C2 Page 12-16

CHAPTER 2: QUADRATIC FUNCTIONS

2.3 QUADRATIC FUNCTIONS. THR EFFECTS ON THE SHAPE AND POSITION OF THE q.f GRAPHS IF THE VALUES
OF , change.
1)A general form of a Q.F is a function in the form of ( ) = 2 + + where
, are constant and ≠ 0. Changes in the shape and position of the graph of function ( ) = 2 + +
2) The shape of the graph of a Q.F is a _____________________ which is
________________________ about the ______________________ that passes through the 1. Only the value > 0 > 0 < 0 < 0
minimum point or maximum point.
of changes. (Smiley graph) (Smiley graph) (Sadly graph) (Sadly graph)

When a ↑ , When a ↓ , When a ↑ , When a ↓ ,

graph width ↓ graph width ↑ graph width ↑ graph width ↓

Change in value of affects the shape and width of the graph, but the y-intercept remain
unchanged.

Changes in the shape and position of the graph of function ( ) = 2 + +

2. Only the value > 0 > 0 < 0 < 1

of changes. (Smiley graph) (Smiley graph) (Sadly graph) (Sadly graph)

When b > 0 , When b < 0 , When > 0 , When < 0 ,

vertex is left of vertex is right vertex is right vertex is left of

y-axis of y-axis of y-axis y-axis

yy y y

Change in value of only affects the position of vertex with respect to the y-axis, however
the__________ of the graph and the _________________ remain unchanged.

Changes in the shape and position of the graph of function ( ) = 2 + +

3. Only the value > 0 > 0 < 0 < 1

of changes. (Smiley graph) (Smiley graph) (Sadly graph) (Sadly graph)

When c > 0 , When c < 0 , When c > 0 , When c < 0 ,

y-intercept is y-intercept is y-intercept is y-intercept is

above of x- below of x-axis above of x- below of x-axis

axis axis

Change in value of only affects the position of graph either vertically upwards or vertically
downwards. The ____________of the graph is unchanged.

Belongs to JSEE 12

CHAPTER 2: QUADRATIC FUNCTIONS

Exercise 13:
1.Sketch of the graph for = 2 − 5 where = 1, = 0 = −5. Make an analysis

and a generalization on the shape and position of the graph when the following values

change.

c) The value of becomes (i) 3 (ii) -8
(i).3 (ii) −8

Hence, sketch the graph. (ii) 1
a) The value of a becomes (i) 3 4
(ii). 1
(i).3 4

Exercise 14:

HOMEWORK: Self practice
Sketch the graph of h(x) = −5x2 − x + 3, where a = −5, b = −1 and c = 3. Then sketch

on the different diagram when the following values chamge.

b) The value of becomes (i) 5 (ii) -5 a) The value a changes to (i) -3 (ii) -7 (iii) 5
(i). 5 (ii).−5
b) The value of b changes to 1

c) The value of c changes to -3

Belongs to JSEE 13

CHAPTER 2: QUADRATIC FUNCTIONS

RELATING THE POSITION OF THE GRAPH OF A QUADRATIC FUNCTION AND THE Exercise 16:
TYPES OF ROOTS.
1. Each of the following diagrams shows the graphs for quadratic functions. By calculating
Exercise 15: b2 −4ac, match each of the graphs with one of the function given:

Determine the shape and position of the graphs of the functions with respect to the a b2 −4ac Graph

x-axis. + 1 (−2)2 − 4(1)(4) = −12 c

Functions a b2 −4ac Position of the graph Types of roots 1. f(x) = x2 – 2x + 4
Cuts x-axis at 2 different
1. points Two real and 2. g(x) = −2x2 – 3x – 4
f(x) = 2x2 – 4x + 3 + 2 distinct roots 3. h(x) = x2 – 6x + 8
4. m(x) = 8x – 16 – x2
•• 5. n(x) = 4x2 – 20x + 25
6. p(x) = 5x – 6 – x2

2. (a) (b)
g(x) = x2 – 2x + 1 yy
x

3. (c) x y
h(x) = 3x2 + x +2 y (d) x

4. (f) y x x
m(x) =3x – x2 (g) y

5.
n(x) = 1 + 2x – x2

x

Belongs to JSEE 14

Exercise 17: c) g(x) = hx2 + 8x – 12 CHAPTER 2: QUADRATIC FUNCTIONS

d) g(x) = 2x2 – 3x + h – 1

1.Determine the values of k if each of the following quadratic functions graph touches the

x-axis at one point only. a) f(x) = 2kx2 – 3kx + 1 b) f(x) = x2 – 3kx + 5k – 1

Example:
f(x) = x2 + 2kx – 4x + 1
a = 1, b = 2k – 4, c = 1
touches 1 point  b2 – 4ac = 0

(2k – 4) 2 – 4 (1) (1) = 0
4k2 −16k + 16 – 4 = 0

4k2 −16k + 12 = 0
k2 – 4k + 3 = 0

(k – 3) (k – 1) = 0

k = 3 or k = 1 h > −4/3 h < 17/2

3.Determine the range of values of m if each of the following quadratic function graphs does

k = 0, 8/9 k = 2/9, 2 not intersect the x-axis.
d) f(x) = x2 + 4x + 3 – 2k e) f(x) = (3x – 2)2 + k – 1
c) f(x) = x2 – 2kx + 4k + 5 Example a) h(x) = x2 – mx2 – 6x + 8 b) h(x) = (m + 3)x2 – 3x + 1
h(x) = mx2 – 12x + 2

a = m, b = − 12, c = 2
No intersection  b2 – 4ac < 0

(− 12)2 – 4(m) (2) < 0
144 – 8m < 0

−8m > 144

m> 18

k = −1, 5 k = −1/2 k=1

2.Determine the range of values of h if each of the following quadratic function graphs

intersect the x-axis at two distinct points m < −1/8 m > −3/4
d) h(x) = 3x2 – 6x + 1 – 2m
Example a) g(x) = x2 – 3x − 6h + 3 b) g(x) = x2 – 2x + 3h c) h(x) = x2 – 2x + 4 – m
g(x) = x2 + 2x + 5 – 3h
a = 1, b = 2, c = 5 – 3h
Intersect 2 points  b2 – 4ac > 0

22 – 4 (1) (5 – 3h) > 0
4 – 20 + 12h > 0

h> 4

3

h > 1/8 h < 1/3

15 m<3 m < −1

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4. Find the possible values of p if each of the following graph touches the x-axis at only one point CHAPTER 2: QUADRATIC FUNCTIONS
(a) ( ) = 2 − + + 3
6.Determine the range of values of m if the graph of the function ℎ( ) = 2 + (6 − 3) + 9 + 1
does not intersect the x-axis.

(b) ( ) = 4 2 + 4(3 − ) + 5(3 − ) 7. Prove that function ( ) = 3 2 − 5 + 3 is always positive for all values of x .

5. Determine the range of values of k for which the graph for quadratic function 8. Prove that the function ( ) = 3 − 4 − 5 2 is always negative for all values of x .
( ) = ( + 3) 2 + 2( − 1) + intersects the x-axis at two distinct points.

Belongs to JSEE 16