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Answers
Teacher Copy
Plan
Pacing: 1 class period
Chunking the Lesson
Example A #1 Example B
Example C #2
Check Your Understanding
Lesson Practice
Teach
Bell-Ringer Activity
Students should recall that an absolute value of a number is its distance from zero on a number line.
Have students evaluate the following:
1. |6| [6]
2. |–6| [6]
Then have students solve the following equation.
3. |x|= 6 [x = 6 or x = –6]
Example A Marking the Text, Interactive Word Wall
Point out the Math Tip to reinforce why two solutions exist. Work through the solutions to the equation algebraically. Remind students that
solutions to an equation make the equation a true statement. This mathematical understanding is necessary for students to be able to check
their results.
© 2014 College Board. All rights reserved.
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Developing Math Language
An absolute value equation is an equation involving an absolute value of an expression containing a variable. Just like when solving
algebraic equations without absolute value bars, the goal is to isolate the variable. In this case, isolate the absolute value bars because they
contain the variable. It should be emphasized that when solving absolute value equations, students must think of two cases, as there are two
numbers that have a specific distance from zero on a number line.
1 Identify a Subtask, Quickwrite
When solving absolute value equations, students may not see the purpose in creating two equations. Reviewing the definition of the absolute
value function as a piecewise-defined function with two rules may enable students to see the reason why two equations are necessary.
Have students look back at Try These A, parts c and d. Have volunteers construct a graph of the two piecewise-defined functions used to
write each equation and then discuss how the solution set is represented by the graph.
Example B Marking the Text, Simplify the Problem, Critique Reasoning, Group Presentation
Start with emphasizing the word vary in the Example, discussing what it means when something varies. You may wish to present a simpler
example such as: The average cost of a pound of coffee is $8. However, the cost sometimes varies by $1. This means that the coffee could
cost as little as $7 per pound or as much as $9 per pound. Now have students work in small groups to examine and solve Example B by
implementing an absolute value equation. Additionally, ask them to take the problem a step further and graph its solution on a number line.
Have groups present their findings to the class.
ELL Support
For those students for whom English is a second language, explain that the word varies in mathematics means changes. There are different
ways of thinking about how values can vary. Values can vary upward or downward, less than or greater than, in a positive direction or a
negative direction, and so on. However, the importance comes in realizing that there are two different directions, regardless of how you think
of it.
Also address the word extremes as it pertains to mathematics. An extreme value is a maximum value if it is the largest possible amount
(greatest value), and an extreme value is a minimum value if it is the smallest possible amount (least value).
Developing Math Language
An absolute value inequality is basically the same as an absolute value equation, except that the equal sign is now an inequality symbol: <,
>, ≤, ≥, or ≠. It still involves an absolute value expression that contains a variable, just like before. Use graphs on a number line of the
solutions of simple equations and inequalities and absolute value equations and inequalities to show how these are all related.
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Example C Simplify the Problem, Debriefing
Before addressing Example C, discuss the following: Inequalities with |A| > b, where b is a positive number, are known as disjunctions
and are written as A < –b or A > b.
For example, |x| > 5 means the value of the variable x is more than 5 units away from the origin (zero) on a number line. The solution is x
< – 5 or x > 5.
See graph A.
This also holds true for |A| ≥ b.
Inequalities with |A| < b, where b is a positive number, are known as conjunctions and are written as –b < A < b, or as –b < A and A < b.
For example: |x| < 5; this means the value of the variable x is less than 5 units away from the origin (zero) on a number line. The solution
is –5 < x < 5.
See graph B.
This also holds true for |A| ≤ b.
Students can apply these generalizations to Example C. Point out that they should proceed to solve these just as they would an algebraic
equation, except in two parts, as shown above. After they have some time to work through parts a and b, discuss the solutions with the
whole class.
Teacher to Teacher
Another method for solving inequalities relies on the geometric definition of absolute value |x – a| as the distance from x to a. Here’s how
you can solve the inequality in the example:
|2x + 3 | + 1 > 6
|2x + 3| > 5
|
2 | x − −3
2
>
5
|x − −3
| >
2
5
2
Thus, the solution set is all values of x whose distance from − 32 is greater than 52 . The solution can be represented on a number line and
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written as x < –4 or x > 1.
2 Quickwrite, Self Revision/Peer Revision, Debriefing
Use the investigation regarding the restriction c > 0 as an opportunity to discuss the need to identify impossible situations involving
inequalities.
Check Your Understanding
Debrief students’ answers to these items to ensure that they understand concepts related to absolute value equations. Have groups of
students present their solutions to Item 4.
Assess
Students’ answers to Lesson Practice problems will provide you with a formative assessment of their understanding of the lesson
concepts and their ability to apply their learning.
See the Activity Practice for additional problems for this lesson. You may assign the problems here or use them as a culmination for the
activity.
Adapt
Check students’ answers to the Lesson Practice to ensure that they understand basic concepts related to writing and solving absolute
value equations and inequalities and graphing the solutions of absolute value equations and inequalities. If students are still having
difficulty, review the process of rewriting an absolute value equation or inequality as two equations or inequalities.
Plan
Pacing: 1 class period
Chunking the Lesson
#1–3 #4–5
#6–7 #8–9
Check Your Understanding
Lesson Practice
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Teach
Bell-Ringer Activity
Have students evaluate the piecewise function
−x,
if x < − 3
f (x) = 2x + 1, if − 3 ≤ x < 2
x + 3,
if x ≥ 2
for each value.
1. x = −4 [f(x) = 4]
2. x = −3 [f(x) = −5]
3. x = 0 [f(x) = 1]
4. x = 2 [f(x) = 5]
5. x = 5 [f(x) = 8]
1–3 Activating Prior Knowledge, Discussion Groups, Think-Pair-Share
Ask students to reflect back on the word constant from earlier algebra courses and define in their own words what this means and
how it applies to the step function definition. Because the pieces of a step function remain constant, each “step” is what type of line
segment? When students are finished discussing, ask a representative from one group or pair to share their responses with the class
and ask other students for feedback.
Developing Math Language
While it is a specific type of piecewise-defined function, a step function is easy to recognize because it looks like a series of
horizontal steps. Point out that a real staircase also has vertical pieces connecting the horizontal landings. This is not the case with a
step function because if there were any vertical pieces, it would not be a function.
4–5 Chunking the Activity, Predict and Confirm, Discussion Groups
Have students work with a partner. Before they use their calculators, ask students to predict what the function of y = int(x) will look
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like. Ask them to think about the places, or parts of the steps, that will be included (closed circle plot) and not included (an open
circle plot). Then have the students follow the steps given to them to graph this function. Be sure to point out the Technology Tip
in the margin for changing mode from “Connected” to “Dot,” in order to keep the calculator from connecting the breaks between
the steps and making it appear that this is not really a function at all. Have them compare their predictions to the results on the
calculator.
For additional technology resources, visit SpringBoard Digital.
Differentiating Instruction
Extend students’ learning by asking students to journal or write some situation in the real world that could represent a step
function. After giving them some time to think and write, ask for some of their examples. Responses will vary. Below are two
examples:
1. The labor cost of hiring an electrician to come to one′s home to do some work. The electrician charges a fee of $75 per hour
(or any fractional part of one hour) for labor. So, the fee for 0 ≤ h ≤ 1 = $75; the fee for 1 < h ≤ 2 = $150, and so on.
2. The cost to ship a parcel has a flat rate of $5, plus an additional cost of $0.20 per ounce up to, but not exceeding, the first 12
ounces. Then, when the weight exceeds 12 ounces, the cost per ounce increases to $0.30 per ounce.
6–7 Create Representations, Quickwrite, Think-Pair-Share
In graphing this piecewise-defined function, students are introduced to the absolute value function. It is likely that students will
have an informal understanding of absolute value, but it is imperative that they also understand absolute value as a piecewisedefined function for future studies in mathematics.
Differentiating Instruction
Students may encounter the function
f ( x) =
| x|
x
in this or future math classes. Use this opportunity to create the piecewise representation of
f (x) = − 1, if x < 0
f ( x) =
f (x) = 1
if x > 0
.
Then use the definition of the absolute value function to make the necessary transformation:
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| x|
x
for x < 0 and
=
|x|
x
−x
= −1
x
=
x
=1
x
for x > 0. Note that the function is undefined at x = 0.
8–9 Marking the Text, Interactive Word Wall
Evaluating absolute values should not be a new experience for students, but understanding this concept in terms of piecewisedefined functions is likely to be new. Item 9 addresses that issue.
Check Your Understanding
Debrief students’ answers to these items to ensure that they understand concepts related to step functions and absolute value
functions.
Assess
Students’ answers to Lesson Practice problems will provide you with a formative assessment of their understanding of the
lesson concepts and their ability to apply their learning.
See the Activity Practice for additional problems for this lesson. You may assign the problems here or use them as a culmination
for the activity.
Adapt
Check students’ answers to the Lesson Practice to ensure that they understand concepts related to step functions and
absolute-value functions. When graphing step functions, watch for students who use closed circles for all of the endpoints.
Remind these students that in order for these graphs to represent functions, they must pass the vertical line test − no vertical line
can intersect the graph in more than one point. Demonstrate that if all of the endpoints are closed, the graph will not pass this
test.
Plan
Pacing: 1 class period
Chunking the Lesson
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#1 #2–3 #4–5
#6 #7–8 #9–10
#11 Example A
Check Your Understanding
Lesson Practice
Teach
Bell-Ringer Activity
Students should be familiar with how to evaluate absolute value functions. Have students evaluate each function for the given
value:
1. f(x) = |x| for x = 9 [f(x) = 9]
2. f(x) = |x| for x = −9 [f(x) = 9]
3. f(x) = |x + 1| for x = 9 [f(x) = 10]
4. f(x) = |x + 1| for x = −9 [f(x) = 8]
Students should discuss the impact of the “+ 1” in Items 3 and 4.
1 Activating Prior Knowledge, Create Representations
With text-based reminders about parent functions and transformations, students are asked to identify transformations of the
parent absolute value function.
Transformations that students are expected to recognize from earlier math courses include the following: vertical translations,
vertical stretches/shrinks, and reflections over the x-axis.
Developing Math Language
A parent function is the most basic function of a particular type. This lesson refers to the parent absolute value function as
f(x) = |x|. Here are a few more examples of common parent functions:
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Linear: f(x) = x, or y = x
Quadratic: f(x) = x2, or y = x2
Cubic: f(x) = x3, or y = x3
Inverse:
1
1
f (x) = , or y =
x
x
Radical:
f (x) = √x, or y = √x
Each of these parent functions can be altered to change its basic position, size, and shape. When this occurs, a new
function, known as a transformation, is produced. The cause of a transformation involves some type of arithmetic
operation(s) to the parent function.
Mini-Lesson: Vertical Translations and Vertical Stretch/Shrink
If students need additional help graphing vertical translations of parent functions or vertical stretches or shrinks of a parent
function, a mini-lesson is available to provide practice.
See the Teacher Resources at SpringBoard Digital for a student page for this mini-lesson.
Differentiating Instruction
If students experience difficulties in describing transformations, or if they have limited experience with transforming parent
functions, then giving them additional opportunities to graph transformations may be appropriate. The practice exercises on
this page and the next may be helpful in reinforcing understanding.
2–3 Create Representations, Predict and Confirm
Horizontal translations are introduced in Item 2 and then expanded in Items 3–5.
4–5 Look for a Pattern
The counterintuitive nature of this translation, x − a moving to the right and x + a moving to the left, may be a surprise to
students. Items 4 and 5 provide the opportunity to reason abstractly to generalize the effects of x ± a.
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Teacher to Teacher
Here is one way of explaining horizontal translations given by x − a: Show students that the y-value for the graph of y =
f(x − a) at a is the same y-value that y = f(x) has at 0. When x = a, then y = f(x − a) = f (a − a) = f(0). When x = 0, then y
= f(x) = f(0). The y-values are equal.
Point out that those functions show a horizontal shift in the x-values from 0 to a. If a is positive, this shift is to the right a
units, and if a is negative, this shift is to the left a units. In general, the y-value for the graph of y = f(x − a) at b + a is the
same y-value that y = f(x) has at x = b. Again, point out that the shift is a units right if a > 0 and left if a < 0.
Mini-Lesson: Reflections over the x-axis
If students need additional help graphing reflections of parent functions over the x-axis, a mini-lesson is available to
provide practice.
See the Teacher Resources at SpringBoard Digital for a student page for this mini-lesson.
Technology Tip
To graph the parent absolute value function on a TI graphing calculator, follow these basic steps:
1. Press the [y=] key in the upper left corner.
2. Press the [CLEAR] key to clear any previous functions.
3. Press the [MATH] key.
4. Press the right arrow one time to highlight the NUM command.
5. Select option 1, “abs (” for absolute value, by pressing [ENTER].
6. Press the [X,T,Θ,n] key.
7. Press the [)] key to close parentheses.
8. Press the [GRAPH] key.
Note: Use this basic process with the transformations of the parent absolute value function.
For additional technology resources, visit SpringBoard Digital.
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6 Predict and Confirm, Create Representations, Chunking the Activity
Before students graph the functions given in Item 6, have them write down their predictions for the transformation
that will take place with the parent absolute value graph. Have them confirm their predictions by graphing. Before
moving on to the next items, sketch a couple of absolute value horizontal transformations, and ask students to write
the equations of the functions you graphed.
7–8 Discussion Groups, Predict and Confirm, Critique Reasoning, Think-Pair-Share
After presenting Items 7 and 8, have students work in small groups or with a partner to try to predict how the value of
k in f(x) = |kx| affects the graph of f(x) = |x|. In other words, what if g(x) = |3x| and
1
h( x ) = x ?
3
Have students collaborate and share their findings.
9–10 Close Reading, Marking the Text
Students probably will not have too much difficulty expressing regularity in repeated reasoning and coming up with a
generalization for Item 9, based upon the previous items. However, Item 10 is written in such a way that you may
want to emphasize some points. The functions in Items 9 and 10 are written the same, but students should look closely
at the values of k. In Item 10, k is really a fractional value. Because of the way this is written, the only difference
between the answers for Items 9 and 10 is the words shrink and stretch. The reason why the factors are both 1k is
because 1k , when k is a fraction, is the inverse of the fractional value.
11 Summarizing
Point out that the phrases “vertical stretch” and “horizontal shrink” can both be used to describe what is taking place
in Item 11a. However, the difference between these descriptions is the values of their factors.
Example A Activating Prior Knowledge
This concept of unraveling multiple transformations to a parent absolute function is similar to following orders of
operations with arithmetic operations.
Universal Access
For students having difficulty with the concept of absolute value, explain that an absolute value represents a distance,
or measure, of a number from the origin of a number line. Furthermore, a measurement cannot be a negative value.
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Keeping that in mind, when applying this concept to absolute value functions and their graphs, this is why they
remain above the x-axis unless written with a negative sign preceding the absolute value bars.
Check Your Understanding
Debrief students’ answers to these items to ensure that they understand concepts related to transformations of
functions.
Assess
Students’ answers to Lesson Practice problems will provide you with a formative assessment of their understanding
of the lesson concepts and their ability to apply their learning.
See the Activity Practice for additional problems for this lesson. You may assign the problems here or use them as a
culmination for the activity.
Adapt
Check students’ answers to the Lesson Practice to ensure that they understand transformations of the absolute value
parent function. If students have trouble graphing a function by using the values in the equation, allow them to
graph by generating ordered pairs.
Learning Targets
p. 65
Identify the effect on the graph of replacing f(x) by f(x) + k, k · f(x), f(kx), and f(x + k).
Find the value of k, given these graphs.
Activating Prior Knowledge (Learning Strategy)
Definition
Recalling what is known about a concept and using that information to make a connection to a new concept
Purpose
Helps students establish connections between what they already know and how that knowledge is related to
new learning
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Look for a Pattern (Learning Strategy)
Definition
Observing information or creating visual representations to find a trend
Purpose
Helps to identify patterns that may be used to make predictions
Debriefing (Learning Strategy)
Definition
Discussing the understanding of a concept to lead to consensus on its meaning
Purpose
Helps clarify misconceptions and deepen understanding of content
Think-Pair-Share (Learning Strategy)
Definition
Thinking through a problem alone, pairing with a partner to share ideas, and concluding by sharing results
with the class
Purpose
Enables the development of initial ideas that are then tested with a partner in preparation for revising ideas
and sharing them with a larger group
Identify a Subtask (Learning Strategy)
Definition
Breaking a problem into smaller pieces whose outcomes lead to a solution
Purpose
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Helps to organize the pieces of a complex problem and reach a complete solution
Suggested Learning Strategies
Activating Prior Knowledge, Create Representation, Look for a Pattern, Debriefing, Think-Pair-Share,
Identify a Subtask
p. 68p. 67p. 66
Math Tip
Transformations include:
vertical translations, which shift a graph up or down
horizontal translations, which shift a graph left or right
reflections, which produce a mirror image of a graph over a line
vertical stretches or vertical shrinks, which stretch a graph away from the x-axis or shrink a
graph toward the x-axis
horizontal stretches or horizontal shrinks, which stretch a graph away from the y-axis or
shrink a graph toward the y-axis
The absolute value function f(x) = |x| is the parent absolute value function. Recall that a parent function
is the most basic function of a particular type. Transformations may be performed on a parent function
to produce a new function.
1. Model with mathematics. For each function below, graph the function and identify the
transformation of f(x) = |x|.
a. g(x) = |x| + 1
vertical translation up 1 unit
b. h(x) = |x| − 2
vertical translation down 2 units
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c. k(x) = 3|x|
vertical stretch by a factor of 3
d. q(x) = −| x |
reflection over the x-axis
Technology Tip
Many graphing calculators use a function called “abs” to represent absolute value.
2. Use the coordinate grid at the right.
a. Graph the parent function f(x) = |x|.
b. Predict the transformation for g(x) = |x − 3| and h(x) = |−2x|.
Predictions may vary.
c. Graph the function g(x) = |x − 3| and h(x) = |−2x|.
d. What transformations do your graphs show?
g(x) is a horizontal translation of f(x) 3 units to the right.
h(x) is a vertical stretch of f(x) by a factor of 2.
3. Reason abstractly and quantitatively. Use the results from Item 2 to predict the
transformation of h(x) = |x + 2|. Then graph the function to confirm or revise your prediction.
Sample prediction: horizontal translation 2 units to the left
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The functions in Items 2 and 3 are examples of horizontal translations. A horizontal translation
occurs when the independent variable, x, is replaced with x + k or with x − k.
4. In the absolute value function f(x) = |x + k| with k > 0, describe how the graph of the function
changes, compared to the parent function.
The function moves k units to the left.
5. In the absolute value function f(x) = |x − k| with k > 0, describe how the graph of the function
changes, compared to the parent function.
The graph moves k units to the right.
6. Graph each function.
a. f(x) = |x − 4|
b. f(x) = |x + 5|
Math Tip
A horizontal stretch or shrink by a factor of k maps a point (x, y) on the graph of the original
function to the point (kx, y) on the graph of the transformed function.
Similarly, a vertical stretch or shrink by a factor of k maps a point (x, y) on the graph of the
original function to the point (x, ky) on the graph of the transformed function.
7. Use the coordinate grid at the right.
a. Graph the parent function f(x) = |x| and the function g(x) = |2x|.
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b. Describe the graph of g(x) as a horizontal stretch or horizontal shrink of the graph of
the parent function.
a horizontal shrink by a factor of
1
2
8. Express regularity in repeated reasoning. Use the results from Item 7 to predict how
the graph of h(x) = 12 x is transformed from the graph of the parent function. Then graph
h(x) to confirm or revise your prediction.
Sample prediction: a horizontal stretch by a factor of 2
9. In the absolute value function f(x) = |kx| with k > 1, describe how the graph of the function
changes compared to the graph of the parent function. What if k < −1?
The graph of f(x) is a horizontal shrink of the graph of the parent function by a factor of
k > 1 or when k < −1.
1
k
when
10. In the absolute value function f(x) = |kx| with 0 < k < 1, describe how the graph of the
function changes compared to the graph of the parent function. What if −1 < k < 0?
The graph of f(x) is a horizontal stretch of the graph of the parent function by a factor of
0 < k < 1 or when −1 < k < 0.
1
k
when
11. Each graph shows a transformation g(x) of the parent function f(x) = |x|. Describe the
transformation and write the equation of g(x).
a.
vertical stretch by a factor of 3 or horizontal shrink by a factor of 13 ; g(x) = 3|x| or g(x) =
|3x|
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b.
vertical translation down 3 units; g(x) = |x| − 3
Example A
p. 69
Describe the transformations of g(x) = 2|x + 3| from the parent absolute value function and use
them to graph g(x).
Math Tip
To graph an absolute value function of the form g(x) = a|b(x − c)| + d, apply the
transformations of f(x) = |x| in this order:
1. horizontal translation
2. reflection in the y-axis and/or horizontal shrink or stretch
3. reflection in the x-axis and/or vertical shrink or stretch
4. vertical translation
Step 1:
Describe the transformations.
g(x) is a horizontal translation of f(x) = |x| by 3 units to the left, followed by a vertical stretch
by a factor of 2.
Apply the horizontal translation first, and then apply the vertical stretch.
Step 2:
Apply the horizontal translation.
Graph f(x) = |x|. Then shift each point on the graph of f(x) by 3 units to the left. To do so,
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subtract 3 from the x-coordinates and keep the y-coordinates the same.
Name the new function h(x). Its equation is h(x) = |x + 3|.
Step 3:
Apply the vertical stretch.
Now stretch each point on the graph of h(x) vertically by a factor of 2. To do so, keep the
x-coordinates the same and multiply the y-coordinates by 2.
Technology Tip
You can check that you have graphed g(x) correctly by graphing it on a graphing
calculator.
Solution: The new function is g(x) = 2|x + 3|.
Try These A
For each absolute value function, describe the transformations represented in the rule and
use them to graph the function.
a. h(x) = −|x − 1| + 2
translate to the right 1 unit, reflect over the x-axis, and translate up 2 units
b. k(x) = 4|x + 1| − 3
translate 1 unit to the left, vertically stretch by a factor of 4, then translate 3 units down
p. 70
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Check Your Understanding
12. Graph the function g(x) = |−x|. What is the relationship between g(x) and f(x) = |x|?
Why does this relationship make sense?
g(x) and f(x) are the same function. This relationship makes sense because |x| = |−x|.
13. Compare and contrast a vertical stretch by a factor of 4 with a horizontal stretch
by a factor of 4.
Sample answer: Both transformations stretch points on the original graph away from an
axis. A vertical stretch maps a point (x, y) on the original graph to point (x, 4y) on the
transformed graph. A horizontal stretch maps a point (x, y) on the original graph to point
(4x, y) on the transformed graph.
14. Without graphing the function, determine the coordinates of the vertex of f(x) = |x
+ 2| − 5. Explain how you determined your answer.
(−2, −5); Sample explanation: The graph of f(x) is a translation of the graph of the
absolute value parent function by 2 units left and 5 units down. The vertex of the graph of
the absolute value parent function is (0, 0), so the vertex of the graph of f(x) must be (−2,
−5).
Lesson 4-3 Practice
15. The graph of g(x) is the graph of f(x) = |x| translated 6 units to the right. Write the
equation of g(x).
g(x) = |x − 6|
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16. Describe the graph of h(x) = −5|x| as one or more transformations of the graph
of f(x) = |x|.
a reflection over the x-axis and a vertical stretch by a factor of 5
17. What are the domain and range of f(x) = |x + 4| − 1? Explain.
Domain: {x | x ∈ ℝ}; range: {y | y ∈ ℝ, y ≥ −1}; Sample explanation: The function is
defined for all real values of x, so the domain is all real numbers. The graph of f(x)
opens upward, and its vertex is at (−4, −1). −1 is the minimum value of f(x), so the
range is all real numbers ≥ −1.
18. Graph each transformation of f(x) = |x|.
a. g(x) = |x − 4| + 2
b. g(x) = |2x| − 3
c. g(x) = −|x + 4| + 3
d. g(x) = −3|x + 2| + 4
19. Attend to precision. Write the equation for each transformation of f(x) = |x|
described below.
a. Translate left 9 units, stretch vertically by a factor of 5, and translate down
23 units.
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f(x) = 5|x + 9| − 23
b. Translate left 12 units, stretch horizontally by a factor of 4, and reflect over
the x-axis.
f ( x) = −
1
4
(x + 12)
c.
f(x) = 3|x − 2| − 4 or f(x) = |3(x − 2)| − 4
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