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CHAPTER 2 Points, Lines, and Functions
Chapter 2
Points, Lines, and Functions
Section 2.1:
An Introduction to the Coordinate Plane
 Points in the Coordinate Plane
Points in the Coordinate Plane
The Rectangular Coordinate System:
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University of Houston Department of Mathematics
SECTION 2.1 An Introduction to the Coordinate Plane
Plotting Points in the Coordinate Plane:
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Example:
Solution:
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University of Houston Department of Mathematics
SECTION 2.1 An Introduction to the Coordinate Plane
Graphing Horizontal and Vertical Lines:
Example:
Solution:
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CHAPTER 2 Points, Lines, and Functions
Graphing Other Lines:
Example:
Solution:
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SECTION 2.1 An Introduction to the Coordinate Plane
Additional Example 1:
Solution:
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Additional Example 2:
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University of Houston Department of Mathematics
SECTION 2.1 An Introduction to the Coordinate Plane
Solution:
Additional Example 3:
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CHAPTER 2 Points, Lines, and Functions
Solution:
Additional Example 4:
Solution:
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SECTION 2.1 An Introduction to the Coordinate Plane
(c) Draw a line through the points.
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Additional Example 5:
Solution:
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University of Houston Department of Mathematics
SECTION 2.1 An Introduction to the Coordinate Plane
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CHAPTER 2 Points, Lines, and Functions
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University of Houston Department of Mathematics
Exercise Set 2.1: An Introduction to the Coordinate Plane
17. If the point (a, b) is in Quadrant I, identify the
quadrant of each of the following points:
(a) (-a, -b)
(b) (-a, b)
(c) (a, a)
Plot the following points in a coordinate plane.
1.
A(3, 4)
2.
B(2, -5)
3.
C(-3, -1)
4.
D(-4, -6)
5.
E(-5, 0)
6.
F(0, -2)
18. If the point (a, b) is in Quadrant I, identify the
quadrant of each of the following points:
(a) (-b, a)
(b) (b, b)
(c) (-b, -a)
19. If the point (a, b) is in Quadrant II, then a  0
and b  0 . Identify the quadrant of each of the
following points:
(a) (-a, -b)
(b) (b, a)
(c) (a, -b)
Write the coordinates of each of the points shown in
the figure below. Then identify the quadrant or axis
in which the point is located.
y
7.
G
8.
H
9.
I
J
21. If the point (a, b) is in Quadrant IV, identify the
quadrant of each of the following points:
(a) (b, -b)
(b) (-a, -a)
(c) (b, a)

I
G

10. J
H


20. If the point (a, b) is in Quadrant III, then a  0
and b  0 . Identify the quadrant of each of the
following points:
(a) (-a, b)
(b) (b, a)
(c) (-a, -b)


x

11. K
22. If the point (a, b) is in Quadrant II, identify the
quadrant of each of the following points:
(a) (-a, b)
(b) (b, b)
(c) (a, -a)

12. L
K

L
Plot each of the following sets of points in a coordinate
plane. Then identify the quadrant or axis in which
each point is located.
13. (a)
(b)
(c)
(d)
A(2, 5)
B(-2, -5)
C(2, -5)
D(-2, 5)
14. (a)
(b)
(c)
(d)
A(4, -3)
B(-4, -3)
C(-4, 3)
D(4, 3)
15. (a)
(b)
(c)
(d)
A(0, -2)
B(-2, 0)
C(2, 0)
D(0, 2)
16. (a)
(b)
(c)
(d)
A(-3, 0)
B(3, 0)
(0, -3)
D(0, 3)
23. If the point (a, b) is in Quadrant III, identify the
axis on which each of the following points lies:
(a) (a, 0)
(b) (0, b)
(c) (-b, 0)
24. If the point (a, b) is in Quadrant IV, identify the
axis on which each of the following points lies:
(a) (0, -b)
(b) (-a, 0)
(c) (b, 0)
Answer True or False.
25. The point (0, 5) is on the x-axis.
26. The point (-4, 0) is in Quadrant II.
27. The point (1, -3) is in Quadrant IV.
28. The point (-2, -5) is in Quadrant III.
29. The point (0, 0) is in Quadrant I.
MATH 1300 Fundamentals of Mathematics
30. The point (-6, 1) is in Quadrant IV.
31. If the point (a, b) is in Quadrant IV, then b  0 .
32. If the point (a, b) is in Quadrant II, then a  0 .
33. If the point (a, b) is in Quadrant I, then the point
(b, a) is also in Quadrant I.
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Exercise Set 2.1: An Introduction to the Coordinate Plane
34. If the point (a, b) is in Quadrant I, then the point
(a, -b) is in Quadrant II.
35. If the point (a, b) is in Quadrant II, then the point
(-a, -b) is in Quadrant III .
47. Graph the line x  2 .
48. Graph the line y  5 .
49. Graph the line y  4 .
36. If the point (a, b) is in Quadrant IV, then the
point (-b, a) is in Quadrant I.
50. Graph the line x  3 .
37. If the point (a, b) is in Quadrant III, then b  0 .
51. On the same set of axes, graph the lines x  1
and y  3 .
38. If the point (a, b) is on the y-axis, then a  0 .
39. If the point (a, b) is on the y-axis, then b  0 .
52. On the same set of axes, graph the lines x  5
and y  2 .
40. If the point (a, b) is on the y-axis, then a  0 .
53. On the same set of axes, graph the lines x 
41. If the point (a, b) is on the y-axis, then the point
(b, a) is on the x-axis.
42. If the point (a, b) is on the x-axis, then the point
(a, 3) lies in Quadrant I .
Answer the following.
43. Given the following points:
A(3, 5), B(3, 1), C(3, 0), D(3, -2)
(a)
(b)
(c)
(d)
and y  0 .
54. On the same set of axes, graph the lines x  0
and y   52 .
Graph the following lines by first completing the table
and then plotting the points on a coordinate plane.
55. y  3x  2
Plot the above points on a coordinate plane.
What do the above points have in common?
Draw a line through the above points.
What is the equation of the line drawn in
part (c)?
Plot the above points on a coordinate plane.
What do the above points have in common?
Draw a line through the above points.
What is the equation of the line drawn in
part (c)?
45. (a) List four points that are on the x-axis.
(b) Analyze the coordinates of the points you
have listed. What do they have in common?
(c) Give the equation of the x-axis.
x
y
-2
-1
0
1
44. Given the following points:
A(-3, 4), B(0, 4), C(1, 4), D(3, 4)
(a)
(b)
(c)
(d)
7
2
2
56. y  2x  5
x
y
-2
-1
0
1
46. (a) List four points that are on the y-axis.
(b) Analyze the coordinates of the points you
have listed. What do they have in common?
(c) Give the equation of the y-axis.
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University of Houston Department of Mathematics
Exercise Set 2.1: An Introduction to the Coordinate Plane
57. y   4x  7
x
y
0
1
4
-5
2
 32
58. y  5x 1
x
y
2
-1
3
5
-6
0
Answer the following.
59. Graph the line segment with endpoints (-7, 0)
and (0, 7).
60. Graph the line segment with endpoints (3, 5) and
and (-5, -3).
61. Graph the line segment with endpoints (1, -4)
and (-1, 4)
62. Graph the line segment with endpoints (-2, 6)
and (6, 2).
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CHAPTER 2 Points, Lines, and Functions
Section 2.2:
The Distance and Midpoint Formulas
 The Distance Formula
 The Midpoint Formula
The Distance Formula
Finding the Distance Between Two Points:
Example:
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University of Houston Department of Mathematics
SECTION 2.2 The Distance and Midpoint Formulas
Solution:
Additional Example 1:
Solution:
Additional Example 2:
Solution:
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Additional Example 3:
Solution:
Additional Example 4:
Solution:
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SECTION 2.2 The Distance and Midpoint Formulas
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CHAPTER 2 Points, Lines, and Functions
Additional Example 5:
Solution:
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SECTION 2.2 The Distance and Midpoint Formulas
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Additional Example 6:
Solution:
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SECTION 2.2 The Distance and Midpoint Formulas
Additional Example 7:
Solution:
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Use the Pythagorean Theorem to determine c.
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SECTION 2.2 The Distance and Midpoint Formulas
The Midpoint Formula
Finding the Midpoint of a Line Segment:
Example:
Solution:
Additional Example 1:
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Solution:
Additional Example 2:
Solution:
Additional Example 3:
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SECTION 2.2 The Distance and Midpoint Formulas
Solution:
Additional Example 4:
Solution:
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Additional Example 5:
Solution:
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SECTION 2.2 The Distance and Midpoint Formulas
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Exercise Set 2.2: The Distance and Midpoint Formulas
Use the Pythagorean Theorem to find the missing side
of each of the following triangles.
6.
Pythagorean Theorem: In a right triangle, if a and b
are the measures of the legs, and c is the measure of
the hypotenuse, then a2 + b2 = c2.
(a) Plot the above points on a coordinate plane.
(b) Draw segment AB. This will be the
hypotenuse of triangle ABC.
(c) Find a point C such that triangle ABC is a
right triangle. Draw triangle ABC.
(d) Use the Pythagorean theorem to find the
distance between A and B (the length of the
hypotenuse of the triangle).
c
a
b
1.
c
Use the distance formula to find the distance between
the two given points. (You can also use the method from
the previous two problems to double-check your answer.)
5
12
2.
Given the following points:
A(3, 1) and B(1,  5)
7
a
7.
(3, 6) and (5, 9)
8.
(4, 7) and (2, 3)
9.
(5, 0) and (2, 6)
10. (9,  4) and (2,  3)
5
3.
11. (4, 0) and (0,  7)
6
2
b
4.
12. (4,  8) and (10,  1)
13.
5,  12  and 3,  56 
14.
 32 ,  1 and  34 , 0
c
6
8
Find the midpoint of the line segment joining points A
and B.
15. A(7, 6) and B(3, 8)
Answer the following.
5.
Given the following points:
A(1, 2) and B(4, 7)
16. A(5, 9) and B(1, 3)
17. A(7, 0) and B(4, 8)
18. A(7,  5) and B(4,  3)
(a) Plot the above points on a coordinate plane.
(b) Draw segment AB. This will be the
hypotenuse of triangle ABC.
(c) Find a point C such that triangle ABC is a
right triangle. Draw triangle ABC.
(d) Use the Pythagorean theorem to find the
distance between A and B (the length of the
hypotenuse of the triangle).
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19. A(3, 0) and B(0,  9)
20. A(6,  7) and B(10,  6)
21. A
 13 ,  5 and B   53 , 7 



22. A 3,  12 and B 8,  56

University of Houston Department of Mathematics
Exercise Set 2.2: The Distance and Midpoint Formulas
Answer the following.
23. (a) Graph the line segment with endpoints
A(2, 6) and B(5,  4) .
(b) Find the distance from A to B.
(c) Find the midpoint of AB .
24. (a) Graph the line segment with endpoints
A(4, 0) and B(2,  5) .
(b) Find the distance from A to B.
(c) Find the midpoint of AB .
25. If M (4, 7) is the midpoint of the line segment
joining points A and B, and A has coordinates
(2, 3) , find the coordinates of B.
26. If M (5, 3) is the midpoint of the line segment
joining points A and B, and A has coordinates
(1, 6) , find the coordinates of B.
27. If M (3,  5) is the midpoint of the line segment
joining points A and B, and B has coordinates
(1,  2) ,
(a) Find the coordinates of A.
(b) Find the length of AB .
28. If M (2, 1) is the midpoint of the line segment
joining points A and B, and B has coordinates
(5,  3) ,
(a) Find the coordinates of A.
(b) Find the length of AB .
29. Determine which of the following points is
closer to the origin: A(5,  6) or B(3, 7) ?
30. Determine which of the following points is
closer to the point (4,  1) : A(2, 3) or
B(6, 6) ?
31. A circle has a diameter with endpoints
A(5,  9) and B(3, 5) .
(a) Find the coordinates of the center of the
circle.
(b) Find the length of the radius of the circle.
32. A circle has a diameter with endpoints A(2,  7)
and B(8, 1) .
(a) Find the coordinates of the center of the
circle.
(b) Find the length of the radius of the circle.
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CHAPTER 2 Points, Lines, and Functions
Section 2.3:
Slope and Intercepts of Lines
 The Slope of a Line
 Intercepts of Lines
The Slope of a Line
Finding the Slope of a Line:
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University of Houston Department of Mathematics
SECTION 2.3 Slope and Intercepts of Lines
Example:
Solution:
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University of Houston Department of Mathematics
SECTION 2.3 Slope and Intercepts of Lines
Additional Example 1:
Solution:
Additional Example 2:
Solution:
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CHAPTER 2 Points, Lines, and Functions
Additional Example 3:
Solution:
Additional Example 4:
Solution:
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University of Houston Department of Mathematics
SECTION 2.3 Slope and Intercepts of Lines
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CHAPTER 2 Points, Lines, and Functions
Intercepts of Lines
Finding Intercepts of Lines:
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University of Houston Department of Mathematics
SECTION 2.3 Slope and Intercepts of Lines
Horizontal Lines:
Vertical Lines:
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CHAPTER 2 Points, Lines, and Functions
Example:
Solution:
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University of Houston Department of Mathematics
SECTION 2.3 Slope and Intercepts of Lines
Example:
Solution:
Additional Example 1:
Solution:
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CHAPTER 2 Points, Lines, and Functions
Additional Example 2:
Solution:
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University of Houston Department of Mathematics
SECTION 2.3 Slope and Intercepts of Lines
Additional Example 3:
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CHAPTER 2 Points, Lines, and Functions
Solution:
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University of Houston Department of Mathematics
Exercise Set 2.3: Slope and Intercepts of Lines
State whether the slope of each of the following lines is
positive, negative, zero, or undefined.
1.
2.
p
q
3.
r
4.
s
5.
6.

y

t
x



w



25. e
26. f

x






w
c
24. d


y
e
23. c
r

t
Find the slope of each of the following lines. If
undefined, state ‘Undefined.’

f

p

q
s
Find the slope of the line that passes through the
following points. If undefined, state ‘Undefined.’
7.
(0, 0) and (3, 7)
8.
(8, 0) and (3, 6)
9.
(2, 5) and (4, 10)
d
For each of the following:
(a) Complete the given table.
(b) Plot the points on a coordinate plane and
graph the line.
(c) Use two points from the table to find the slope
of the line.
27. y  4x  1
10. (7, 3) and (5, 9)
x
11. (6, 4) and (2, 4)
0
12. (5, 1) and (5, 8)
2
y
13. (2, 3) and (6,  7)
3
14. (2,  6) and (5, 10)
0
 12
15. (3,  8) and (3,  4)
16. (8,  7) and (1,  7)
17. (2,  8) and (0,  3)
18. (1,  4) and (7, 2)
28. y  3x  2
19.
 12 , 1 and  32 , 16 
20.
 2, 34  and  15 , 85 
21.

22.
  53 ,  107  and  14 ,  87 
2,
7

4
9
 and 
 56 ,
x
y
2
2
4

1
2

MATH 1300 Fundamentals of Mathematics
 34
3
149
Exercise Set 2.3: Slope and Intercepts of Lines
29. y  32 x  4
x
y
4
5
9
8
3
2
For each of the following graphs:
(a)
(b)
(c)
(d)
(e)
State the x-intercept.
State the y-intercept.
State the coordinates of the x-intercept.
State the coordinates of the y-intercept.
Find the slope of the line.
y
33.


30. y   53 x  6
x
y
x
5




0

7
8
0
 y
34.

Answer the following.
31. Examine the relationship in numbers 27-30
between each of the equations and the
corresponding slope that you found for each line.
Do you see any pattern? Can you determine the
slope of the line from simply looking at its
equation?



x







32. Based on the pattern found in the previous
problem, state the slope of the following lines
without graphing the line or performing any
calculations:
(a) y  2 x  9
(b) y  7 x  5
(c) y   54 x  2
(d) y  73 x  4
For each of the following equations:
(a) Find the x- and y-intercepts of the line.
(b) State the coordinates of the intercepts.
(c) Plot the x- and y-intercepts on a coordinate
plane.
(d) Graph the line, based on the intercepts.
35. y  2x  8
36. y  3x  6
37. y  4 x  5
38. y  3x  7
39. 5x  2 y  20
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University of Houston Department of Mathematics
Exercise Set 2.3: Slope and Intercepts of Lines
40. 2x  3 y  18
Answer the following.
41. 3x  5 y  30
55. Examine the relationship in numbers 53 and 54
between each of the equations and the
corresponding y-intercept that you found for
each line. Do you see any pattern? Can you
determine the y-intercept of the line from simply
looking at its equation?
42. 3x  24  4 y
43. 2x  3 y  10
44. 4x  6 y  9
45. 5x  3 y  21  0
56. Based on the pattern found in the previous
problem, state the y-intercept of the following
lines without graphing the line or performing any
calculations:
(a) y  2 x  9
(b) y  7 x  5
46. 4x  7 y  8  0
47. 2 x  2 y  7
48. 3x  15
49. 4 y  12
(c) y   54 x  2
50. 4x  4 y 15
(d) y  73 x  4
51. 6 x  24
52. 2 y  14
For each of the following:
(a) Complete the given table.
(b) Plot the points on a coordinate plane and
graph the line.
(c) Find the x- and y-intercepts of the line.
(d) Find the slope of the line.
53. y  2 x  8
x
y
0
0
2
6
0.5
54. y   x  3
x
y
0
0
3
1.5
2
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CHAPTER 2 Points, Lines, and Functions
Section 2.4:
Equations of Lines
 Writing Equations of Lines
Writing Equations of Lines
Different Forms for Equations of Lines:
Example:
Solution:
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University of Houston Department of Mathematics
SECTION 2.4 Equations of Lines
Example:
Solution:
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Example:
Solution:
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SECTION 2.4 Equations of Lines
Example:
Solution:
Additional Example 1:
Solution:
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CHAPTER 2 Points, Lines, and Functions
Additional Example 2:
Solution:
To sketch the graph, begin by using
the y-intercept to plot the point  0,1 .
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SECTION 2.4 Equations of Lines
Additional Example 3:
Solution:
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CHAPTER 2 Points, Lines, and Functions
Additional Example 4:
Solution:
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University of Houston Department of Mathematics
SECTION 2.4 Equations of Lines
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Exercise Set 2.4: Equations of Lines
Write an equation in slope-intercept form for each of
the following lines.
y
1.

x



For each of the following equations,
(a) Write the equation in slope-intercept form.
(b) Identify the slope and the y-intercept of the
line.
(c) Graph the line.
5.
2x  y  5
6.
y  4x  0
7.
5x  y  1
8.
3x  y  6
9.
x  4y  0




10. x  3 y  9
y
2.
11. 5x  4 y  12

x


12. 2x  5 y  10

13. 5 y  2x  30  0

14. 3x  2 y  8  0

15.
3.

5
4
x  12 y  1
y
16.  23 x  12 y  1
x



Each set of conditions below describes the properties
of a particular line. Using these conditions,
(a) Graph the line.
(b) Write an equation for the line in point-slope
form.
(c) Write an equation for the line in slopeintercept form. (Do this algebraically, and
then check to see if your result matches your
graph.)


4.

y
17. Slope

x





160

2
; passes through  6, 4 
3
18. Slope 
5
; passes through  4,  3
2
19. Passes through
 8, 2
20. Passes through
 4, 7 
and  4,  7 
and  1,  3
University of Houston Department of Mathematics
Exercise Set 2.4: Equations of Lines
Write an equation in slope-intercept form for the line
that satisfies the given conditions.
4
21. Slope  ; y-intercept 3
7
22. Slope  4 ; y-intercept 5
23. Slope
4
; passes through  5,  3
5
3
24. Slope  ; passes through 12,  5
4
25. Slope 
26. Slope
2
; passes through  3, 2
9
1
; passes through  4,  2 
5
27. Passes through  10,  2 and  5, 7 
28. Passes through  6, 1 and  9, 4 
29. Passes through  4, 5 and 1,  2 
30. Passes through
 7, 0
and  3,  5
31. x-intercept 7 ; y-intercept 5
32. x-intercept 2 ; y-intercept 6
33. Slope 
34. Slope
3
; x-intercept 4
2
1
; x-intercept 6
5
Answer the following, assuming that each situation
can be modeled by a linear equation.
35. If a company can make 21 computers for
$23,000, and can make 40 computers for
$38,200, write an equation that represents the
cost C of x computers.
36. A certain electrician charges a $40 traveling fee,
and then charges $55 per hour of labor. Write an
equation that represents the cost C of a job that
takes x hours.
MATH 1300 Fundamentals of Mathematics
161
CHAPTER 2 Points, Lines, and Functions
Section 2.5:
Parallel and Perpendicular Lines
 Pairs of Lines – Parallel and Perpendicular Lines
Pairs of Lines - Parallel and Perpendicular Lines
Parallel Lines:
Perpendicular Lines:
Two lines with slopes m1 and m2 perpendicular if and only if m1m2  1 .
162
University of Houston Department of Mathematics
SECTION 2.5 Parallel and Perpendicular Lines
Example:
Solution:
MATH 1300 Fundamentals of Mathematics
163
CHAPTER 2 Points, Lines, and Functions
Example:
Solution:
164
University of Houston Department of Mathematics
SECTION 2.5 Parallel and Perpendicular Lines
Additional Example 1:
Solution:
Additional Example 2:
Solution:
MATH 1300 Fundamentals of Mathematics
165
CHAPTER 2 Points, Lines, and Functions
Additional Example 3:
Solution:
Additional Example 4:
Solution:
166
University of Houston Department of Mathematics
SECTION 2.5 Parallel and Perpendicular Lines
MATH 1300 Fundamentals of Mathematics
167
Exercise Set 2.5: Parallel and Perpendicular Lines
State whether the following pairs of lines are parallel,
perpendicular, or neither.
1.
y  3x  5
y  3x  7
2.
y   52 x  1
3.
y  73 x  5
18. x  5
x  5
y  x7
2
3
5.
y  2x  5
y  2x  5
6.
y  5x  7
y   15 x  3
8.
9.
16. x  3
y 3
17. y  2
x0
y  32 x  5
7.
y   14
y   52 x  3
y   73 x  4
4.
15. y  4
2x  5 y  7
5x  2 y  6
3x  4 y  8
3x  4 y  8
2x  3 y  5
4x  6 y  11
10. x  y  5  0
x y  2
11. The line passing through (2, 5) and (7, 9)
The line passing through (2,  6) and (2,  1)
12. The line passing through (4, 7) and (0, 5)
The line passing through (3, 8) and (5, 9)
13. The line passing through (6, 0) and (4, 10)
The line passing through (3,  7) and (7,  11)
14. The line passing through (1,  7) and (2, 5)
The line passing through (6, 6) and (2, 5)
19. The line passing through (4, 5) and (1, 5)
The line passing through (2,  3) and (0,  3)
20. The line passing through (2, 6) and (2, 8)
The line passing through (3,  4) and (5,  4)
Each set of conditions below describes a particular
line. Using these conditions, write an equation for each
line in the following two forms:
(a) Point-slope form
(b) Slope-intercept form
21. Passes through (4, 7) ; parallel to the line
y  2x  5
22. Passes through (4, 7) ; perpendicular to the line
y  2x  5
23. Passes through (12, 5) ; perpendicular to the
line y  6x  1
24. Passes through (12, 5) ; parallel to the line
y  6x  1
25. Passes through (3,  7) ; parallel to the line
y   54 x  2
26. Passes through (3,  7) ; perpendicular to the
line y   54 x  2
27. Passes through (1, 6) ; perpendicular to the line
2x  3 y  7
28. Passes through (1, 6) ; parallel to the line
2x  3 y  7
168
University of Houston Department of Mathematics
Exercise Set 2.5: Parallel and Perpendicular Lines
Write an equation for the line that satisfies the given
conditions. With the exception of vertical lines, write
all equations in slope-intercept form.
29. Passes through (1, 4) ; parallel to the x-axis
30. Passes through (1, 4) ; parallel to the y-axis
31. Passes through (2,  6) ; parallel to the line
x4
32. Passes through (2,  6) ; parallel to the line
y4
33. Passes through (2, 3) ; and is
(a) parallel to the line y  23 x  5
(b) perpendicular to the line y  23 x  5
34. Passes through (20,  2) ; and is
(a) parallel to the line y  5x  3
(b) perpendicular to the line y  5x  3
35. Passes through (2, 3) ; parallel to the line
5x  2 y  6
36. Passes through (1, 5) ; parallel to the line
4x  3 y  8
37. Passes through (2, 3) ; perpendicular to the line
5x  2 y  6
38. Passes through (1, 5) ; perpendicular to the
line 4x  3 y  8
39. Passes through (4,  6) ; parallel to the line
containing (3,  5) and (2, 1)
40. Passes through (8, 3) ; parallel to the line
containing (2,  3) and (4, 6)
41. Perpendicular to the line containing (3, 5) and
(7,  1) ; passes through the midpoint of the line
segment connecting these points
42. Perpendicular to the line containing (4,  2) and
(10, 4) ; passes through the midpoint of the line
segment connecting these points
MATH 1300 Fundamentals of Mathematics
169
CHAPTER 2 Points, Lines, and Functions
Section 2.6:
An Introduction to Functions
 Definition of a Function
 Domain of a Function
Definition of a Function
Definition:
170
University of Houston Department of Mathematics
SECTION 2.6 An Introduction to Functions
Defining a Function by an Equation in the Variables x and y:
The Function Notation:
Example:
Solution:
MATH 1300 Fundamentals of Mathematics
171
CHAPTER 2 Points, Lines, and Functions
Example:
Solution:
172
University of Houston Department of Mathematics
SECTION 2.6 An Introduction to Functions
Additional Example 1:
Solution:
Additional Example 2:
Solution:
MATH 1300 Fundamentals of Mathematics
173
CHAPTER 2 Points, Lines, and Functions
Additional Example 3:
Solution:
Additional Example 4:
174
University of Houston Department of Mathematics
SECTION 2.6 An Introduction to Functions
Solution:
MATH 1300 Fundamentals of Mathematics
175
CHAPTER 2 Points, Lines, and Functions
Additional Example 5:
Solution:
176
University of Houston Department of Mathematics
SECTION 2.6 An Introduction to Functions
Domain of a Function
Finding the Domain of a Function:
Example:
Solution:
Example:
Solution:
MATH 1300 Fundamentals of Mathematics
177
CHAPTER 2 Points, Lines, and Functions
Additional Example 1:
Solution:
178
University of Houston Department of Mathematics
SECTION 2.6 An Introduction to Functions
Additional Example 2:
Solution:
Additional Example 3:
Solution:
MATH 1300 Fundamentals of Mathematics
179
CHAPTER 2 Points, Lines, and Functions
180
University of Houston Department of Mathematics
Exercise Set 2.6: An Introduction to Functions
For each of the examples below, determine whether
the mapping makes sense within the context of the
given situation, and then state whether or not the
mapping represents a function.
1.
Erik conducts a science experiment and maps the
temperature outside his kitchen window at
various times during the morning.
Express each of the following rules in function
notation. (For example, “Subtract 3, then square”
would be written as f ( x )  ( x  3)2 .)
7.
(a) Divide by 7, then add 4
(b) Add 4, then divide by 7
8.
(a) Multiply by 2, then square
(b) Square, then multiply by 2
9.
(a) Take the square root, then subtract 6 squared
(b) Take the square root, subtract 6, then square
57
9
62
10
65
Temp. (oF)
Time
2.
Dr. Kim counts the number of people in
attendance at various times during his lecture this
afternoon.
1
85
2
10. (a) Add 4, square, then subtract 2
(b) Subtract 2, square, then add 4
Complete the table for each of the following functions.
11. f ( x)  x3  5
f ( x)
2
87
3
Time
x
1
# of People
0
1
State whether or not each of the following mappings
represents a function.
2
3.
7
9
-3
A
0
5
4
12. g ( x)  ( x  4)2  1
f ( x)
3
B
1
4.
-6
8
4
-7
A
B
9
5.
1
4
6
9
-2
-6
1
A
Find the domain of each of the following functions.
Write the domain first as an inequality, and then
express it in interval notation.
B
13. f ( x) 
6.
x
1
x
0
8
2
4
A
14. f ( x )  
4
x
B
MATH 1300 Fundamentals of Mathematics
181
Exercise Set 2.6: An Introduction to Functions
5
x 3
15. f ( x) 
16. f ( x) 
7
x8
17. f ( x) 
x6
x4
18. h( x) 
x4
x6
8
19. f (t ) 
2t  5
20. h(t ) 
2
3t  4
35. h( x)  2 x  9
36. h(t )  3t  2
37. g ( x)  1  5x
38. f ( x)  4  x
39. f ( x)  8  5  2 x
40. f ( x)  2  7 x  4
41. H ( x) 
x2
x6
3 x
x
21. g ( x) 
4x 1
4x  9
42. G( x) 
22. f ( x ) 
5x  7
3x  7
43. f (t )  3 t  1
23. g ( x) 
x 1
x2  9
x2
24. h( x)  2
x  25
25. f ( x)  x 2  2 x  24
26. f ( x)  7  2 x
27. g ( x)  3x  5
28. h( x)  x2  16
29. f (t )  t
30. h( x)  3 x
31. f ( x)  x  5
32. g ( x)  x  7
33. f ( x)  3 x  5
182
34. g ( x)  3 x  7
44. g ( x)  3 2x  9
45. h(t ) 
3
46. f ( x) 
t 1
t 5
3
2x  9
4x  7
47. h( x)  5 x
48. h( x)  4 x
49. g ( x)  6 3x  5
50. g ( x)  5 2 x  7
51. f ( x)  x
52. g ( x)  x  2
53. H ( x)  2x  6
54. f ( x)  3x  5
University of Houston Department of Mathematics
Exercise Set 2.6: An Introduction to Functions
55. f ( x) 
56. f ( x ) 
2
x7
62. If g ( x)  x  7 ,
(a) Find g (0)
(b) Find x when g ( x)  0
5
x
(c) Find g  2 
(d) Find x when g ( x)  2
57. f ( x) 
x 3
x4
58. f ( x) 
x9
x 1
(e) Find g  3
(f) Find x when g ( x)  3
63. If h( x)  x  2 , find
(a) h(7)
(b) h(25)
(c) h
Evaluate the following.
59. If f ( x)  5 x  4 ,
64. If h( x)  x  2 , find
(a) Find f (3)
(b) Find x when f ( x)  3
 
(c) Find f  12
(a) h(7)
(b) h(25)
(c) h
(d) Find x when f ( x)   12
(e) Find f  0
60. If f ( x)  3x  1 ,
 34 
(e) Find f  0
(a)
f (16)
(b) f (12)
(c)
(a) Find f (5)
(b) Find x when f ( x)  5
(d) Find x when f ( x) 
 14 
65. If f ( x)  x  3 , find
(f) Find x when f ( x)  0
(c) Find f
 14 
f 9
66. If f ( x)  x  3 , find
(a)
3
4
(f) Find x when f ( x)  0
61. If h( x)  x  3 ,
f (16)
(b) f (12)
(c)
f 9
67. If g ( x)  x2  5x  6 ,
(a) Find g (3)
(a) Find h(1)
(b) Find g  4 
(b) Find x when h( x)  1
(c) Find g  12
(c) Find h  2
(d) Find x when h( x)  2
(e) Find h  7 
(f) Find x when h( x)  7
 
(d) Find g  0 
68. If h(t )  t 2  2t  15 ,
(a) Find h(0)
(b) Find h(6)
(c) Find h  5
 
(d) Find h  32
MATH 1300 Fundamentals of Mathematics
183
Exercise Set 2.6: An Introduction to Functions
69. If f ( x) 
2 x
,
x 3
(a) Find f (7)
(b) Find f (0)
(c) Find f  5
(d) Find f  3
(e) Find f  2
70. If g ( x) 
5  2x
,
x4
(a) Find g (2)
(b) Find g (4)
(c) Find g
 52 
(d) Find g  3
(e) Find g (0)
184
University of Houston Department of Mathematics
SECTION 2.7 Functions and Graphs
Section 2.7:
Functions and Graphs
 Graphing a Function
Graphing a Function
The Graph of a Function:
The Vertical Line Test:
MATH 1300 Fundamentals of Mathematics
185
CHAPTER 2 Points, Lines, and Functions
Example:
Solution:
\
186
University of Houston Department of Mathematics
SECTION 2.7 Functions and Graphs
Example:
Solution:
MATH 1300 Fundamentals of Mathematics
187
CHAPTER 2 Points, Lines, and Functions
Example:
Solution:
188
University of Houston Department of Mathematics
SECTION 2.7 Functions and Graphs
Additional Example 1:
Solution:
MATH 1300 Fundamentals of Mathematics
189
CHAPTER 2 Points, Lines, and Functions
Additional Example 2:
The graph of y  f  x  is shown below.
(a) Find the domain of f.
(b) Find the range of f.
(c) Find the following function values: f  3 ; f  1 ; f  0 ; f 1 .
(d) For what value(s) of x is f  x   2 ?
Solution:
Part (a):
190
University of Houston Department of Mathematics
SECTION 2.7 Functions and Graphs
Part (b):
Part (c):
MATH 1300 Fundamentals of Mathematics
191
CHAPTER 2 Points, Lines, and Functions
Part (d):
Additional Example 3:
Solution:
192
University of Houston Department of Mathematics
SECTION 2.7 Functions and Graphs
Additional Example 4:
Solution:
MATH 1300 Fundamentals of Mathematics
193
CHAPTER 2 Points, Lines, and Functions
194
University of Houston Department of Mathematics
SECTION 2.7 Functions and Graphs
Additional Example 5:
Solution:
MATH 1300 Fundamentals of Mathematics
195
CHAPTER 2 Points, Lines, and Functions
Additional Example 6:
Solution:
196
University of Houston Department of Mathematics
SECTION 2.7 Functions and Graphs
MATH 1300 Fundamentals of Mathematics
197
CHAPTER 2 Points, Lines, and Functions
Additional Example 7:
Solution:
198
University of Houston Department of Mathematics
SECTION 2.7 Functions and Graphs
MATH 1300 Fundamentals of Mathematics
199
Exercise Set 2.7: Functions and Graphs
Determine whether or not each of the following graphs
represents a function.
9.
y
x
1.
y
x
y
10.
2.
y
x
x
3.
y
x
4.
y
x
5.
For each set of points,
(a) Graph the set of points.
(b) Determine whether or not the set of points
represents a function. Justify your answer.
11.
(1, 5), (2, 4), (3, 4), (2, 1), (3, 6)
12.
(3, 2), (1, 2), (0,  3), (2, 1), (2, 1)
13.
(2, 0), (4, 1), (6, 0), (3, 1), (5, 2)
14.
(1,  4), (2, 3), (4, 1), (4, 2), (2,  3)
y
x
Answer the following.
6.
y
x
7.
y
x
15. Analyze the coordinates in each of the sets
above. Describe a method of determining
whether or not the set of points represents a
function without graphing the points.
16. Determine whether or not each set of points
represents a function without graphing the
points. Justify each answer.
(a) (7, 3), (3,  7), (1, 5), (5, 1), (2, 1)
(b)
8.
(c)
y
(d)
(6, 3), (4, 3), (2, 3), (3, 3), (5, 3)
(3, 6), (3,  4), (3, 2), (3,  3), (3, 5)
(2,  5), (5, 2), (2, 5), (5,  2), (5, 2)
x
200
University of Houston Department of Mathematics
Exercise Set 2.7: Functions and Graphs
Answer the following.
17. The graph of y  f (x) is shown below.
19. The graph of y  g (x) is shown below.
(a) Find the domain of the function. Write your
answer in interval notation.
(a) Find the domain of the function. Write your
answer in interval notation.
(b) Find the range of the function. Write your
answer in interval notation.
(b) Find the range of the function. Write your
answer in interval notation.
(c) Find the following function values:
(c) Find the following function values:
g (2); g (0); g (2); g (4); g (6)
f (2); f (0); f (4); f (6)
(d) For what value(s) of x is f ( x)  9 ?
 y
(d) Which is greater, g (2) or g (3) ?
y
f


g




x


x














18. The graph of y  g (x) is shown below.
20. The graph of y  f (x) is shown below.
(a) Find the domain of the function. Write your
answer in interval notation.
(a) Find the domain of the function. Write your
answer in interval notation.
(b) Find the range of the function. Write your
answer in interval notation.
(b) Find the range of the function. Write your
answer in interval notation.
(c) Find the following function values:
g (2); g (0); g (1); g (3); g (6)
(c) Find the following function values:
(d) For what value(s) of x is g ( x)  2 ?
(d) Which is smaller, f (0) or f (3) ?
f (3); f (2); f (1); f (1); f (4)
y
y

f
g





x





MATH 1300 Fundamentals of Mathematics
x





201
Exercise Set 2.7: Functions and Graphs
For each of the following functions:
(a) State the domain of the function. Write your
answer in interval notation.
(b) Choose x-values corresponding to the domain
of the function, calculate the corresponding yvalues, plot the points, and draw the graph of
the function.
38.
x  3y  4
39. 2 y  5 x  7  0
40. 3x  4 y  8  0
21. f ( x)   32 x  6
22. f ( x)  23 x  4
23. h( x)  3x  5, 1  x  3
24. h( x)  2x,  3  x  2
25. g ( x)  x  3
26. g ( x)  x  4
27. f ( x)  x  3
28. f ( x)  5  x
29. F ( x)  x 2  4 x
30. G ( x)  ( x  3)2  1
For each of the following equations,
(a) Solve for y.
(b) Determine whether the equation defines y as a
function of x. (Do not graph.)
31. 3 y  5x  8
32. 2 x  9  6 y  2
33. 2 y  3x2  7
34. y 2  1  5x
35. x  3  y 2
36. x 2  y  3
37.
202
y 2 x
University of Houston Department of Mathematics