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Precalculus
Chapter 1
Functions and Relations
© McGraw Hill LLC. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill LLC.

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Chapter Outline
1.1 The Rectangular Coordinate System and
Graphing Utilities
1.2 Circles
1.3 Functions and Relations
1.4 Linear Equations in Two Variables and Linear
Functions
1.5 Applications of Linear Equations and Modeling
1.6 Transformations of Graphs
1.7 Analyzing Graphs of Functions and Piecewise-
Defined Functions
1.8 Algebra of Functions and Function
Composition
1.1 B-2

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Concepts
1. Plot Points on a Rectangular Coordinate
System
2. Use the Distance and Midpoint Formulas
3. Graph Equations by Plotting Points
4. Identify x- and y-Intercepts
5. Graph Equations Using a Graphing Utility
1.1 B-3

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Concept 1
Plot Points on a Rectangular
Coordinate System
1.1 B-4

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Example 1
Plot each point.
a. (2, 3)
b. (−4, 6)
c. (0, 2.5)
d.
3
, 0
4
 
 
 
-
e.  
2, 2
-
f. ( , 4)
-

Access the text alternative for slide images. 1.1 B-5

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Concept 2
Use the Distance and Midpoint
Formulas
1.1 B-6

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Distance Formula
The distance between points 1 1
( , )
x y and 2 2
( , )
x y
is given by:
   
2 2
2 1 2 1
d x x y y

= - -
1.1 B-7

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Example 2
Find the distance between the points (−3, −4)
and (3, 0). Give the exact distance and an
approximation to 2 decimal places.
Solution:
   
2 2
3 3 0 4
d = + + +
2 2
6 4 52 2 13 7.12
 
= = =
1.1 B-8

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Skill Practice 1
Find the distance between the points (−1, 4) and
(3, −6). Give exact distance and an approximation
to 2 decimal places.
1.1 B-9

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Example 3
Find the distance between the points (−1.2, 4.5)
and (3.8, 1.5). Give the exact distance and an
approximation to 2 decimal places.
   
2 2
3.8 1.2 1.5 4.5
d = + + -
 
2
2
5 3 34 5.83

= + - =
1.1 B-10

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Example 4 (1 of 2)
Determine if the given points form the vertices
of a right triangle: ( 2, 1), (2, 5), (5, 2).
A B C
-
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Example 4 (2 of 2)
Solution:
2 2
( , ) (2 2) (5 1) 32
d A B = + + - =
2 2
( , ) (5 2) (2 1) 50
d A C = + + - =
2 2
( , ) (5 2) (2 5) 18
d B C = - + - =
2 2 2
a b c

+ =
     
2 2 2
32 18 50 32 18 50

+ = + = Yes
1.1 B-12

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Skill Practice 2
Determine if the points ( 6, 4), (2, 2),
X Y
- - - and
(0, 5)
Z form the vertices of a right triangle.
1.1 B-13

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Midpoint Formula
The midpoint of the line segment with endpoints
1 1
( , )
x y and 2 2
( , )
x y is:
1 2 1 2
,
2 2
x x y y
 
 
 
+ +
1.1 B-14

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Example 5
Find the midpoint of the line segment with
endpoints (−1.8, −3) and (4.5, −1).
Solution:
 
1.8 4.5 3 1 27
, 1.35, 2 or , 2
2 2 20
M
   
   
   
- + - -
= = - -
1.1 B-15

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Skill Practice 3
Find the midpoint of the line segment with
endpoints (−1.5, −9) and (−8.7, 4).
1.1 B-16

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Concept 3
Graph Equations by Plotting
Points
1.1 B-17

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Example 6 (1 of 2)
Graph the equation by plotting points.
2
3
y x
= +
Solution:
x y Ordered Pairs
0 3 (0, 3)
1 4 (1, 4)
2 7 (2, 7)
3 12 (3, 12)
−1 4 (−1, 4)
−2 7 (−2, 7)
1.1 B-18

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Example 6 (2 of 2)
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1.1 B-19

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Example 7 (1 of 2)
Graph the equation by plotting points.
3
= +
y x
Solution:
x y Ordered pairs
3
- 0 ( 3,0)
-
2
- 1 ( 2,1)
-
1
- 2 ( 1,1.4)
-
0 3 (0,1.7)
1 4 (1, 2)
2 5 (2, 2.2)
1.1 B-20

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Example 7 (2 of 2)
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1.1 B-21

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Skill Practice 4
Graph the equation by plotting points.
2
4
x y
+ =
1.1 B-22

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Example 8 (1 of 2)
Graph the equation by plotting points.
2
2 0
x y
- + =
2
2
x y
= -
x y Ordered Pairs
−2 0 (−2,0)
−1 1 (−1,1)
2 2 (2,2)
7 3 (7,3)
−1 −1 (−1,−1)
2 −2 (2,−2)
1.1 B-23

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Example 8 (2 of 2)
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1.1 B-24

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Skill Practice 5
Graph the equation by plotting points.
2
2
x y
+ =
1.1 B-25

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Concept 4
Identify x- and y-Intercepts
1.1 B-26

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Determining x- and y-Intercepts from
an Equation
Given an equation in x and y,
• Find the x-intercept(s) by substituting 0 for y
in the equation and solving for x.
• Find the y-intercept(s) by substituting 0 for x
in the equation and solving for y.
1.1 B-27

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Example 9
Estimate the x- and y-intercept(s) from the
graph.
x-int:Blank____________
y-int:Blank____________
Solution:
x-int: (3, 0)
y-int: (0, 3)
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1.1 B-28

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Example 10
Estimate the x- and y-intercept(s) from the graph.
x-int:Blank____________
y-int:Blank____________
Solution:
x-int: (2, 0) (−2, 0)
y-int: (0, −4)
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1.1 B-29

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Example 11
Find the x- and y-intercept(s).
2
9
y x
- =
Solution: 2
int: 9
x x
- = -
No real solution, No x-intercept.
int: 0 9
y y
- - =
y=9
(0,9)
1.1 B-30

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Example 12
Find the x- and y-intercepts.
| 3| 2
x y
= - +
Solution:
x-intercept:
|0 3| 2
x = - +
| 3| 2
= - +
= 3 + 2
= 5
(5,0)
y-intercept:
0 | 3| 2
y
= - +
2 | 3|
y
- = -
No Solution.
No y-intercept
1.1 B-31

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Example 13 (1 of 2)
Find the x- and y-intercepts.
2 2
( 3)
1
9 4
x y +
+ =
Solution: x-intercept:
2 2
(0 3)
1
9 4
x +
+ =
2
9
1
9 4
x
+ =
2
5
9 4
x -
=
2
4 45
x = -
2 45
4
x
-
=
No x-intercept 1.1 B-32

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Example 13 (2 of 2)
y-intercept:
2
0 ( 3)
1
9 4
y +
+ =
2
( 3) 4
y + =
3 4
y + = ±
3 2 5, 1
y = - ± = - -
)(0, 5)
(0, 1 -
-
1.1 B-33

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Skill Practice 6
Given the equation 2
4,
y x
= -
a. Find the x-intercept(s).
b. Find the y-intercept(s).
1.1 B-34

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Concept 5
Graph Equations Using a
Graphing Utility
1.1 B-35

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Graph Equations Using a Graphing
Utility (1 of 3)
Enter the equation into the graphing editor in
the form Y =Blank____________
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1.1 B-36

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Graph Equations Using a Graphing
Utility (2 of 3)
The standard viewing window is from −10 to
10 on the x-axis and −10 to 10 on the y-axis.
The tick marks on each axis are 1 unit apart.
This viewing window would be listed:[−10, 10, 1]
by [−10, 10, 1].
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1.1 B-37

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Graph Equations Using a Graphing
Utility (3 of 3)
To set up a table, enter the starting value for x
and the increment of change for x.
In this case, Tbl = 1 means the x-value will
change in steps of 1 unit. 1, 2, 3, ... etc.
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1.1 B-38

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Example 14
Use a graphing utility to graph y = 0.5x + 2
on the standard viewing window.
Make a table beginning at x = 0 with Tbl = 2.
Solution: X Y1
0 2
2 3
4 4
6 5
8 6
10 7
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Example 15
Use a graphing utility to graph 2
8 2
y x x
= - -
on the standard viewing window.
Make a table beginning at x = −2 with Tbl = 1.
Solution:
X Y1
−2 8
−1 9
0 8
1 5
2 0
3 −7
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Example 16
Use a graphing utility to graph 3 2
6 3 4
y x x x
= - + -
on the viewing window defined by [−3, 8, 1] by
[−30, 25, 5].
Solution:
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Example 17
Graph on the given viewing window [−5, 5, 1]
by [−1, 10, 1].
2
2 8
y x
= -
Solution:
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Skill Practice 7
Use a graphing utility to graph y = −x + 2 and
2
0.5 2
y x
= - - on the viewing window
[−6, 6, 1] by [−4, 8, 1].
1.1 B-43

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