Reference for Basic Level Mathematics.

You can also graph a line if you know only one point and
the slope. You have all the information you need-a starting
point, the number of units it rises, and the number of units
it runs.

EXAMPLE: Draw the line that goes through ( 0, −4 ) and
has a slope of  -23  .

Begin by plotting

the given point ( 0, −4 ).

2
1

-2 -1 -1 1   2   3   4   5
-2
-3 2 3

Then, use the RISE -4 ( 0, −4 )

and RUN of the slope -5

to find the next point. -23  

SLOPE =

Lastly, connect the points

that you created, and draw

arrows at the end of the line

to infer that the line goes on forever.

440

1. Is the slope positive, 3
negative, zero, 2
or undefined? 1

-3   -2 -1 1   2   3
-1
-2 
-3

2. Is the slope positive, 3 1   2   3
negative, zero, 2
or undefined? 1

-3   -2 -1
-1
-2 
-3

3. Is the slope positive, 3 1   2   3
negative, zero, 2
or undefined? 1

-3   -2 -1
-1
-2 
-3

4. Is the slope positive, 3 1   2   3
negative, zero, 2
or undefined? 1

-3   -2 -1
-1
-2 
-3

441

5. Use a slope triangle to 1    2  3  4  5  6

find the slope of this line:

6
5
4
3
2
1

-6   -5   -4   -3 -2   -1
-1
-2 
-3
-4
-5
-6

6. Use a slope triangle to 6
5
find the slope of this line: 4
3
2
1

-6   -5   -4   -3 -2   -1 1    2  3  4  5  6
-1
-2 
-3
-4
-5
-6

442

7. Use the slope formula to find the slope
of the line that passes through the points

( 2, 8 ) and ( 5, 7 ).

8. Use the slope formula to find the slope
of the line that passes through the points

( −3, 2 ) and ( −6, 10 ).
9. Draw the line that goes through ( 3, 5 )

and has a slope of 4.
10. Draw the line that goes through ( 6, −2 )

and has a slope of 0.

W H EW. . . TH OS E
SLOPES WERE PRETTY

INTENSE!

answers 443

 1. Zero 6
2. Positive 5
3. Negative 4
4. Undefined
23
5. Slope = -26  = 3 2
1

-6   -5   -4   -3 -2   -1 -1 1 
-2 
6

-3

-4

-5

-6

6. Slope = - -67  

4
3
2
1

-6   -5   -4   -3 -2   -1 -1 1    2  3  4  5 
-2 
-6

7 -3
-4

444

7. m = -57  ---  28 = - -31
8. m = --106 - -+  23 = --83 = - -83

9.

9 1    2  3  4  5  6  7  8  9
8
7
6
5
4
3
2
1

-2   -1
-1
-2 

10.

3
2
1

-3   -2   -1 1    2  3  4  5  6 
-1
-2
-3 

445

Chapter 58

LINEAR EQUATIONS

AND

FUNCTIONS

A LINEAR  EQUATION is an equation whose graph is a line.

A linear equation always has the form:
y = mx + b

y represents every y- value on the line, m represents
tYh-e IsNloTpEeR(tChEe PrTati(owohfer-ReRIUaSNElin),eacnrdosbsersepthreesye-n atsxitsh).eIf you

know the y- intercept, you know the value of y at the point
where the line crosses the y-axis, and, therefore, you also

know the x- value there because it is always 0! If you know

both the y- intercept and the slope of a line, you can graph
the line.

446

Because the graph of a linear equation is a straight line, this
means that all linear equations are functions . . . except for
vertical lines!

EXAMPLE: Graph y = 2x + 1.

The equation of a linear function is y = mx + b , so in
case:
this m = -21  
b=1



First, plot the y-intercept (in this equation, where ( -21  )

y = 1 and x = 0 ). From there, we know from the slope
that we must rise 2 and run 1. Then we continue to

plot points until we have a line.

Lastly, we connect the 6 2
points and draw an 5 1
arrow on the ends of 4 2
the line to show that 3 1
it keeps going in both 2
directions. 1 1    2  3  4  5  6

-3 -2   -1
-1
-2 
-3

447

EXAMPLE: Graph y = -21   x - 3.

The equation of a Pay close attention to positive and
negative symbols so you don’t
linear function is 3
2 mistakenly plot the wrong point!
y  =  mx + b, 1
so in this case: 1    2  3  4  5  6  7

b = - 3 1
12
m = -21   -3 -2   -1 2
-1
-2 

-3

-4

EXAMPLE: Graph y = x.

The equation of a 3
linear function is
2 1
y  =  mx + b, 1 11
so in this case: 1 1   2   3
-3   -2 -1
b=0 -1
m = 1 or -11  
-2 

-3

Because y = x is the same as y = 1x + 0

448

Even if the given equation doesn’t look like this: y = mx + b ,

we can manipulate it so y is isolated on one side of the
equal sign.

EXAMPLE: Graph y + x = 4 .

y+x=4
y+x−x=4−x
y = 4 − x (We can put the -x  before the +4,

so it is in y = mx + b  form.)

y = −x + 4 (Now we know the y -intercept and

slope is --11  .)

m = −1

b=4 6

5

4

3 -1 1
2

1

-3 -2   -1 1    2  3  4  5  6
-1
-2 
-3

449

EXAMPLE: Graph 2y - 4x - 2 = 0.

2y - 4x - 2 = 0 Think: How can I get y
2y - 4x - 2 + 2 = 0 + 2
2y - 4x = 2 isolated on one side
2y - 4x + 4x = 2 + 4x of the equal sign?
2y = 4x + 2

-22y  = -42x  + -22   6 2
5 1
4 2

y = 2x + 1 3 1
2
1    2  3  4  5  6
m = -21   1
b =  1
-3 -2   -1
-1
-2 

-3

450

SHORTCUT ALERT! Any time you have an equation that looks

like: “y = (a number),“ plot a point on the y -axis at that

number, and trace a horizontal line going through the point.

EXAMPLE: Graph y = 2.

Every point on this line has a y -coordinate equal

to 2: ( 0, 2) ( 1, 2) ( 2, 2), etc.

5 1   2   3   4   5   6
4
3
2
1

-6 -5 -4   -3   -2 -1
-1
-2
-3 

REMEMBER: The slope of a horizontal line is zero.

451

Similarly, any time you have an equation that looks like

“x = (a number),” plot a point on the x -axis at that number

and trace a vertical line going through the point.

EXAMPLE: Graph x = −5.

Every point on this line has an x -coordinate

equal to −5.

5 1   2   3   4   5   6
4
3
2
1

-6 -5 -4   -3   -2 -1 -1
-2
-3 
-4
-5

REMEMBER: The slope of a vertical line is undefined.

452

1. What is the slope and y -intercept of y = 2x + 3 ?

2. What is the slope and y -intercept of y = - -23  x + 1 ?
3. What is the slope and y -intercept of y = -65   x ?
4. What is the slope and y -intercept of y + x = −2 ?
5. Graph y = -21 x  - 3.
6. Graph y = −3x + 7.

7. Graph y + 2x = 0.

8. Graph 3y = 9x - 6.

9. Graph y = −5.

10. Graph x = 6.

answers 453

 1. Slope = -21  , y -intercept = ( 0, 3 )
2. Slope = - -23  , y -intercept = ( 0, 1 )
3. Slope = -65  , y -intercept = ( 0, 0 )
4. Slope = - -11  , y -intercept = ( 0, -2)

5. 5
4
3
2
1

6.   -5   -4   -3   -2 -1 -1 1   2   3   4   5    6 

-2

10 -3

9 -4
8 -5

7
6

5
4
3
2
1

  -5   -4   -3   -2 -1 -1 1  2  3  4  5  6 7  8  
-2
-3
-4

454

7.
5
4
3
2
1

  -5   -4   -3   -2 -1 -1 1   2   3   4   5  

--32 8. 5
4
-4 3
-5 2
1

  -5   -4   -3   -2 -1 -1 1   2   3   4   5  
-2
5 -3

4 -4
-5
9. 3

2

1

  -5   -4   -3   -2 -1 -1 1   2   3   4   5  

-2

-3

-4
-5

10. 5
4
3
2
1

  -5   -4   -3   -2 -1 -1 1   2   3   4   5    6  
-2
-3

-4
-5

455

Chapter 59

SIMULTANEOUS

LINEAR EQUATIONS

AND FUNCTIONS

But why stop at one line? We can take two linear equations

and study them together, such as:

ax + by = c

dx + ey = ƒ

This is known as SIMULTANEOUS  LINEAR  EQUATIONS.

Because each of the linear questions represents a line,
we can ask, “If I draw two lines, where do they
intersect?” The process of finding the answer is called

SOLVING  SIMULTANEOUS  LINEAR  EQUATIONS.

There are three ways of doing it.

456

GRAPHING

We can solve simultaneous equations by graphing
each linear equation and then finding where they intersect.

EXAMPLE: Graph the simultaneous equation to

find the solution.

x+y=5
2x - y = 4

First rewrite each of the equations into the y = mx + b

form in order to graph each equation bet ter.

We can change the first equation from  x + y = 5
to y = -x + 5, and change the second equation from  
2x - y = 4 to y = 2x - 4.

Then, we graph the equations

by using the y-intercept

and slope from each. 5
4

From the graph, 3 1   2   3   4   5
we can see that 2
the two lines 1

intersect at ( 3, 2). -5 -4   -3   -2 -1
-1
-2 

So, the solution to the -3
-4
simultaneous equation is ( 3, 2). -5

457

EXAMPLE: Graph the simultaneous equations to find

the solution.

2x + y = -2
2x + y = 3

In order to graph these more easily, change the

first equation from 2x + y = -2 to y = -2x - 2,
and change the second equation from 2x + y = 3
to y = -2x + 3.

Then, we graph the 5
two equations. 4
3

There are no 2
1

intersection -5 -4   -3   -2 -1 -1 1   2   3   4   5

points, so there is -2 
-3
NO  SOLUTION to the

simultaneous equation. -4

-5

How can there be NO solution? Let’s look back at the original
two linear equations. The first linear equation is 2x + y = -2
and the second linear equation is 2x + y = 3. In other words, the
equations tell us that the expression 2x + y will equal -2 and 3
AT THE SAME TIME. Of course, this logically doesn’t make
sense because something cannot equal two different numbers!
That’s why there is no solution to this simultaneous equation.

458

EXAMPLE: Graph the simultaneous equation to

find the solution.

4x - 2y = 6
2x - y = 3

We can change the first equation from 4x - 2y =  6
to y = 2x - 3, and change the second equation
from  2x - y = 3 to y = 2x - 3.

Graphing the 5 1   2   3   4   5
two equations, 4
we realize that 3
they are the 2
exact same line! 1

-5 -4   -3   -2 -1
-1
-2 
-3
-4
-5

In this case, we can see that the two lines overlap
each other and intersect at every point along the line,

so there are an INFINITE  NUMBER  OF  SOLUTIONS

to the simultaneous equation.

459

SUBSTITUTION METHOD

We can also use algebra to solve simultaneous linear

equations-one way is known as the SUBSTITUTION
METHOD. In the substitution method, we find the solution

by rewriting one equation and then substituting it into

the other equation.

EXAMPLE: Solve the simultaneous equations by

using the substitution method.

4x + y = 7 O1 First, number your
3x + 2y = 9 O   2 equations.

OThen, rewrite equation 1   as: y = -4x + 7

and substitute it

into equation O2 : 3x + 2 ( -4x + 7 ) = 9

Now we can solve for x : 3 x - 8x + 14 = 9
-5x = - 5
x =  1

O OSubstituting x =  1 into either 1 or 2 ‚

or y = -4x + 7, we find that y = 3 ‚

so the solution is (x, y ) = ( 1, 3 ).

You can plug either the x- or y-value into

both original equations to check your work.

460

EXAMPLE: Solve the simultaneous equations by

using the substitution method.

-2x + 6y = 1 O1
x - 4y = 2 O   2
OWe can rewrite equation 2  as: x = 4y + 2

and substitute it

Ointo equation 1 : -2(4y + 2) + 6y = 1

Now we can solve for y : -8y - 4 + 6y = 1
-2y = 5
y = - -25  

O OSubstituting y = - -25  into either 1 or 2 ‚

or x = 4y + 2, we find that x = -8,
so the solution is: (x, y ) = (-8, --25   ).

ADDITION METHOD

The third way to solve simultaneous equations is by using the

ADDITION  METHOD. The goal of the addition method is to
eliminate either the x or y variable by adding their opposite.

First, we multiply all the terms of one equation by a constant
that will get one of the variables to add up to zero. Next, add
the two equations together, in order to eliminate a variable,
and then solve the simultaneous equation.

461

EXAMPLE: Solve the simultaneous equations by

using the addition method.

4x - y = -7 O1 First, number the
-3x + 2y = 9 O   2 equations.

OThen, multiply each term in equation 1 by 2 WE MULTIPLY BY 2
O’and call it equation 1 : BECAUSE THEN -2y IN
O 8x - 2y = -14 1 ’
O’ ONow add equation 1 and equation 2 WILL CANCEL

OUT 2y IN IN

THE NEXT STEP.

together: 8x - 2y = -14  O1 ’
-3x + 2y = 9    O2


8x + (-3x ) + (-2y) + 2y = -14 + 9 OR TRY ALL THREE
IF YOU WANT TO
5 x = -5 TRIPLE-CHECK YOUR WORK.
x = -1
O O O’Substituting x = -1 into either 1 , 2 , or 1 ,

we find that y = 3,

so the solution is: (x, y ) = (-1, 3 ).

462

We can multiply the terms of both equations if that is

what is necessary to solve the problem. Just find the least
common multiple of the x’s or the y’s and multiply each
term accordingly to eliminate the variable.

EXAMPLE: Solve by using the addition method.

2x + 5y = 3  O1
3x + 4y = 1  O2

The LCM of 2x and 3x is 6x. Therefore, we must

O O’multiply equation 1 by  3 and call it equation 1 :
O’ 6x + 15y = 9 1
O O’Next, multiply equation 2 by −2 and call it equation 2 :
O’
-6x - 8y = -2 2
’1 and equation O 2 O21’ t’’ogether:

6x + 15y = 9
-6x - 8y = -2
ONow add equation
O



 6x + (-6x ) + 15y + (-8y ) = 9 + (-2)
 7y = 7
 y = 1

O O O’ O’Substituting y = 1 into either 1  , 2 , 1 , or 2 ,

we find that, x = -1,
so the solution is: (x, y ) = ( -1, 1 ).

463

1. Solve the simultaneous equation by using the graphing

method:

x+y=3
2x + y = 7

2. Solve the simultaneous equation by using the graphing

method:
-x + y = 4
x + y = 2

3. Solve the simultaneous equation by using the graphing

method:

x + 2y = 0
 x - y = 6


4. Solve the simultaneous equation by using the graphing

method:

 - 4x + 3y = 6
2x - 4y = 2


464

5. Solve the simultaneous equation by using either the addition

method or the substitution method:

x-y=9
x+y=7


6. Solve the simultaneous equation by using either the addition

method or the substitution method:

x + y = 10
2x - y = -4

7. Solve the simultaneous equation by using either the addition

method or the substitution method:
3x - 2y = 10
2x + 3y = 11

8. Solve the simultaneous equation by using either the addition

method or the substitution method:
5x - 3y = 9
 -x + y =  -2

answers 465

 1. Solution: ( 4, −1) 10
9
8
7
6
5
4
3

2

1

-10  -9  -8  -7  -6  -5  -4  -3  -2 -1 -1 1  2  3  4  5  6 7  8  9  10
-2
-3
-4
-5
-6 
 -7
-8
-9

-10 

2. Solution: ( −1, 3 )

10
9
8
7
6
5
4
3

2

1

-10  -9  -8  -7  -6  -5  -4  -3  -2 -1 -1 1  2  3  4  5  6 7  8  9  10
-2
-3
-4
-5
-6 
 -7
-8
-9

-10 

466

3. Solution: = ( 4, −2) 10
9
8
7
6
5
4
3

2

1

-10  -9  -8  -7  -6  -5  -4  -3  -2 -1 -1 1  2  3  4  5  6 7  8  9  10
-2
-3
-4
-5
-6 
 -7
-8
-9

-10 

4. Solution: = ( −3, −2 ) 10
9
8
7
6
5
4
3

2

1

-10  -9  -8  -7  -6  -5  -4  -3  -2 -1 -1 1  2  3  4  5  6 7  8  9  10
-2
-3
-4
-5
-6 
 -7
-8
-9

-10 

5. Solution: = ( 8, −1 ) 7. Solution: = ( 4, 1 )
6. Solution: = ( 2, 8 ) 8. Solution: = (-23  , − -21  )

467

Chapter 60

NONLINEAR

FUNCTIONS

NONLINEAR  FUNCTIONS are NOT straight lines when

they are graphed, and they are NOT in the form y = mx  + b.

One example of a nonlinear function is a QUADRATIC

FUNCTION. In a quadratic function, the input variable (x )
is squared like so: x 2. The result is a PARABOLA, which is a

U-shaped curve.

EXAMPLE: Create an input/output chart

and graph y = x 2.

468

INPUT FUNCTION: OUTPUT COORDINATE
( y  )
(x ) y  =  x 2 POINTS ( x, y )
9 ( -3, 9 )
-3 y  =  ( -3 ) 2 4 ( -2, 4 )
-2 y  =  ( -2 ) 2 1 ( -1, 1 )
-1 y  =  ( -1  ) 2 0 ( 0, 0 )
0 y  =  ( 0 ) 2 1 ( 1, 1 )
1 y  = ( 1 ) 2 4 ( 2, 4 )
2 y  =  ( 2 ) 2 9 ( 3, 9 )
3 y  =  ( 3 ) 2

10
9
8
7
6
5
4
3
2
1

-5 -4   -3   -2 -1 -1  1   2   3   4   5 

469

EXAMPLE: Create an input/output chart

and graph y = -2x 2 + 1.

INPUT FUNCTION: OUTPUT COORDINATE

(x ) y  =  -2x 2  + 1 ( y  ) POINTS ( x, y )
-2 -7 ( -2, -7 )
-1 y  = -2 ( -2 ) 2  +  1 -1 ( -1, -1 )
0 y  = -2 ( 4 ) +  1 1 ( 0, 1 )
1 y  =  - 8  +  1 -1 ( 1, -1 )
2 -7 ( 2, -7 )
y  = -7
y  = -2 ( -1 ) 2  +  1
y  = -2 ( 1) +  1

y  =  - 2  +  1
y  = -1

y  = -2 ( 0 ) 2  +  1
y  = -2 ( 0 ) +  1

y  =  0  +  1
y  = 1

y  = -2 ( 1  ) 2  +  1
y  = -2 ( 1  ) +  1
y  =  - 2  +  1

y  = -1
y  = -2 ( 2 ) 2  +  1
y  = -2 ( 4 ) +  1

y  =  - 8  +  1
y  = -7

470

2
1

-5   -4   -3   -2 -1 1   2   3   4   5
-1
-2
-3
-4
-5
-6
-7
-8

In the first example
( y = x 2 ),

the coefficient in front of the x (which is 1 )
is positive, so the parabola opens upward.

In the second example
( y = -2x 2 + 1 ),

the coefficient in front of the x (which is -2)
is negative, so the parabola opens downward.

This pattern is true whenever you
graph nonlinear functions.

471

Another example of a nonlinear function is an ABSOLUTE
VALUE  FUNCTION. When graphed, the result is shaped

like a “V.”

EXAMPLE: Create an input/output chart

| |and graph y =   x  .

INPUT FUyN  C=T  | xIO  |N: OUTPUT COORDINATE
y  = | -2 | ( y  )
(x ) | |y  =  -1   2 POINTS ( x, y )
y  = | 0 | 1 ( -2, 2 )
-2 y  = | 1 | 0 ( -1, 1 )
-1 y  = | 2 | 1 ( 0, 0 )
0 2 ( 1, 1 )
1 ( 2, 2 )
2

3
2
1

-3 -2     -1 1   2   3

472 -1 

| |EXAMPLE: Create an input/output chart

and graph y = -   x  + 1 .

INPUT FyU  N= C- T| xI O | +N 1: OUTPUT COORDINATE
| | y  = -  -2  + 1
(x )  y  = -2 + 1 ( y  ) POINTS ( x, y )
-1 ( -2, -1 )
-2 y  =  - 1 0 ( -1, 0 )
-1 1 ( 0, 1 )
0 | | y  = -  -1  + 1 0 ( 1, 0 )
1  y  = - 1 + 1 -1 ( 2, -1 )
2
y  =  0

| | y  = -  0  + 1
 y  = 0 + 1

y  =  1

| | y  = -  -1  + 1
 y  = -1 + 1

y  =  0

| | y  = -  -2  + 1
 y  = -2 + 1

y  =  -1

2 1   2   3
1

-3   -2 -1
-1
-2 

473

1. Complete the input/output chart and graph y = x 2 + 1.

INPUT FUNCTION: OUTPUT COORDINATE

(x ) y  =  x 2 +  1 ( y  ) POINTS ( x, y )

-2

-1

0

1

2

2. Complete the input/output chart and graph y = -2x 2.

INPUT FUNCTION: OUTPUT COORDINATE

(x ) y  =  -2x 2  ( y  ) POINTS ( x, y )

-2

-1

0

1

2

474

| |3. Complete the input/output chart and graph y =  x  + 2.

INPUT FyU  N= C | xT |I +O  2N: OUTPUT COORDINATE

(x ) ( y  ) POINTS ( x, y )

3

-1

0

2

4

| |4. Complete the input/output chart and graph y = - x  - 1.

INPUT FyU  =N  C-T | xI O|  -N 1: OUTPUT COORDINATE

(x ) ( y  ) POINTS ( x, y )

-5
-3
0
2
5

answers 475

 1. INPUT FUNCTION: OUTPUT COORDINATE

(x ) y  = x 2 + 1 ( y  ) POINTS ( x, y )
-2 5 ( -2, 5 )
-1  y  =  ( -2 )2 + 1 2 ( -1, 2 )
0 y  = 4 + 1 1 ( 0, 1 )
1 y  =  5 2 ( 1, 2 )
2 5 ( 2, 5 )
 y  =  ( -1 )2 + 1
y  = 1 + 1
y  =  2

 y  =  ( 0 )2 + 1

y  = 0 + 1
y  =  1

 y  =  ( 1 )2 + 1

y  = 1 + 1
y  =  2

 y  =  ( 2 )2 + 1
y  = 4 + 1
y  =  5

5 1   2   3   4   5
4
3
2
1

-5 -4   -3   -2 -1
-1
-2 

476

2. INPUT FUNCTION: OUTPUT COORDINATE

(x ) y  = -2x 2 ( y  ) POINTS ( x, y )
-2 -8 ( -2, -8 )
-1  y  =   -2  ( -2 )2  -2 ( -1, -2)
0 0 ( 0, 0 )
1 y  =  -2  ( 4 ) -2 ( 1, -2 )
2 y  =  -8 -8 ( 2, -8 )

 y  = -2 ( -1 )2

y  =  -2( 1 )
y  =  -2

 y  = -2 ( 0 )2

y  =  -2 ( 0 )
y  =  0

 y  = -2 ( 1 )2

y  =  -2 ( 1 )
y  =  -2

 y  = -2 ( 2)2

y  =  -2 ( 4 )
y  =  -8

2 1   2   3   4   5
1

-5 -4   -3   -2 -1
-1
-2 
-3
-4
-5
-6
-7
-8

477

3. FUNCTION: OUTPUT COORDINATE

INPUT y  = | x  | + 2 ( y  ) POINTS ( x, y )
5 ( -3, 5 )
(x )  y  = | -3 | + 2 3 ( -1, 3 )
-3 2 ( 0, 2 )
-1 y  = 3 + 2 4 ( 2, 4 )
0 y  =  5 6 ( 4, 6 )
2
4  y  = | -1 | + 2

y  = 1 + 2
y  =  3

 y  = | 0 | + 2

y  = 0 + 2
y  =  2

 y  = | 2 | + 2

y  = 2 + 2
y  =  4

 y  = | 4 | + 2

y  = 4 + 2
y  =  6

6 1   2   3   4   5
5
4
3
2
1

-5 -4   -3   -2 -1
-1
-2 

478

4. FUNCTION: OUTPUT COORDINATE

INPUT y  = - | x  | - 1 ( y  ) POINTS ( x, y )
-6 ( -5, -6 )
(x )  y  = - | -5 | - 1 -4 ( -3, -4 )
-5 -1 ( 0, -1 )
y  = - ( 5 ) - 1 -3 ( 2, -3 )
-3 y  =  -6 -6 ( 5, -6 )

0  y  = - | -3 | - 1

2 y  = - ( 3 ) - 1
y  =  -4
5
 y  = - | 0 | - 1

y  = - ( 0 ) - 1
y  =  -1

 y  = - | 2 | - 1

y  = - ( 2 ) - 1
y  =  -3

 y  = - | 5 | - 1

y  = - ( 5 ) - 1
y  =  -6

2 1   2   3   4   5
1

-5 -4   -3   -2 -1
-1
-2 
-3
-4
-5
-6
-7

479

Chapter 61

POLYGONS
AND THE

COORDINATE

PLANE

Besides plot ting points, drawing lines, and creating various
other graphs, we can also use the coordinate plane to draw
shapes. We just plot points and connect them with lines.

EXAMPLE:

Plot the points (2,−3), (4,−3), (4,1), and (2,1).

Then identify the shape that is formed by
the points.

480

First plot the points:

5 1   2   3   4   5 6 7
4
3
2
1

-7 -6   -5  -4   -3   -2 -1
-1
-2 
-3
-4
-5

Then create the shape

by connecting the dots:

5
4
3
2
1

-7 -6   -5  -4   -3   -2 -1 1   2   3   4   5 6 7
-1
-2 
-3
-4
-5

It’s a rectangle!

481

EXAMPLE: Plot the points (−2,3), (2,3), (0,−3), (4,0), and
(−4,0). Then identify the shape that is formed by the points.

First plot the points: 5
4
3
2
1

-7 -6   -5  -4   -3   -2 -1 1   2   3   4   5 6 7
-1
-2 
-3
-4
-5

Then connect the dots:

5
4
3
2
1

-7 -6   -5  -4   -3   -2 -1 1   2   3   4   5 6 7
-1

The points create -2 
a pentagon. -3
-4
-5

482

DON’T  FORGET: We can find the length of the side

of a figure if the x -coordinate or the y -coordinate is
the same. So, we can solve problems related to shapes
on a coordinate plane.

EXAMPLE: A square is plotted on a coordinate plane.

Three of its vertices are A ( 1, −3), B ( 5,−3), and C ( 1,  1).

What are the coordinates of point D, the 4th vertex?

First, plot points 5

A, B, and C on the 4

coordinate plane: 3

2C

1

-7 -6   -5  -4   -3   -2 -1 1   2   3   4   5 6 7
-1
-2  AB
-3
-4
-5

Because we know a square has four equal sides,
we only have to find the length of one side.

Because Point A and Point B share the same

y -coordinate, we can calculate the distance between
the two points by finding the difference of the

x-coordinates, which is 5 - 1 = 4.

483

Therefore, the distance between Point C and Point D

must also be 4 units, and the distance between
Point B and Point D must also be 4 units.

So, the coordinates

for Point D are ( 5, 1 ).

5 CD
4
3 1   2   3   4   5 6 7
2
1 AB

-7 -6   -5  -4   -3   -2 -1
-1
-2 
-3
-4
-5

484

1. P lot the points (1, −3), ( 5,  2), and ( 7, −2). Then identify

 the shape that is formed by the points.

2. P lot the points (−3, 1 ), ( 1, 2), (−2,−3 ), and ( 2,−2 ).

 Then identify the shape that is formed by the points.

3. P lot the points (−1, 2), ( 2, −1 ), ( 1, 4), and (4, 1 ). Then identify

 the shape that is formed by the points.

4. P lot the points ( 1, 0), ( 1, 4), (3, 2), (3, 4), (−1, 2), and (−1, 0).

 Then identify the shape that is formed by the points.

5.   A square is plot ted on a coordinate plane. Three of

its vertices are at A (−5, 4 ), B ( 2, 4 ), and C ( 2, −3).
What are the coordinates of Point D, the 4th vertex?

6. A rectangle is plot ted on a coordinate plane. Three of

its vertices are at A ( 1, 3), B ( 3, −2), and C. ( 3, 3).
What are the coordinates of Point D, the 4th vertex?

7. A square is plot ted on the coordinate plane. Three of

its vertices are at A ( 2, 4 ), B ( −2, 4 ), and C ( 2, 0 ).
What are the coordinates of Point D, the 4th vertex?

8. A rectangle is plot ted on the coordinate plane. Three of

its vertices are at A (−5, −2), B (−5, 3), and C ( 6, 3).
What are the coordinates of Point D, the 4th vertex?

answers 485

 1. The shape is a triangle.  2. T he shape is a square.

3 1   2   3   4   5 6 7 2 1   2   3 
2 1
1
-3   -2  -1
-1 -1
-1 -2 
-2  -3
-3

3. The shape is a rectangle.    4. T he shape is a hexagon.

5 1   2   3   4   5 5 1   2   3   4   5
4 4
3 3
2 2
1 1

  -2 -1 -2 -1
-1 -1 
-2 

 5. ( −5, −3 )   6. ( 1, −2 )
7. (−2, 0 )  8. ( 6, −2 )

486

Chapter 62

TRANSFORMATIONS

Besides plotting points and shapes, we can also use
the coordinate plane to work with shapes and figures.

A TRANSFORMATION is a change of position or size

of a figure. When we transform a figure, we create a
new figure that is related to the original.

TRANSLATIONS

One type of transformation is a TRANSLATION, which

is a transformation that moves all the figure’s points the
same distance and direction. However, the orientation
and size remain the same. The newly translated figure is

called the IMAGE, and the new points are written with a

PRIME  SYMBOL ( ‘ ).

487

EXAMPLE:

7 If you translate a figure
more than once, you
B6 can label the new figures

5
4 with DOUBLE PRIME (ll),
B ’ tripLE PRIME (lll),
3 C or more!
2

A1

-7   -6   -5   -4   -3   -2 -1 C ’1   2   3   4   5   6   7
-1

-2  A ’
-3

-4

-5  ABC ≅  A ’B’C ’
-6

-7

≅DON’T FORGET THE SYMBOL FOR CONGRUENT IS .

These congruent triangles have corresponding sides that
are the same length and corresponding angles that are
the same number of degrees. They are simply plotted on
different parts of the coordinate plane, so only the locations
of the triangles are different.

To do a translation, move each point according to the
given criteria.

488

EXAMPLE: G iven  ABC, translate it as follows:
(x + 4, y + 3 ).

5 Al is First, write the original
4 coordinates.
3 X = -2 + 4 = 2
Y = -2 + 3 = 2 Then calculate each
’2 Al = ( 2 , 1 ) translated point by

1 C  A ’ adding 4 units to the
x-value (x + 4 ) and
-5   -4   -3   -2 -1 1   2   3   4   5 then adding 3 units to
-1 the y-value (y + 3 ).
A -2  B ’
C -3
-4
B -5

ORIGINAL IMAGE ’ ’ ’Lastly, plot and label
ACB ’  ’’ (((022,,,
A (-2, -2) 1) the image as A B C .
B (-2, -4) -1)
C (-4, -3) 0)

If it’s a simple translation like this, you can simply move
each point by counting the units on the coordinate
plane. If it’s more complex, you can calculate the
translation separately and then plot your new points.

489