1
+1
10. g(x) = __
-0.5(x - 3)
VISUAL CUES
After horizontal and vertical asymptotes of a graph
have been identified, visual learners may benefit from
drawing horizontal and vertical arrows to find the
reference points.
y
4
The transformations are:
•a horizontal stretch by a factor of 2
2
x
•a reflection across the y-axis
0
•a translation of 3 units to the right and 1 unit up
-2
Domain of g(x):
Range of g(x):
Set notation: {x∣x ≠ 3}
Set notation: {y∣y ≠ 1}
Inequality: x < 3 or x > 3
Interval notation: (-∞, 3) ⋃ (3, +∞)
(
2
4
6
2
4
Inequality: y < 1 or y > 1
Interval notation: (-∞, 1) ⋃ (1, +∞)
)
y
2
1
11. g(x) = -0.5 _
-2
x-1
x
The transformations are:
1
•a vertical compression by a factor of _
2
-2
•a reflection across the x-axis
-2
-4
•a translation of 1 unit to the right and 2 units down
-6
Domain of g(x):
Range of g(x):
Inequality: x < 1 or x > 1
Inequality: y < -2 or y > -2
Set notation: {x∣x ≠ 1}
Set notation: {y∣y ≠ -2}
Interval notation: (-∞, 1) ⋃ (1, +∞)
0
Interval notation: (-∞, -2) ⋃ (-2, +∞)
1
+3
12. g(x) = _
2(x + 2)
y
6
© Houghton Mifflin Harcourt Publishing Company
4
The transformations are:
1
•a horizontal compression by a factor of _
2
•a translation of 2 units to the left and 3 units up
Range of g(x):
Inequality: x < -2 or x > -2
Inequality: y < 3 or y > 3
Set notation: {x∣x ≠ -2}
Set notation: {y∣y ≠ 3}
Interval notation: (-∞, -2) ⋃ (-2, +∞)
Exercise
A2_MTXESE353947_U4M09L2.indd 485
24–25
Lesson 9.2
-6
Domain of g(x):
Module 9
485
2
-4
Interval notation: (-∞, 3) ⋃ (3, +∞)
Lesson 2
485
Depth of Knowledge (D.O.K.)
3 Strategic Thinking
2x
0
2/21/14 1:06 AM
Mathematical Processes
1.G Explain and justify arguments
1
1
Rewrite the function in the form g(x) = a ______
+ k or g(x) = ______
__1 (x - h) + k and graph it.
(x - h)
b
Also state the domain and range using inequalities, set notation, and interval notation.
13.
3x - 5
g(x) = _
y
x-1
3
x - 1 ⟌ 3x - 5
–––––––
Horiz. Asymp: y = 3
1
g(x) = -2(____
)+3
x- 1
Domain of g(x):
Inequality: x < 1 or x > 1
Set notation: {x∣x ≠ 1}
Interval notation: (-∞, 1) ⋃ (1, +∞)
x+5
_
Relate the term reference point to the noun reference,
meaning a source of information. A reference on a job
application is person who serves as a source for
information about the applicant.
6
Vert. Asymp: x = 1
3x - 3
―――
-2
14. g(x) =
CONNECT VOCABULARY
4
(–1, –1) becomes (0, 5)
(1, 1) becomes (2, 1)
2
x
0
-2
Range of g(x):
2
4
Inequality: y < 3 or y > 3
Set notation: {y∣y ≠ 3}
Interval notation: (-∞, 3) ⋃ (3, +∞)
––––––
4
x+4
――
1
1
+ 2.
g(x) = _______
0.5(x + 4)
(–1, –1) becomes (–6, 1)
(1, 1) becomes (–2, 3)
Domain of g(x):
Inequality: x < -4 or x > -4
Set notation: {x∣x ≠ -4}
y
6
0.5x + 2
2
0.5x + 2 ⟌ x + 5
2
x
Vert. Asymp: x = –4
-8
-6
-4
-2
Horiz. Asymp: y = 2
0
-2
Range of g(x):
Inequality: y < 2 or y > 2
Set notation: {y∣y ≠ 2}
Interval notation: (-∞, -4) ⋃ (-4, +∞) Interval notation: (-∞, 2) ⋃ (2, +∞)
-4x + 11
_
––––––––––
-2
-4x + 8
――――
3
(
)
1
-4
g(x) = 3 ____
x-2
(–1, –1) becomes (1, –7)
(1, 1) becomes (3, –1)
Domain of g(x):
Inequality: x < 2 or x > 2
Set notation: {x∣x ≠ 2}
Interval notation: (-∞, 2) ⋃ (2, +∞)
Module 9
A2_MTXESE353947_U4M09L2 486
x
y
x-2
-4
x - 2 ⟌ -4x + 11
0
2
4
6
-2
-4
Vert. Asymp: x = 2
Horiz. Asymp: y = –4
-6
-8
© Houghton Mifflin Harcourt Publishing Company
15. g(x) =
Range of g(x):
Inequality: y < -4 or y > -4
Set notation: {y∣y ≠ -4}
Interval notation: (-∞, -4) ⋃ (-4, +∞)
486
Lesson 2
23/02/14 2:56 PM
Graphing Simple Rational Functions
486
4x + 13
16. g(x) = _
-2x - 6
-2
-2x - 6 ⟌ 4x + 13
AVOID COMMON ERRORS
––––––––
For real-world problems, students may not check to
see whether the domain is continuous or discrete,
and if it is discrete, how a solution should be
rounded. Stress the importance of checking whether
solutions are defined and whether it is appropriate to
round up or down.
y
2
x
-6
4x + 12
―――
1
-4
-2
1
-2
g(x) = _______
-2(x + 3)
Vert. Asymp: x = -3
0
-2
-4
-6
Horiz. Asymp: y = –2
–5
, –3)
(–1, –1) becomes (__
2
–7
, –1)
(1, 1) becomes (__
2
Domain of g(x):
Range of g(x):
Set notation: {x∣x ≠ -3}
Set notation: {y∣y ≠ -2}
Inequality: x < -3 or x > -3
Interval notation: (-∞, -3) ⋃ (-3, +∞)
Inequality: y < -2 or y > -2
Interval notation: (-∞, -2) ⋃ (-2, +w∞)
(
)
1
17. Write the function whose graph is shown. Use the form g(x) = a ____
+ k.
x-h
h=3
8
k=4
Use the point (4, 7):
7= a
y
(4, 7)
6
1
+4
(_
4 - 3)
4
2
3=a
1
g(x) = 3
+4
x-3
(_)
(2, 1)
0
x
2
4
6
1
+k
18. Write the function whose graph is shown. Use the form g(x) = ______
1
_
© Houghton Mifflin Harcourt Publishing Company
b
Use the point (-2, 1):
1
1=
+2
1
(-2 + 4)
b
b = -2
1
g(x) =
+2
1
- (x + 4)
2
(-2, 1)
__
_
-8
-6
y
4
(-6, 3)
k=2
-4
-2
2
x
0
-2
__
_
A2_MTXESE353947_U4M09L2.indd 487
Lesson 9.2
6
h = -4
Module 9
487
(x - h)
487
Lesson 2
2/21/14 1:06 AM
1
19. Write the function whose graph is shown. Use the form g(x) = ______
+ k.
1
_
b
y
(x - h)
h=3
(3.5, 0)
0
k = -1
2
-2
Use the point (3.5, 0):
1
0=
-1
1(
3.5 - 3)
b
1
b=
2
1
g(x) =
-1
2(x - 3)
__
_
_
_
4
x
6
Students may lightly connect the reference points to
see that they are symmetric about (h, k).
(2.5, -2)
-4
(
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Math Connections
2
)
1
+k
20. Write the function whose graph is shown. Use the form g(x) = a ____
x-h
h = -2
x
y
-6 -4
(-3, -2)
k = -4
Use the point (-1, -6):
-6 = a
1
-4
(_
-1 + 2 )
-2 = a
g(x) = -2
0
2
-2
(-1, -6)
-8
1
)-4
(_
x+2
_
100
C(s) = s + 15
The graphs appear to intersect between s = 13
100
and s = 14. Since C(13) =
+ 15 ≈ 22.69 and
13
100
C(14) =
+ 15 ≈ 22.14, the minimum number
14
of suncatchers that brings the average cost below
_
_
$22.50 is 14.
Module 9
A2_MTXESE353947_U4M09L2.indd 488
Average cost (dollars)
Let s be the number of suncatchers that Maria
makes. Let C(s) be the average cost (in dollars) of a
suncatcher when the cost of the kit is included.
40
35
30
25
20
15
10
5
0
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Bernie
Pearson/Alamy
21. Maria has purchased a basic stained glass kit for $100. She plans to make
stained glass suncatchers and sell them. She estimates that the materials for
making each suncatcher will cost $15. Model this situation with a rational
function that gives the average cost of a stained glass suncatcher when the
cost of the kit is included in the calculation. Use the graph of the function
to determine the minimum number of suncatchers that brings the average
cost below $22.50.
C
s
10
20
30
40
Number of suncatchers
488
Lesson 2
2/21/14 1:06 AM
Graphing Simple Rational Functions
488