)
x
1
-18 You can evaluate the function fat any real value
of x, so the domain is the set of all real numbers.
Domain of fis {x e IR).
The vertex of the graph is (7.5, 10.125), so the
RRange of fis {y e I 10.125).
maximum value of the function fis 10.125
Exercise 3L
Use your GDC to plot the graph of 3 f(x) =-2x2 -7x+3 EXAM HINT
the quadratic function. Then use 4 f(x) = 1.25x2 - 12.4x In examination
the graph to find the coordinates questions where the
Use your GDC to plot the graph domain is restricted
of the x-intercept, the y-intercept for a given function,
of the quadratic function over you must not sketch
and the vertex of the graph. the given domain. Sketch the points outside of
that domain.
1 f(x) = 3x2 + 7x—4 graph on your paper, labelling the
coordinates of the key features of
Use your GDC to plot the graph of the graph. Then write down the
the quadratic function. Sketch the range of the function.
graph on your paper, labelling S f(x) =-3.6x2+ 8.1,
the coordinates of key features of for -1.5 s x <1.5
the graph. Then write down the 6 f(x) = 3x2 + 7x— 4, for —2 x S 4
domain and range of the function.
When sketching the graph of a quadratic without the use of a GDC,
different forms of the equation will help you to identify the key
features of the graph.
Vertex form
In the previous section, you graphed functions of the form
f(x) = a(x— h)2 + k by transforming the graph of f(x) . Complete the
following investigation to learn what the parameters of the equation
tell you about the graph of the function.
143
MODELLING RELATIONSHIPS: LINEAR AND OUADRATIC FUNCTIONS
Investigation 5
1 Graph each function by transforming the graph ofy = x2.
a f(x) =2(x+ 1)2—4
b f(x) 1
2
c f(x) = —3(x- 1)2-6
k2 Copy the table and record the parameters a, h and Use your graphs
from question 1 to determine whether the graph is concave up or down,
the equation of the axis of symmetry and the coordinates of the vertex,
f(x) = — + k Up/ Axis of
a h k down symmetry Vertex
a f(x) =2(x+
b f(x)
2
c f(x) = —3(x- 1)2-6
Look for patterns in your table and answer the following questions.
Conce tual Which parameter in the functionf(x) = a(x — h)2 + k
determines whether the graph of the function is concave up or concave
down? Explain how you determine whether it is up or down.
a Conce tual What is the equation of the axis of symmetry and what are
the coordinates of the vertex of the graph off(x) = a(x — h)2 + k?
5 Consider the functionf(x) = 4(x— 3)2 + 2. Without using your GDC, say
whether the graph is concave up or concave down. Write down the equation
of the axis of symmetry and the coordinates ofthe vertex of the graph. Then
graph the equation on your GDC to verify your answers.
A quadratic function written in the formf(x) = a(x — h)2 + k, a 0, is said
to be in vertex form. The coordinates of the vertex are (h, k) and the equation
of the axis of symmetry xis = h.
Example 21
Write down the coordinates of the vertex and the equation of the axis of symmetry
for each function:
a f(x) = —3(x— 4)2+ b f(x) = 2 (x +
144
3.4 Malaysian Rnnit
O
a f(x) = —3(x— 4)2+1
vertex: (4, l) The vertex is (h, k), where h = 4 and k = l.
The axis of symmetry is x = h.
axis of symmetry: x = 4 f(x) = 2(x— (—3))2 + (—6), so h =—3 and
b 3)2-6
vertex: —6)
axis of symmetry: x = —3
General form
A quadratic function in the form f(x) = ax2 + bx+ c, a 0, is said to be in
general form. You will learn about the parameters of this form in the
following investigation.
Investigation 6
1 Expand each of the following equations to write them in the general form,
f(x) = ax2 + bx + c.
b f(x) 1
2
c f(x) = —
2 Complete a table like the one below. Record your answers from question International-
bb mindedness
1 in the first column. Find the values of—— and f
Recall that you How do you use the
2a 2a Babylonian method of
found the equation of the axis of symmetry and coordinates of the vertex multiplication?
of each graph in investigation 5. Try 36 x 14
f(x)=ax2+bx+c b Axis of Vertex
2a
a symmetry
b
c
3 Conce tual What is the equation of the axis of symmetry of the graph
f(x) = ax2 + bx + c?
Conce tual What are the coordinates of the vertex of the graph of
f(x) = ax2 + bx + c?
5 Conce tual What is the x-coordinate ofthe y-intercept ofthe graph of any
= +function? Hence, find they-intercept of the graph off(x) ax2 bx + c.
145
MODELLING RELATIONSHIPS: LINEAR AND OUADRATIC FUNCTIONS
f(x) = ax2 + bx + c, a 0, is called the general form of the equation fora
quadratic function.
b
The equation of the axis of symmetry is x = ——-. The coordinates of the
vertex are b
and the y-intercept is (0, c).
Example 22
Find the equation of the axis of symmetry, the coordinates of the vertex, and the
y-intercept. Use these features of the graph of the quadratic and a point of symmetry to
the y-intercept to sketch a graph.
a f(x) = —x2+6x—4 b f(x) =2x2+4x— I
a f(x) = —x2 + 6x — 4
—b —6 = 3 the axis of symmetry is
Fa- 2(-1)
—b = — (3)2 + 6(3) —4 = 5 the
2a
vertex is (3, 5).
X
c = the y-intercept is (0, —4).
The point symmetric to the y-intercept is
a = —l < the parabola is concave down
(0,-4) and the vertex is a maximum point.
b f(x) = 2x2 + 4x— I
x *= —l the axis of symmetry is
(0,-1) Fa- 2(2)
—b
+4(-1) 1
the vertex is (—1,—3).
c = —l the y-intercept is (0, —l).
The point symmetric to the y-intercept is
a = 2 > 0 the parabola is concave up
and the vertex is a minimum point.
3.4 Malaysian Rnnit
37
Factorized form
For the graph of a quadratic function y the x-intercepts are the
x-values where y =f(x) = 0.
Writing a quadratic function in factorized form allows us to easily find
the values of x for which f(x) = 0.
A quadratic function which is written in the form f(x) = a(x — p) (x — q),
a 0, is saidto be written in factorized form (or intercept form). The
coordinates ofthex-intercepts of the graph are (p, 0) and (q, 0).
Here,f(x) is written as the product of three linear factors: a, (x —p) and (x— q).
Sof(x) = 0 only when one or more of the three factors is itself equal to zero.
Since a 0, it follows that either:
(x — p) = 0 and hence x = p, or
(x — q) = 0 and hence x = q
(p, 0) and (q, 0) are the x-intercepts of the graph y=f(x).
You can also find the equation of the axis of symmetry from the
factorized form. Due to the symmetry of a parabola, the axis of symmetry
will pass through the midpoint between the x-intercepts.
Ifthe x-coordinates of the x-intercepts are p and q then the equation of the
axis of symmetry is x = P+q
2
and the vertex of the graph has coordinates
2
Example 23
Find the equation of the axis of symmetry and the coordinates of the x-intercepts, the
y-intercept and the vertex. Use these features to sketch the graph of the parabola.
a f(x) = (x —4) (x— 2) b f(x) =
a f(x) = (x— 4) (x — 2)
(0,8) a= I > the parabola is concave up and the
vertex is a minimum point.
p and q = 2 the x-intercepts are (4, 0) and
= 3 the axis of symmetry is x = 3.
22
—l the
(4,0) 2
vertex is (3,—1).
Continued on next page
MODELLING RELATIONSHIPS: LINEAR AND OUADRATIC FUNCTIONS
O
Note that you find the y-intercept of the graph
of any function by finding f(0).
f(0) = (0— 2) = 8 the y-intercept is
b f(x) =
(-21, z) a =—2 < 0 the parabola is concave down
and the vertex is a maximum point.
p =—3 and q the x-intercepts are (—3, 0)
and (—1, 0).
3+ = —2 the axis of symmetry is
22
x=-2.
f(-2) = -2(-2 + + 1) = 2 the
2 the y-intercept is
vertex is (—2, 2).
2(0 + 3)(0 + l) = —6
Reflect What is the largest possible domain of a quadratic function?
What is the range of the function y = a(x— h)2 + k?
A summary of the relationship between the parameters of a quadratic
function and key features of the graph of the function is shown in the table.
Quadratic functions and their graphs TOK
General form Vertex form Intercept form q), How would you choose
f(x) = ax2 + bx + c, f(x) = a(x— + k, f(x) = which formula to use?
y-intercept: (0, c) axis of symmetry: x-intercepts: (p, 0), When is intuition
axis of symmetry: x=h axis of symmetry: helpful and harmful in
b mathematics?
vertex: (h, k)
vertex: b 2
b
vertex: P +q
2
2
a>0 the parabola is concave up and the vertex is a minimum point.
a<0 the parabola is concave down and the vertex is a maximum point.
Reflect What do the parameters in each of the different forms tell you about
the graph?
148
3.4 Malaysian Rnnit
37
Reflect Why is it useful to have more than one form for representing a
quadratic function?
Exercise 3M
1 Write down the equation of the axis of 3 Find the coordinates of the x-intercepts, the
equation of the axis of symmetry and the
symmetry and the coordinates of vertex for coordinates of the vertex for the graph of
the graph of each function: each function:
a f(x) = 2(x— 3) 2 + 4
a f(x) = (x— 2) (x— 4)
b f(x) = (x— — 5
c f(x) 3)2+2 b f(x) l)
d c f(x) = —
2 Find the coordinates of the y-intercept, the d f(x) = 2(x + 3)(x+ 2)
equation of the axis of symmetry and the 4 Sketch the graph of each function and label
coordinates of the vertex for the graph of
each function: the key features of the graph:
a f(x) = x2— 8X+ 5 a f(x) = 2(x+ 1)
b f(x) =
c f(x) = —2x2-8x- 11 b f(x) = —2x2- 8x- 11
d f(x) = c f(x) =
d f(x) = 3x2+12x+8
Changing to general form or factorized form
You now know the three forms of a quadratic function. At times you
may want to change one form to another to determine certain key
features of the graph of the functon.
If a quadratic expression will factorize, then you can factorize it in
order to write in factorized form.
Example 24
Each of the following functions can be written in the form f(x) = a(x— p) (x— q).
Find the values of a, p and q and then write down the coordinates of the x- and y-intercepts
of the graph of y
a f(x) =x2+6x— 16 b f(x) = —4x2 + 2x
a f(x) =x2+6x— 16 Factorize.
f(x) = (x + 8) (x — 2) Note that a = l, p = 2, q = —8 is also correct.
x-intercepts: (—8, 0) and (2, 0) f(x) = x2 + 6x— 16 —16
y-intercept: (0, —16)
Continued on next page
149
MODELLING RELATIONSHIPS: LINEAR AND OUADRATIC FUNCTIONS
O Factorize.
b f(x) = —4x2+ 2x
f(x) = —2x(2x— l) Factor out the coefficient of Xin the first
f(x) =—4x x —— factor.
21) 11
1
—40—0) x
2
22
x-intercepts: (0, 0) and
0)
y-intercept: (0, 0)
—4X2 + = ++ C= O
A quadratic function given in factorized form can be expanded to
change it to general form.
Example 25
a The function f(x) = 3(x— l) (x + 2) can be written in the formf(x) = ax2+ bx + c. Find the
values of a, b and c and then write down the coordinates of the x- and y-intercepts of the
graph y =f(x).
1
b The function f(x) = ——(x —4)2 —2 can be written in the formf(x) = ax2+ bx + c. Find the
2
values of a, b and c and the coordinates of the vertex and the y-intercept of the graph
f(x) = 3(x2 + x —2) Expand the brackets.
f(x) = 3x2 + 3x— 6
Multiply out the bracket.
x-intercepts: (1,0) and (—2,0)
f(x) = 3(x— l)(x+2) p = I and q = —2
y-intercept: (0,—6) f(X) = 3X2 + 3X — 6 C = —6
Expand the brackets.
b f(x) = 1 -4)2-2
2
f(x) (x2 -8X+16) -2
f(x) 1 Multiply out and simplify.
+4 x —10
2
1
4, c=-10
2
vertex: (4, —2) 1 h = 4 and k
-10
f(x) = ——(x —4)2 —2
2
y-intercept.• (0, -10) f(x) 1
2
150
3.4 Malaysian Rnnit
Exercise 3N
1 Each function can be written in the form S Let fix) = (x— — 2. Part of the graph of
f(x) = a(x—p) (x—q), where p > q. y is shown.
Find the values of a, p and q and then c
write down the coordinates of the x- and
y-intercepts of the graph of y =f(x).
a f(x) + 7x — 18
b f(x) = 3x2 - 1 IX+ 10
d f(x) = —4x2 + 18x— 8
2 Each function can be written in the form x
f(x) = ax2 + bx+ c. Find the values of a, b and
a The vertex of the graph of y=f(x) is A.
c and then write down the coordinates of the
i Write down the coordinates of A.
x- and y-intercepts of the graph of y
ii Write down the equation of the axis
a f(x) =4(x— 5) of symmetry for the graph of y = f(x).
b
3 Each function can be written in the form b Find the equation for the function fin
f(x) = ax2 + bx+ c. Find the values of a, b and
c. Find the coordinates of the vertex and of the form f(x) = ax2 + bx + c.
the y-intercept of the graph of y =f(x).
C The coordinates of B are (O,q). Write
a f(x) down the value of q.
b f(x) e The coordinates of C are (p,q). Find the
2
value of p.
4 The function f(x) = x2 — 2x— 8 can be written
in the form f(x) = a(x— p)(x — q), where p > q. 6 Let f(x) — 2x— 3 andg(x) = x— 2.
a Find the values of: a Let h(x) = (fo g) (x). Show that
h(x) = x2 — 6x+ 5.
b Find the equation of the axis of
symmetry for the graph of h.
pii C Find the coordinates of the vertex of the
qiii graph of h.
b Write down the coordinates of the: d Find an equation for h in the form
h(x) = (x—p)(x — q), where p and q are
i x-intercepts
ii y-intercept integers.
c Find the coordinates of the vertex of the e Sketch a graph of y = —h(x), for
1 SXS5.
graph of y = f(x).
d Sketch the graph of y =f(x).
Fitting a quadratic function to a graph
Now you will find the equation of a quadratic function given some
information about its graph.
151
MODELLING RELATIONSHIPS: LINEAR AND OUADRATIC FUNCTIONS
Reflect What are three different forms of an equation of a quadratic
function?
For each form of an equation ofa quadratic function, which features of the
graph of the function help you determine the parameters in the equation?
Example 26
Use the information shown in the graph to find an equation for the quadratic function.
Write your final answer in general form, f(x) = ax2 + bx + c.
a
1
10 1
x 6
10
o,
o, 24)
wnn x
a f(x) = a(x— p) (x— q)
Since both x-intercepts are given, you can start with
a(x + — 3) intercept form and substitute —4 for p and 2 for q.
-24 = - 3) From the point (0,—24) you know that y=—24 when x = 0.
-24
Substitute and solve for a.
+ 4)(x— 3)
You now have the function in intercept form. Expand to
f(x) = 2(x2 + x- 12)
write fin general form.
f(x) = 2x2 + 2x- 24 You can check your answer by graphing this answer
b f(x) = a(x— h)2+k on your GDC and verifying the graph has the correct
intercepts.
Since the vertex is given, you can start with vertex form
and substitute 2 for h and —2 for k.
From the point (0, 4) you know that y when x = 0.
4=4a 2 Substitute and solve for a.
a=1.5
f(x) = 1.5(x- 2)2-2 You now have the function in vertex form. Expand to
1.5(x2- -2 write fin general form.
1.5x2- You can check your answer by graphing this answer on
your GDC and verifying the graph has the correct vertex
and y-intercept.
152
galayvan
Exercise 30
1 Use the information shown in the graph
to find an expression for the quadratic
function. Write your final answer in the
form f(x) = ax2 + bx+ c.
*lemn
(2 -16 (1,3)
16
(0,3
menu. ( 10,6ö)
45)
10 10
153
MODELLING RELATIONSHIPS: LINEAR AND OUADRATIC FUNCTIONS
2 The graph of the quadratic function a Write down the coordinates of the vertex
y=f(x) has x-intercepts (—1, 0) and (3, 0).
of the graph and then explain its meaning
The function has a maximum value of 4.
in terms of the context of the graph.
a Find the equation of the axis of
symmetry for the graph of y b Find an equation for h(t) and give the
domain.
b Write down the coordinates of the
C Find the predicted height of the model
vertex for the graph of y =f(x).
rocket 2.4 seconds after launch.
c Find an equation for fin the form
f(x) = a(x — + k, where a, h and k are Predicted model rocket height
constants to be determined.
100
Ad translation of the graph of y =f(x) right 90
80
4 units and down 5 units results in the
70
graph of y =g(x). Find an expression for
the function g(x) in the form 60 omaaoo•o
f(x) = ax2 + bx + c.
50
3 The table and graph are representations
of the predicted height, h m, of a model 40 o:aaaoaoo
rocket, t seconds after it is launched. 30
20
Time Height (m)
after launch (s) 35 10
1 80 67 x
3
Time after launch (s) 8
4
5
8
Developing inquiry skills
In the opening scenario for this chapter you looked at how crates of
emergency supplies were dropped from a plane. The functions give
the height of the crate.
During free fall: h(t) = —4.9t2 + 720
With parachute open: g(t) = — 5 t + 670
Without using your GDC, sketch a graph of this piecewise function
and label the key features.
NPp
TOK
How can you deal with the ethical dilemma of using mathematics to plot the course
of a missile?
154
3.5 Malaysian Rnnit
37
3.5 Solving quadratic equations by
factorization and completing the
square
Break-even point, profit, revenue and costs are four related business
concepts.
What does each of these terms mean?
Why would business owners want to find break-even points?
The publisher of a newsletter uses the following models to estimate
monthly revenue, R, and monthly cost, C. Cost and revenue are in
thousands of euros and x is the number of subscribers in thousands.
R(x) = 35x - 0.25x2
C(x) = 300 + 15x
How would you find the number of subscribers that determines the
break-even point?
In sections 3.3 and 3.4 you saw that any equation that can be written in
the form ax2+ bx + c = 0, where a * 0 is called a quadratic equation. In
this section you will study two methods of solving quadratic equations.
Solving by factorization International-
If the left-hand side of ax2+ bx + c = 0 can be factorized to the form
a(x — p)(x — q) = 0, then the equation can be solved using the zero- mindedness
product property. You have already met this concept in section 3.4
when you learned to write a quadratic equation in factorized form. Ancient Babylonians
and Egyptians studied
a(x —p) (x — q) is the product ofthree linearfactors:a, (x —p) and (x — q). quadratic equations
like these thousands
So a(x — p) (x — q) = 0 only when one or more of the three factors is itself ofyears ago to find, for
example, solutions to
equal to zero. problems concerning
the area ofa rectangle.
Since a (), it follows that either:
(x — p) = 0 and hence x = p, or
(x — q) = 0 and hence x = q
Example 2?
Solve each equation using the factorization method:
a x2 + 5x+ 6=0 b 2x2-7x-4=o C 9x2 +6x+l
a x2+ 5x+6=0
Factorize the left-hand side of the equation.
0 or x +3=0 Use the zero-product property to set each factor equal to
zero and solve.
x = —2 or x = —3
—2 and —3 are called the solutions or roots of the equation.
Continued on next page
155
MODELLING RELATIONSHIPS: LINEAR AND OUADRATIC FUNCTIONS
O Factorize the left-hand side of the equation.
b 2x2-7x-4=o
Use the zero-product property to set each factor equal to
2x+ or x-4=o zero and solve.
1 HINT
In parts b and c, the expression
or has not been factored into the form
2
a(x —p) (x — q), though it could be.
1
Sometimes (like in these examples)
2
it is better not to factor out the "a".
Factorize the left-hand side of the equation.
This can be written as (3x+ 0 and so
3X+1=o 9x2 + 6x+ I is called a perfect square trinomial.
1 When a quadratic is a perfect square you can say the
equation has two equal roots.
3
Exercise 3P 2 Solve by factorization:
1 Solve by factorization: a 2x2+x-3=o b 3x2+5x- 12=0
c 4x2+11x+6=o d 9x2 — 49 = 0
a b x2-x-20=o e 4x2-16x+7=o f 12x2+ lix-5=o
ec - 8x+ 12 = o d x2-121=o
e x2 + x — 42 = 0 f x2 -8X+ 16=0
Quadratic equations not given in the form ax2 + bx + c = 0 must
re-written before solving by factorization.
Example 28
Find the roots of these equations using the factorization method:
a 3x2-4= 16x+8 b n(n+ 8) = 5(n+2)
a 3x2-4=16X+8
3x2 - 16x- 12 = o Collect like terms on one side of the equation and write
or X -6=0 the quadratic in general form.
2 Factorize and solve for x.
3
Remember that roots is another term for the solutions
of an equation.
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3.5 Malaysian Rnnit
37
O Quadratic equations do not need to be written in terms
of x. The variable in this equation is n.
= 511+10 Expand the brackets and collect like terms.
n2+ 3n -10=0 Factorize and solve for n.
Exercise 30
1 Solve the following quadratic equations by e
-15
factorization:
2 Let f(x) x
a x2-x-20=2x+8
— 2, g(x) = 2x+ I and
b 2x2 - Bx- 8 = -x2 + 2x
h(x) + 5x + 3.
d 3x(x + 5) = —(x + 5)
a Show that (fog) (x) = 4x2 + 4x -1.
b Find the values of x for which
Quadratic equations involving perfect squares
Consider the equation x2 = 4. You could rewrite the equation as
—4=x2 0 and factorize, (x + 2) (x— 2) = 0. The solutions are x = —2,
x = 2. Alternatively, you could take the square roots of both sides of
x2 = 4. You must take both the positive and negative square root of 4 to
obtain both solutions, = ±v'fi. The solutions are x = ±2. The plus or
minus symbol allows you to show both roots in a condensed form.
The quadratic expression x2 + + 25 is called a perfect square
becausex2+ 25 = (x + 5)2.
Quadratic equations which involve a perfect square can be solved by
taking square roots.
Example 29
Find the solutions to each equation:
a x2+mox+25=11
a x2+10X+25=11 Notice that if you collect like terms, x2 + + 14 = 0
x +5 = ±UiT does not factorize.
Factorize the perfect square trinomial on the left-hand
side of the equation.
Take the square roots of each side of the equation.
Solve for x. The answer represents the two solutions,
_5 and —5
Factorize the perfect square trinomial on the left-hand
side of the equation.
Continued on next page
MODELLING RELATIONSHIPS: LINEAR AND OUADRATIC FUNCTIONS
O Take the square roots of each side of the equation and
x = 2±2Vä solve for x.
The solution may be rewritten as 2± 2E, since
Points of intersection
You can use your GDC to find points of intersection of curves and
lines. You can also solve equations by finding points of intersection.
Example 30
a Find the points of intersection of the graphs of
f(x) = 4x2 — 2x— 5 and g(x) = 3x + 2.
b Solve x2 + IOx+ 25 = 11 by using your GDC to find points of
intersection.
a Graph f(x) = 4x2 — 2x— 5 and
g(x) = 3x + 2.
There are two points of
intersection. Find the
coordinates of both points.
neumo
x
6
The answers are shown
correct to 3 s.f.
10
n—uma
x
unoaa
The points of intersection
are (2.09, 8.26) and (-0.838,
-0.514).
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3.5 Malaysian Rnnit
37
O x2+ 25 = 11
b
2
Graph the left- and right-
hand sides of the equation: y
=x2+ andy= 11.
x
d6å6äå The x-coordinates the points
x of intersection are the
x = -1.68, x = -8.32 solutions to the equation.
You must find both
solutions.
The answers are shown
correct to 3 s.f.
Notice that you found the
exact values to the solution
of the equation x2 + + 25
= II in example 29 part a.
You can check to see that
your exact answers have
the approximate values you
found here.
x = -5+ü--1.68
x -8.32
Alternatively, you could
rewritex2 + + 25 = II as
x2 + I()x+ 0 and solve
by finding the x-intercepts
of the graph of f(x) = x2 +
10x+ 14.
Exercise 3R 3X2+ 12X+36=12 4x2-10X+25=27
Find the exact value of the solutions to Find the points of intersection for the following
pairs of functions.
each equation, without using your GDC.
5 f(x) = —x + 5 and g(x) = 4x2 + 3x+ 2
Then use your GDC to approximate and
graphically check your solutions.
lx2-8x+16=10 2x2+20x+100=15
159
MODELLING RELATIONSHIPS: LINEAR AND OUADRATIC FUNCTIONS
6 f(x) = 4.25x2 + 5.35x- 4.81 and Solve each equation by graphing two functions
g(x) = 2x+ 5
and using your GDC to find the points of
2x2 + 3x + 4 and g(x) = —x + 6
intersection.
8 f(x) = 2x2 — 3x+ I and g(x) = —x2+7x-4
9 -3.6x2+ 5.4x-2= 1.8x-7.2
10 0.5x2+ 3x=-x2 +4
Solving equations by completing the square
Many quadratic equations that you will come across do not involve
a perfect square. However, they can easily be transformed into an
expression which does involve a perfect square by a process called
"Completing the square".
You can then use the technique you have just studied in order to solve
them.
First you will find the value of c that can be added to X2 + bx to form a
perfect square trinomial.
The following area models will show you how to do this.
x 6 The area of the region shaded in
x x2 green isx2 and the area of the region
shaded in pink is 6x. Thus, the area
of the whole rectangle is x2 + 6x.
x 33 Divide the pink region into two equal
x
parts.
x 3 Rearrange the parts.
x The large square formed has area
So the expressionx2 + 6xbecomes
a perfect square when you add 9.
39
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37
Investigation ?
For each ofthe following diagrams, find the value of c which makes the expression
into a perfect square, and then write the expression as a perfect square.
1 x2 + 6x + c 2 x-2 + 8x + c
x3 x
xx
3 4 x2 + bx + c
3
S Copy and fill in the blank: To complete the square forx2 + bx, add
6 How does the area model help you understand
the process of completing the square and why is the process called
completing the square?
To solve a quadratic equation in the formx2 + bx + c = 0 by completing the
square:
i Write the equation in the formx2 + bx = c.
2 Add to both sides and write the equation as:
2 2
b b
2 2
3 Factorize the perfect square on the left-hand side: 2 2
b b
2 2
4 Take the square root of both sides and solve for x.
161
MODELLING RELATIONSHIPS: LINEAR AND OUADRATIC FUNCTIONS
Example 31
Solve these quadratic equations by completing the square:
a 10 b x2+4x-21=o c x2-5X+3=o
a 10 2
X2 — 6x + 9 = 10+9 Add b to both sides of the equation.
2
The coefficient of x is —6.
1
= —3 and
2
Complete the square by adding 9 to
both sides of the equation.
(x— 19 Factorize the perfect square on the
left-hand side.
x _ 3 = ±Vi3 Solve for x.
b x2+4x-21=o Add 21 to both sides of the equation.
X2 + 4X = 21 The coefficient of x is 4.
x2 + 4x + 4 = 21+4 44) = 2 and (2)2 = 4.
(x + = 25 2
X = —2±5 Complete the square by
adding 4 to both sides
of the equation.
Factorize the perfect
square on the left-hand
side.
Solve for x.
C x2— Subtract 3 from both sides of the HINT
equation. When the solutions
are rational numbers
x2 — 5x = — 3
you could also solve
the equation by
factorization.
The coefficient of x is —5.
5 2
( 5) = and 5 25
2 24
—x
25 25 25
Complete the square by adding to
4
4 4 both sides of the equation.
2 Factorize the perfect square on the
5 13 left-hand side.
24
5 Solve for x.
22
2
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37
Exercise 3S 3 International-
Solve these quadratic equations 4 x2-12x+4=o mindedness
by completing the square: s x2 + 5x — 4 = 0 Over one thousand years
ago, Arab and Hindu
1 x2+12x=2 6 mathematicians were
2 developingmethods
similar to completing the
To solve ax2 + bx + c = 0 by completing the square, when the coefficient square to solve quadratic
equations. They were
of x-2 is not l, you must first divide both sides of the equation by the
finding solutions to
coefficient of x2. mathematical problems
Example 32 such as 'What must be
the square which, when
Solve by completing the square: increased by 10 of its
own roots, amounts to
39?" This is written as
x2+ 10x= 39
a 3x2+12x=18 b 4x2-8x-6=o
a 3x2+ 12x= 18 The coefficient of x2 is 3.
x2 + 4x=6 Divide both sides of the equation by 3.
Now the coefficient of x is 4.
1
= 2 and (2)2 = 4.
2
Complete the square by adding 4 to
both sides of the equation.
Factorize the perfect square on the
left-hand side.
Solve for x.
b 4x2 — 8x -6=0 Divide both sides of the equation by
4 and add the constant term to both
3
sides.
2
Now the coefficient of x is —2.
5-2) = —l and I
2 Complete the square by adding 1 to
both sides of the equation.
25
2 Factorize the perfect square on the
left-hand side.
5 Solve for x.
2
The answer could also be written as
5
2
2
163
MODELLING RELATIONSHIPS: LINEAR AND OUADRATIC FUNCTIONS
Reflect What can be said about the roots of a quadratic equation that can be
solved by the factorization method?
What can be said about the roots of a quadratic equation that can be solved by
completing the square?
Can you use factorization to find the roots of any quadratic equation? Can you
use completing the square to find the roots of any equation?
Exercise 3T C and R are both given in thousands of
euros. The values of x for which revenue is
Solve each quadratic equations by completing
equal to cost are called "break-even" points.
the square:
a Write down an equation, in terms of x,
1 2x2+ 16x= 10
2 5x2 - 30x= 10 to find the break-even points.
3 6x2- 12x-3=o b Solve your equation in a by completing
4 6x(X+ 8) = 12 the square.
s 2x2+x-6=o c Write down the number of subscribers at
6 2x(x+8) +12=0
which newspaper sales break even.
The publisher of a newsletter uses the
following models to estimate monthly Profit is equal to revenue minus cost.
sales revenue (R) and the monthly cost of d Find the number of subscribers that
production (C), in terms of the number of
yields maximum profit.
people who subscribe to the newspaper (x). e Find the maximum profit.
R(x) = 35x- 0.25x2
C(x) = 300 + 15x
Developing inquiry skills
In the opening scenario for this chapter you looked at how crates of
emergency supplies were dropped from a plane. The functions h(t) and
g(t) give the height of the crate:
During free fall: h(t) = —4.9t2 + 720
With parachute open:g(t) = —5t + 670
How long after the crate leaves the plane does the parachute open? Write
a quadratic equation you could solve to answer this question.
How could you solve this equation?
3.6 The quadratic formula and the
discriminant
Investigation 8
Copy the following table and fill in the missing steps or explanations to complete the square for the general
quadratic equation ax2 + bx + c = 0.
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3.6 Malaysian Rnnit
37
O 2 2 Solve by completing the square.
ax2 + bx + c 1 Divide both sides of the equation by a.
1 —x—
2 2 —Subtract from both sides of the equation.
b 3
3
b 172 4
4 5
6 Factorize the left-hand side of the equation.
a 4a 4a2
+—5 x2 +—x b2 — 4ac
a
6
2a 8 Solve for x.
8
9 Conce tual What is the result of completing the square for the general equation ax2 + bx + c = 0?
Conce tual Could you use this formula to find the roots of any quadratic equation?
11 Conce tual How does completing the square for the general form allow you to find the roots of any
quadratic equation?
The solutions to any quadratic equation given in the form ax2 + bx+ c = 0,
2
where a 0, are given by the quadratic formula
Example 33
Find the exact solutions to each equation by using the quadratic formula:
a x2 —6x+4=0 b 3x2 + 6x c 7X+5 d x2 —4x+5=0
2(1) Use the quadratic formula with a = l, b = —6 and
6±47) 6±26 Give your solution in exact form and simplified as
22 far as is possible.
Continued on next page
165
MODELLING RELATIONSHIPS: LINEAR AND OUADRATIC FUNCTIONS
O Write the equation in general form.
b 3x2 + 6x = -2
Use the quadratic formula with a = 3, b = 6 and
3x2 + 6X+ 2 = o c=2.
2(3) Give your solution in exact form and simplified as
-6±Uiä far as is possible.
6 Write the equation in general form.
63 Use the quadratic formula with a = 6, b = —7 and
c 6x2=7X+5 The solutions are not surds, so this equation could
also have been solved by factorization.
6x2 - 7x- 5=0
2(6)
7±13
12 12
51
2(1) Use the quadratic formula with a = l, b = —4 and
2 There is no real number with a square of —4, so this
equation has no real solutions.
no real solutions
The solutions ofa quadratic equation ax2 + bx + c = 0 are also known as the
roots of the equation. These values are also called the zeros of the quadratic
function y = ax2 + bx+ c.
If p and q are the real zeros ofy = ax2 + bx + c, then the points (p, 0) and
(q, 0) arex-intercepts of the graph of y = ax2 + bx+ c.
Exercise 31.1 2 Solve each equation using the quadratic
formula:
1 Use the quadratic formula to find the roots
of each equation: a 9
a x2+4x-2=o c 2+ 2x-x2=o x
b 3x2-8X+5=o 2x
c 2x2- 5x-2=o
d 3x2+4x=-10
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37
3 Use the quadratic formula to find the zeros 4 Let f(x) = 2x2 — 4x + c. The y-intercept of the
of each function: graph y =f(x) is (0, —2).
a Y = 6x2 + 5x-6 a Write down the value of c.
b Y = 2x2 -4x+1
b Find the vertex of the graph y =f(x).
c
c The x-intercepts of the graph y are
(r + (s, 0) and (r —G, 0). Find the value
of r and of s.
Investigation 9
A quadratic equation may have two distinct real roots, two equal real roots or no real roots. Complete the
investigation to explore the nature of the roots a quadratic equation.
—1 For each equation write down the value of b2 4ac, use the quadratic formula to find the real
roots, and then describe the nature of the roots. Use your GDC to make a rough sketch of the graph
of the related function. Record your answers in a table like the one below. The first row of the table has
been completed for you
ax2 + bx+c= O — 4ac Roots Nature of Sketch of
roots
y=ax2 + bx + c
a —5X2 + -F 2 = O two distinct
-10 5 real roots
x
b 3x2 + 2x- 1 = o
d -x2-6x-9=o no real
e 4x2 -F 12x 9 = O roots
g 4x2- 2x+ 3
2 The quadratic equations in parts fand g have no real roots. Use their graphs to explain why.
Conce tual What are the three types of roots that a quadratic equation can have?
What is the relationship between the nature of the roots of the equation ax2 + bx + c = 0 and
the value of b2 — 4ac?
What is the relationship between the value of b2 — 4ac and the number of x-intercepts of the
graph of the function. y = ax2 + bx+ c?
MODELLING RELATIONSHIPS: LINEAR AND OUADRATIC FUNCTIONS
A quadratic equation ax2 + bx+c= 0 has discriminant A = b2 — 4ac. HINT
AThe symbol is used to represent the discriminant.
A quadratic equation
The discriminant can be used to determine (or discriminate) the number
and nature of the roots of the equation ax2 + bx+c=0, or the number may have complex
of x-intercepts of the graph of the equation y = ax2+ bx+c.
roots, which include
For... The equation ax2+ bx+c=0 The graph ofy=ax2+bx+c a real and imaginary
part, but complex
has . has ... numbers are beyond
two distinct real roots two x-intercepts the scope of the
two equal real roots (one one x-intercept SL syllabus. So, we
repeated root)
no real roots nox-intercepts say "no real roots",
because "no roots"
may not be accurate.
Reflect Why does a quadratic equation with a discriminant of O have two
equal real roots?
Why does a quadratic equation with a discriminant greater than O have two
distinct real roots?
Why does a quadratic equation with a discriminant less than O have no
real roots?
How can you use the discriminant to determine when the roots ofa quadratic
equation will be rational?
Why does the discriminant determine the nature of the roots?
Example 34
Find the value of the discriminant of the following quadratic equations, and then state the
number and nature of the roots of each equation.
a 2x2+5x-1=o b x2+ 10x=-25
a 2x2+5x-1=o This equation is already in general form,
= 25+8= 33
with a = 2, b = 5 and c
Calculate A = b2 — 4ac.
A = 33 > 0 two distinct real roots You must state the value of the discriminant,
and whether it is greater than, equal to or
b x2+ 10x=-25 less than zero.
X2+ 10X+25=o Write the equation in general form, with
100 - 100=0 a = l, b = 10 and c = 25.
two equal real roots Calculate A = b2 — 4ac.
A=0
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3.6 Malaysian Rnnit
37
Example 35
Find the value(s) of k for which the graph of y = x2 + 9x + k has two distinct real roots.
a y = + 9x+k Find the discriminant of x2 + 9x + k = 0.
A = 92 = 81 -4k A = b2 — 4ac where a = l, b = 9 and c = k.
The graph has two distinct real roots when
81-4k>O
81 Remember to reverse the inequality symbol
4 when you divide by a negative value.
Exercise 3V
1 Use your GDC to sketch the following graphs. 3 For each equation, find the value(s) of p such
that the equation has two equal real roots.
Find the value of the discriminant, and then
state the nature of the roots of each equation. a x2 + 5x + p = 0 b 3x2 - 12x + p: O
a b 6x2+7x-3=o c 2x2— 2px+4=0 d x2 — 3px — 2p = 0
3x2 + 4x = 8 m4 For each equation, find the value(s) of
e such that the equation has no real roots.
2 For each equation, find the value(s) of k such a x2 — 2x + m = 0
that the equation has two distinct real roots.
c
a x2 k = b kX2 5
Solving quadratic inequalities International
You have now solved quadratic equations by factorizing, completing mindedness
the square, using a GDC, and using the quadratic formula. You can also Over 2000 years
use any of these methods to help solve a quadratic inequality.
ago, Babylonians
Investigation 10 and Egyptians used
quadratics to work
1 For each of these quadratic equations: with land area.
x2+10x+25=o x2+5x+6=o xe-3x+17=o 169
x2-4x+11=o 5x2 _ 6x+ 9 = O
a Choose a method to solve the equation.
b Was your method quick or efficient?
c How else could you have solved it?
2 Draw a flow diagram that helps you decide when to use each method,
depending on the type of quadratic equation you are faced with.
Conce tual How do you choose which method to use to solve a
quadratic equation?
Conce tual Which methods of solving a quadratic equation work for any
quadratic equation?
Conce tual How is technology beneficial to help solve quadratic
equations?
MODELLING RELATIONSHIPS: LINEAR AND OUADRATIC FUNCTIONS
Example 36
Solve each inequality:
b 2x2 — 52x-3 To solve this inequality, you can sketch
a x2-6x+8<0 the graph of y = x2 — 6x + 8.
When (x — 2)(x — 0, then x = 2 or First, find the x-intercepts of the graph
by factorizing x2 — 6x + 8.
positive positive x Make a rough sketch of y = (x — 2)(x— 4)
02 4 by plotting the x-intercepts and using the
fact that the parabola is concave up.
negative
Since you are looking for values where
The solution is 2 < x < 4. y is less than zero, the solution to the
b 2x2-52x-3 inequality consists of the x-values for
2x2 -x- 2 0
which the graph lies below the x-axis.
When 2x2 — x— 2 = 0, then
Write the quadratic inequality in general
4 negative positive form and consider the graph of
positive 4 2x2 -x-2.
4 To find the x-intercepts of the graph,
use either the quadratic formula or
completing the square.
You can use a number line to analyse the
x
inequality, by plotting the x-intercepts
and considering the concavity of the
graph or by testing a value in each
interval.
The solution is x Since you are looking for values where
4 4
y is greater than or equal to zero, the
solution to the inequality consists of the
x-values for which the graph lies above
the x-axis.
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3.6 Malaysian Rnnit
37
Example 3?
Solve each inequality graphically using your GDC. Give your answers correct to
3 significant figures.
a 2x227x-4
b 3x2-7X<2x-2
a 2x2> 7x—4 Write the quadratic inequality in general
- 7x +42 0 form and plot the graph of y = 2x2 — 7x + 4
on your GDC.
Use your GDC to find the x-intercepts
correct to 3 s.f.
Since you are looking for values where y is
greater than or equal to zero, the solution
to the inequality consists of the x-values for
x which the graph lies above the x-axis.
The solution is x S 0.719 or x 2 2.78. Rather than writing the inequality
in general form, you can graph the
b 3x2-7X<2x-2
expressions on each side of the inequality.
onoomo
ownmuo Use your GDC to graph y = 3x2 — 7x and
x y = 2x — 2. Find the x-coordinates of the
points of intersection.
The solution consists of the values of x for
which the graph of y = 3x2 — 7x lies below
(that is, is less than) the graph of y = 2x— 2.
oneo,oo
rnoooo
The solution is 0.242 < x < 2.76.
171
MODELLING RELATIONSHIPS: LINEAR AND OUADRATIC FUNCTIONS
Example 38 + kx + 9 has no x-intercepts.
Find the value(s) of k for which the graph of y
A=P = k2 - 36 Find the discriminant of x2 + kx + 9 = 0.
k2 — 36 < O A = b2 — 4ac where a = 1, b = k and c = 9.
The graph has no x-intercepts when A < 0.
Make a rough sketch of y = k2 — 36 =
(k + 6) (k — 6), by plotting the x-intercepts
and using the fact that the parabola is
concave up.
Positive Positive
-6 6
Negative Since you are looking for values where y is
less than zero, the solution to the inequality
consists of the k-values for which the graph
lies below the k-axis.
Therefore,
Exercise 3W
1 Solve each inequality: 5 Find the value(s) of k for which the
quadratic equation kx2 — 6kx+ 2 + k = 0 has
a 3x2 + 5x-2>0 b
c 2x2 + 6x—6 < x2+ 2x two distinct real roots.
2 Solve each inequality graphically using 6 The graph of f(x) = 3x2 + px + 4 has no
your GDC. x-intercepts.
a x2-12x>3 b a Find an expression for the discriminant
of f(x) = 0 in terms of p.
C 8x2— 9
3 For each equation, find the value(s) of k b Find the possible values of p.
such that the equation has two distinct real mc Let be the largest possible integer
roots. value of p. Write down the value of m.
a x2+ 2kx+3=O d The function h(x) = 3x2 + mx + 4
m4 Find the value(s) of such that the can be written in the form
equation x2 + 6mx + m = 0 has no real roots.
h(x) = — h)2 + k. Find the values of
a, h and k.
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37
Developing inquiry skills
The opening problem for this chapter considered the height of a crate
dropped from a cargo plane. The functions h andg give the crate's
height above the ground, measured in metres, t seconds after the
crate leaves the plane.
-=During free fall: h(t) 4.9t2 + 720
With parachute open:g(t) 5t+ 670
Now that you have leamed several methods for solving a quadratic
equation you can answer the following questions.
How long after the crate leaves the plane does it reach the
ground?
What is the domain of h and g in the context of the problem?
3.7 Applications ofquadratics
In a suspension bridge, a supporting cable is
attached to two towers. The road deck hangs on
vertical cables attached to the supporting cable. The
supporting cable is in the shape of a parabola.
Tmver Tmver
Supporting cable
t Vertical cable
Road deck
What information is needed to write a function that
models the shape of the supporting cable?
Investigation 11
mThe main span of the Clifton Suspension Bridge in Bristol, UK is about 194 long. The height of the towers is
26.2 m. The supporting cable for the main span of the bridge touches the road deck at the centre of the span.
1 The parabola shown in the graph models x
the supporting cable of the main span of the
Clifton Suspension Bridge and the green line
segments represent the towers.
Write down the values of r, s and t.
2 Consider the three forms of the equation ofa quadratic function you have studied. For each of these
forms the values of some of the parameters are equal to r, s or t.
Y = a(x — + k Y = ax2 + bx + c Y = a(x— — q)
Replace the appropriate parameters in each equation with the values of r, s and t.
3 Choose one ofthe equations from question 2 and find any other missing parameters to complete a quadratic
function that models the supporting cable.
Continued on next page
173
MODELLING RELATIONSHIPS: LINEAR AND OUADRATIC FUNCTIONS
O4 Factual Which form of the quadratic function did you use and why?
5 Define the variables in your function.
How can the quadratic function help you answer questions about the
supporting cable of the Clifton Suspension Bridge?
? Find the height of the supporting cable from the road deck at a point 24m from a tower.
8 Suppose there are vertical cables with length 2.?5 m, attaching the supporting cable to the road deck.
Find the distance of these cables from the left-hand tower.
Reflect How can quadratic relationships be represented? What type of
real-life relationships can be modelled by quadratic equations?
How do you decide which form of quadratic function to use for your model?
You have learned about different forms of quadratic functions and
different methods for solving quadratic equations. Now you will use this
knowledge to solve a variety of real-life problems. While one solution
method is shown in each example, other methods may be possible.
Example 39
mA rancher plans to use 120 of fencing to build a rectangular pen.
a Let x represent the width of the pen. Find the length and area of the pen in terms of x.
b Find the dimensions of the pen if the area is 800 m2.
c Find the maximum possible area of the pen.
a Let I represent the length of the pen Use the fact that the perimeter of the pen is
and A represent the area. m120 to write an equation for the perimeter in
21+2x = 120 terms of I and x.
= 60
By drawing a simple diagram of the pen, it is
1 = 60 —x
clear that the area is length x width.
x
60-x Set the A = 800 in your expression from part a.
A =x(60 — x) Multiply out the bracket and write the quadratic
equation in general form.
b x(60 —x) = 800
Solve by factorization.
x(60 —x) = 800 If the width is 20, the length is 40.
If the width is 40, the length is 20.
60x -x2 = 800
x2 - 60x+ 800 = o
(x - 40) = o
20, 40
mThe pen is 20 by 40 m.
Malaysian Rnnit
3.7
O The maximum occurs at the vertex of the graph
— x) = —x2 + 60x
b = Z6_0 = 30 of the area function.
A = 30(60 - 30) = 302 = 900
The maximum area occurs when the width of
the pen is 30.
The maximum possible area is 900 m2. Substitute 30 for the width in the area equation.
Example 40
A bakery sells apple pies for $12 each, and sells an average of 40 apple pies per day.
The owner estimates that for each $0.50 increase in price, the average sales will decrease
by one pie a day.
Suppose there have been x increases of $0.50 in the price of an apple pie above the initial
price of $12.
a Write down an expression for the price, $P, of an apple pie after x increases of $0.50
above the initial price of $12.
b Write down the number of apple pies sold per day in terms of x.
c Find a function that gives the estimated revenue, R, from apple pie sales in terms of x.
d Use the revenue function to find the sale price the bakery owner should charge to
maximize the revenue from apple pie sales.
a P=12+O.50x Think about the pattern.
One increase = 12 + 0.50(1)
Two increases = 12 + 0.50(2)
x increases = 12 +0.50x
b number Of pies sold a day = 40 —x After x increases of $0.50, the owner will sell
40 —x apple pies per day.
c R(x) = (12 -x) The revenue equals the selling price times the
number of pies sold.
d
You can find the maximum as you did in
(8,512) example 39 or you can use your GDC to graph
the revenue function and find the maximum.
ÅooÄoo
—OOO"O
x The maximum revenue of $512 occurs when
-5 10 20 30 40 50 there are 8 increases of $0.50. 12 + 0.50(8) = 16.
The sale price that maximizes revenue
is $16.
175
MODELLING RELATIONSHIPS: LINEAR AND OUADRATIC FUNCTIONS
Example 41
The height, h, of a ball t seconds after it is thrown is modelled by the function
h(t) = 1.5 + 23t — 4.9t2, where h is the height of the ball in metres.
a Write down the height of the ball at the time it is released from the thrower's hand.
b Find the number of seconds it takes the ball to reach the ground.
c Find the length of time the ball is higher than 16 m.
a The height of the ball is 1.5 m. When t = 0, h = 1.5.
b 1.5 +23t- 4.9B = O
When the ball reaches the ground, the height
h
is 0.
Use your GDC to graph the function
h(t) = 1.5 + 23t— 4.9t2 and find the positive
x-intercept.
10
(4. 5821, 0)
t
10
The ball reaches the ground 4.76 Find the values of t where h is greater than 16.
seconds after it is thrown.
c 1.5 +23t-4.9t2>16 Use your GDC to graph h(t) = 1.5 + 23t — 4.9t2
h and h(t) = 16.
20 3.94347 16) Find the points of intersection.
10 ( 75 399 16 mThe ball is higher than 16 between
svncnnnu 0.750399 seconds and 3.94347 seconds and
3.94347 - 0.750399 = 3.193071.
mThe ball is higher than 16 for 3.19
seconds.
Reflect When solving a real-life problem modelled by a quadratic equation,
how do you determine which method of solution to use?
Exercise 3X
(9)
1 The length of the base of a triangle is four more than twice the height.
The area of the triangle is 24 m2. Find the lengths of the base and height TOK
of the triangle.
2 The height of a ball t seconds after it is thrown is modelled by the How accurate
function h(t) = 2 + 20t — 4.9t2, where h is the height of the ball in metres.
is a visual
a Find the height of the ball 3 seconds after it is thrown. representation of
b Find the times at which the ball has a height of 6 m. a mathematical
concept?
c Find the maximum height of the ball.
176
Malaysian Rnnit
37
A3 bus transports people from an airport to a city centre. It transports 800
people a day at a cost of €5.50 per person. Research has shown that for
every decrease of €0.05 in the fare, 10 more people will ride the bus.
Suppose there have been x decreases of €0.05 below the initial cost of
€5.50.
a Find an expression, in terms of x, for the bus fare.
b Find an expression, in terms of x, for the number of people who ride
the bus in a day.
C Find an expression, in terms of x, for the daily revenue generated by
people riding the bus.
d Find the number of fare decreases that result in a revenue of €4500.
e Find an appropriate domain for this context.
4 The shape of an archway is modelled by the graph
of a quadratic function. The maximum height of the
marchway is 4 and the maximum width is 4 m. The
graph shows a model of the archway.
a Find a function to model the archway.
b Use the function to determine whether an x
m mobject 3 wide and I .6 tall will fit through
the archway.
mS A rectangle has length x and perimeter 310 m.
a Find a function for the area of the rectangle in terms of x.
b Use the function to find the dimensions of the rectangle with the TOK
maximum possible area. We have seen the
FIFA (Fédération Internationale de Football Association), the governing involvement of
body for international football (soccer), sets restrictions on the several nationalities
dimensions of the rectangular playing field. in the development
of quadratics in the
FIFA's rules give the following limitations on the dimensions of a chapter.
To what extent do
regulation football field. you believe that
mathematics is a
The length of the touch line must be greater than the length of the goal
product of human
line.
social collaboration?
mTouch line: minimum 90 m maximum 120
Goal line: minimum 45 m maximum Developing
90 m your toolkit
c Suppose a football field has Goal line Now do the
perimeter 310 m. Determine
Modelling and
whether the dimensions you found
in part b meet the requirements investigation
for a FIFA regulation field width. Touch line activity on page
Explain your reasoning. 182.
d Suppose the function you found in
part a is used to model the area of a regulation FIFA football field
mwith perimeter 310 and a touch line of length x. Write down the
appropriate domain for this function.
e Find the maximum area of a football field with perimeter
m310 that meets FIFA's regulations.
MODELLING RELATIONSHIPS: LINEAR AND OUADRATIC FUNCTIONS
Chapter summary
Graphs of linear functions
m• The gradient ofa line passing through the points (XI,YI) and (X2,Y2) is given by =
• The gradients of parallel lines are equal.
• The product of the gradients of two perpendicular lines is—I.
mx+=• For a graph of a linear function in the form y mandc, the gradient is they-intercept is (0, c).
—y m(x—x• Fora graph ofa linear function in the formy 1 = ), the line passes through the point
(XI,YI) and has gradient m.
• The general form ofthe equation ofa line is ax+ bY+d=0.
Transformations of the graph of y =f(x)
—fThe graph ofy = (x) is a reflection in the x-axis.
The graph of y =f(—x) is a reflection in the y-axis.
=The graph of y af(x) is a vertical dilation with scale factor lal.
vertical stretch when lal>l and vertical compression when
The graph of y = f(qx) is a horizontal dilation with scale factor1—
<horizontal stretch for 0 Iql and horizontal compression for Iql > 1
The graph of y = f(x) + kis a vertical translation.
up k units for k > 0 and down 11<1 units for k < 0
The graph of y — h) is a horizontal translation.
right h units for h > 0 and left 1/11 units for h < 0
Graphs of quadratic functions
• Forthe graph ofa quadratic function in the formf(x) h)2+k, the vertex is (h,k) andthe
equation of the axis of symmetry isx= h. Transformations of the graph of y=x2 can also be used to
graph this form of the equation.
• For the graph ofa quadratic function in the formf(x) =ax2 + bx+c, the equation of the axis of
symmetry is x = -—b and they-intercept is (0, c).
• For the graph ofa quadratic function in the formf(x) (x—q), the x-intercepts are (p, 0)
and (q, 0) and the equation of the axis of symmetry is x = 2-±-L.
2
Solving quadratic equations
• The quadratic equation c, c > O, has two solutions, x = (c and x = —Cc. The solutions can be
written as x ±Cc.
• To solve quadratic equations using factorization, use the zero-product property to set each factor
equal to 0 and solve.
• To solve ax2 + bX+ c = 0 by completing the square: (1) divide both sides of the equation by a;
2
1
—(2) subtract the constant term from both sides of the equation; (3) add
the coefficient of x
2
Oto both sides of the equation; (4) factorize the left-hand side of the equation; and (5) solve for x.
178
O Malaysian Rnnit
• The solutions ofax2 + bX+ c= 0, given by the quadratic formula are x = 3
37
— >If b2 4ac 0, the equation will have two distinct real roots.
If b2 — 4ac = 0, the equation will have two equal real roots. 2
— <If b2 4ac 0, the equation will have no real roots.
Developing inquiry skills RELÆF
In the opening scenario forthis chapter you looked at how crates
of emergency supplies were dropped from a plane. The functions
h(t) andg(t) give the height of the crate:
During free fall: h(t) = —4.9t2 + 720
With parachute open:g(t) _ 5t + 670
How realistic is this model in representing this real-life situation? State at
least two advantages of the model, and give at least one criticism.
Suppose a heavier crate was dropped, but its parachute was the same. How
would the model be different for this heavier crate?
Suggest suitable functions to model the path of a javelin through the air, or the path
of a basketball as it leaves a player's hands and passes through a hoop. What do you
need to consider in finding a model for each? How could each model help to make
predictions in a real-life scenario?
Chapter review Click here for a mixed
review exercise
1 Sketch each line, marking on your sketch
2x+1, -2 < x
the axial intercepts: 3 Consider the function f(x) =
a Y = 2x+4 3,
c 2x+3y-6=O a Find f(l) andf(2). b Graph y
2 Find an equation for each line: 4 Describe the series of transformations of
a the line which passes through the the graph of y = f(x) that lead to the graph
points (—4, 2) and (8, —l) of the given functions.
b the line which is parallel to the line
ab
y = —x +3 and has y-intercept (0, —5)
c y = —f(x+ 2) —l d
c the line which is perpendicular to
S In each case, find the indicated features of
2
the graph of y = f(x).
y = ——x +7 and passing through the
a f(x) = 2(x — 7); x-intercepts,
3
equation of the axis of symmetry
point (2, 4)
b f(x) = -3(x — 4)2 + 2; equation of the
d the line which is passes through
(—3, —4) with gradient 0 axis of symmetry, vertex
C f(x) = —x2 — 4x + 6; equation of the axis
of symmetry, y-intercept
MODELLING RELATIONSHIPS: LINEAR AND OUADRATIC FUNCTIONS
6 The functionf(x) = 3x2+ 18x+ 20
can be written in the form
f(x) = a(x— + k.
a Find the values of:
hii iii
b Write down the vertex of the graph of
Y = fix).
c The graph of y = g(x) is a translation of c
x
the graph of y right 5 units and
The altitude of triangle ABC from B to side
down 3 units. Find the vertex of the AC is 7 cm and AC = 8 cm. The coordinates
graph of y = g(x). of one of the vertices of the inscribed
7 Solve each equation: rectangle are (p, 0).
a (X — = 64 a Write down the coordinates of points A,
B and C.
C X2 + 49 = O d
b Find the equation of the line passing
e 3x2+4x-7=o
through points B and C.
8 The equation—x2 + 3kx—4=0 has two
c Find the dimensions of the rectangle
equal real roots. Find the possible values of k. inscribed in the triangle, in terms of p.
9 The y-intercept of the graph of a quadratic d Write down an expression for the
function is (0, —16) and the x-intercepts
are (—4,0) and (2,0). Find the equation of area of the inscribed rectangle in
the function in the form f(x) = ax2 + bx + c terms of p.
where a, b and c are constants. e Find the dimensions of the rectangle with
10 Solve each equation. Give your answers maximum possible area.
f Find the maximum possible area of the
correct to 3 significant figures.
inscribed rectangle.
a 2x2-6x-5=o b -x2-3x=o.5x-7
11 The height, h metres above the water, of a Exam-style questions
stone thrown from a bridge is modelled by
13 P2: A line has equation —7x —12y + 168 = 0
the function h(t) = 18 + 13t — 4.9t2, where
a Write down the equation of the line
t is the time in seconds after the stone is
thrown. in the form y = mx + c. (2 marks)
a Find the initial height from which the b Given that the line intersects the
stone is thrown.
x-axis at point A and the y-axis at
b Find the maximum height reached by point B, find the coordinates of A
the stone. and B. (2 marks)
C Find the amount of time it takes for c Calculate the area of triangle OAB.
the stone to hit the water below the
(2 marks)
bridge.
14 Using your GDC, sketch the curve of
d Write down the domain of the y = -2.9x2 +4.1x +5.9 for -1 < x 2.
function h in the context of this (2 marks)
real-life scenario.
b Write down the coordinates of the
e Find the length of time for which the
height of the stone is greater than 23 m. points where the curve intersects
A12 rectangle is inscribed in isosceles the x- or y-axis. (2 marks)
triangle ABC as shown in the diagram. c Write down the range of y. (2 marks)
180
Malaysian Rnnit
3
37
IS P2:a Show that the solutions to the 19 P 1: The quadratic curve y = x2 + bx + c
equation x2 —6x —43 = 0 may be
intersects the x-axis at (10, 0) and has
written in the form x = p ± qqß
5
where p and q are positive integers.
equation of line of symmetry x = —.
(4 marks)
b Hence, or otherwise, solve the 2
a Find the values of b and c. (4 marks)
inequality x2 —6x —43 0
b Hence, or otherwise, find the other
(2 marks)
two coordinates where the curve
16 P 1: A function fis defined by
intersects the coordinate axes.
—18, x e R.
(2 marks)
a Write f(x) in the form ax2 + bx + c, 20 P 1: Consider the function
where a, b and c are constants.
f(x) = 2x2 —4x —8, x e R.
(2 marks) a Show that the function fcan
b Find the coordinates of the vertex of be expressed in the form
the graph of f. (l mark) f (x) = a(x— + k, where
a,h and k are constants. (3 marks)
c Find the equation of the axis of b The function f(x) may be
symmetry of the graph of f.
(1 mark) obtained through a sequence of
transformations of g(x) = x2.
d State the range of f. (2 marks) Describe each transformation in
e The graph of fis translated through turn. (3 marks)
2 to form a curve
the vector 21 P 1: Consider the equation
-1 f (x) = 2kx2 +6x + k, x e R.
representing a new function g(x).
a In the case that the equationf(x) = 0
Findg(x) in the form px2 +qx +r, has two equal real roots, find the
where p, q and r are constants. possible values of k. (4 marks)
(3 marks)
b In the case that the equation of the
1? Pl:a Solve the equation 8x2+ 6x— 5 = 0
line of symmetry of the curve y = f(x)
by factorization. (4 marks) is x + I = 0, find the value of k.
b Determine the range of values of k c Solve the equation f(x) = 0 when
for which 8x2 +6x—5 = k has no (3 marks)
real solutions. (3 marks)
18 P 1: Consider the function 22 P i: A curve y = f (x) passes through
f (x) = x 2 —l Ox + 27, x e R. the points with coordinates
and
a Show that the function f can be 14, -10)
expressed in the form a Write down the coordinates of
f (x) + k, where each point after the curve has been
transformed by f (x) f(2x).
a,h and k are constants. (3 marks)
b Hence write down the coordinates (4 marks)
of the vertex of the graph of b Write down the coordinates of
Y = f(x). (1 mark) each point after the curve has been
c Hence write down the equation of +transformed by f (x) + 3.
the line of symmetry of the graph of (4 marks)
Y = f(x). (1 mark)
181
Hanging around Approaches to learning: Thinking skills:
Create, Generating, Planning, Producing
Exploration criteria: Presentation (A), Personal
engagement (C), Reflection (D)
1B topic: Quadratic modelling, Using technology
Investigate
Hang a piece of rope or chain by its two ends. It must be free hanging
under its own weight. It doesn't matter how long it is or how far apart the
ends are.
1 What possible shaped curve does the hanging chain resemble?
2 What form might its equation take?
3 How could you test this?
Import the curve into a graphing package
A graphing package can fit an equation of a curve to a photograph.
Take a photograph of your hanging rope/chain.
What do you need to consider when taking this photo?
Import the image into a graphing package.
Carefully follow the instructions for the graphing package you are
using.
The image should appear in the graphing screen.
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Malaysian Rnnit
3
37
Fit an equation to three points on the curve
Select three points that lie on the curve.
Does it matter what three points you select?
Would two points be enough?
In your graphing package, enter your three points as x- and y-coordinates.
Now use the graphing package to find the best fit quadratic model to your
three chosen points.
Carefully follow the instructions for the graphing package you are using.
Test the fit of your curve
Did you find a curve which fits the shape of your image exactly?
What reasons are there that may mean that you did not get a perfect fit?
The shape that a free-hanging chain or rope makes is actually a catenary and
not a parabola at all. This is why you did not get a perfect fit.
Research the difference between the shape of a catenary and a parabola.
International-mindedness
The word "catenary" comes from the Latin word for "chain".
Extension A well-known
Explore one or more ofthe following — are they quadratic? landmark —
The cross section of a perhaps the
Sydney Harbour
football field. Bridge or the
arches at the
bottom of the
Eiffel Tower.
The curve of The path ofa football when
a banana.
kicked in the air. Here you
would need to be able to Other objects that look like a
use available software to
trace the path of the ball parabola — for example, the arch
that is moving. ofa rainbow, water coming from
a fountain, the arc of a Satellite
dish.
183
Equivalent representations:
rational functions
From business to music, astronomy to architecture, Concepts
rational functions are used to represent relation- Representations
ships between two real-life variables and hence Equivalence
help make predictions and influence decisions. In Microconcepts
this chapter you will learn about reciprocal and
rational functions, their graphs, and where they Domain and range of a rational function
appear in the world around you. u Features of reciprocal and rational functions:
symmetry, intercepts, horizontal and venical
asymptotes
Modelling with reciprocal and rational
functions
How does the concentration of medicine in a How can a business determine the
patient's blood stream change with time? best price to sell their products?
How does the length ofa guitar string
affect the frequency of vibrations?
What is the optimum file size to save
photos to your phone in order to get
the best compromise between photo
quality and available space?
A water park is designing a new water slide.
The slide will be very steep to begin with, and then will
level out so that a person is travelling almost horizontally
when they hit the water at the bottom.
• How could you describe the shape of this slide? Sketch
a path that a person on the slide would travel, from
setting off at the top to reaching the water at the bottom.
How gradual should the curve in the slide be to ensure
a person's safety, but to make sure they travel very fast
to begin with?
• How could you model the shape of the waterslide using
mathematics?
• How might a suitable model help the slide's designer
to choose the best overall shape for the slide?
Developing inquiry skills
Think of other physical objects that might have a similar shape to this water slide — for example,
the scoop on a bulldozer.
Think of any abstract quantities that are related in such a way that the graph showing one quantity
plotted against the other has the same shape as the water slide.
Think about the questions in this opening problem and answer any you can. As you work through
the chapter, you will gain mathematical knowledge and skills that will help you to answer them all.
Before gou start
You should know how to: Skills check Click here for help
with this skills check
1 Solve simple equations 1 Solve these equations:
egx—3 = 0 eg 2x+ I
1 b 6-x=o
c 2x-5=o
2
2 Sketch these lines on the same set of
2 Sketch horizontal and vertical lines
eg sketch the lines x = 2, x = —l, y = 3 axes: x = 3, x = —2, y y = 4.
and y =—2 on the same axes.
5
4
1 34 5
185
EQUIVALENT REPRESENTATIONS: RATIONAL FUNCTIONS
4.1 The reciprocal function
Investigation 1
Think of pairs of positive numbers whose product is 12, for example
1 x 12,
i Copy this table and add more pairs of positive numbers (integers or
fractions) whose product is 12.
x 1 2 34
12 6 4 3
2 Use your GDCto graph a scatter plot of the coordinate pairs in the table.
3 Now make a similar table, but find pairs of negative numbers whose
product is 12, for example -1 x —12 = 12. Use your GDC to graph these
points on the same set of axes.
4 Explain what you notice about
a the value of x asy gets bigger
b the value of y as x gets bigger.
The end behaviour of a graph is the appearance of a graph as it is followed
further and further in either direction.
S What do you notice about the end behaviour of your graph? Can you
explain why this is the case?
The reciprocal of a number is 1 divided by that number. TOK
—For example, the reciprocal of 4 is 1 Is zero the same as
4 nothing?
Taking the reciprocal of a fraction "turns it upside down".
For example, the reciprocal of — is I +—2 = lx
3
The product of any number and its reciprocal is equal to l.
Reflect Why does zero not have a reciprocal?
Example 1 Write as an improper fraction.
Turn the improper fraction upside down.
1
Find the reciprocal of 4—•
2
The reciprocal of — is —2
186
Exercise 4A 4.1
1 Find the reciprocal of each number: EXAM HINT
Remember that the
a3 b5 c -2 d -1 1B command term
e3 22 8 h 2—3 show that requires
you to show all the
5 79 4
steps leading to the
2 Find the reciprocal of each expression:
a 1.5 b x C 2x answer. You may not
simply write down
e d 3h
the final answer.
4
3 Show that the product of each number and its reciprocal
is equal to l:
b7 c 2 d
11 x
x-2
Domain and range of the reciprocal function
The reciprocal function is f (x) = —t where k is a non-zero constant.
x
Reflect Are there any values of x for which the reciprocal function is not
defined?
Recall that the domain of a function is all the values of x for which the
functionf(x) is defined.
In your studies, you have already found that you cannot divide a
number by zero. If you try to calculate I +0, you get an undefined
1
answer. Therefore, 0 is not in the domain of f (x) =
x
However, when x is any real number other than 0, then — exists. So,
1
the domain of f (x) = is all real numbers other than zero.
x
1
xeThe domain of the reciprocal function f (x) = —is
IR\O.
x
This means "x can be any value within the set of real numbers, other than
zero".
EQUIVALENT REPRESENTATIONS: RATIONAL FUNCTIONS
Investigation 2
Look again at investigation 1, where you found pairs of numbers whose
product was 12. This function can be written as 12, or as a reciprocal
function y = —12e
x
12
In this investigation you will consider the range of the function y =
x
15
x
15
10
First, you will consider what happens to the value of y as the absolute value of
x becomes very large. i 24 ii 240 iii 2400 iv -24 v -240
1 a Findywhenxis
vi -2400
b What happens to the value ofy as x gets larger?
Factual Is there a value of x which would make the value ofy zero?
d Conce tual What can you say about the value ofy when the
absolute value of x becomes infinitely large?
Now consider what happens to the y value as the x value gets closer to zero
(remember, x cannot equal zero, but it can get close to it!).
2 a Find the value ofy when i x= 0.1 ii x=o.001
Fiii 0.00001 iv -0.1 v -0.001 vi 0.00001
Conce tual Explain what is happening to y as the absolute value of
x approa ches zero.
c Factual Given any large value ofy, is it possible to find a value of x
1
for which y = —? What does this tell you about the range off(x) ?
x
The range of the reciprocal function f (x) 1
x
This means "f (x) can be any value y within the set of real numbers, other
than zero".
188