Lesson 3-4 Per. 2-5

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Transformations of Linear Functions

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Transformations of Linear Functions

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A family of people is a group that includes people related by birth,
marriage, or adoption, who often share characteristics. Similarly, a
family of graphs includes graphs and equations of graphs that have
at least one characteristic in common. The parent function is the
simplest of the functions in a family.

The family of linear functions includes all lines, with the parent
function f(x) = x, also called the identity function. Its graph contains
all points of the form (a, a). The domain and range of the identity
function are all real numbers. A transformation moves the graph on
the coordinate plane, which can create new linear functions.

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Translations of Linear Functions

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Transformations of Linear Functions

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One type of transformation is a translation. A translation is a
transformation in which a figure is slid from one position to
another without being turned. A linear function can be slid up,
down, left, right, or in two directions.

When a constant k is added to a linear function f(x), the result is
a vertical translation. The y-intercept of f(x) is translated up or
down.

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Translations of Linear Functions

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Translations of Linear Functions

If k > 0, the graph of f(x) is
translated k units up.

Every point on the graph of
f(x) moves k units up.

If k < 0, the graph of f(x) is
translated |k| units down.

Every point on the graph of f(x)
moves |k| units up.

Key Concept: Vertical Translations of Linear Functions

g(x) = x + k

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When a constant h is subtracted from the x-value before the
function f(x) is performed, the result is a horizontal translation.
The x-intercept of f(x) is translated right or left.

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Translations of Linear Functions

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Translations of Linear Functions

If h > 0, the graph of f(x) is
translated h units right.

Every point on the graph of
f(x) moves h units right.

If h < 0, the graph of f(x) is
translated |h| units left.

Every point on the graph of f(x)
moves |h| units left.

Key Concept: Horizontal Translations of Linear Functions

g(x) = (x – h)

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Translations of Linear Functions

Translation

Change to Parent Function

f(x) + k, k > 0

Translates the graph k units up.

f(x) + k, k < 0

Translates the graph |k| units down.

f(x – h), h > 0

Translates the graph h units right.

f(x – h), h > 0

Translates the graph |h| units left.

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Example 1 Watch as I do it
Vertical and Horizontal Translations
of Linear Functions

Describe the translation in each function as it
relates to the graph of the parent function.

a. g(x) = x − 2

b. g(x) = (x + 5)

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Example 1
Vertical and Horizontal Translations
of Linear Functions

a. g(x) = x − 2

Graph the parent graph for
linear functions.

Because f(x) = x, g(x) = f(x) + k
where k = −2.

g(x) = x − 2 → f(x) + (−2)

x

f(x)

f(x) − 2

(x, g(x))

−2

−2

−4

(−2, −4)

0

0

−2

(0, −2)

1

1

−1

(1, −1)

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Example 1
Vertical and Horizontal Translations
of Linear Functions

The constant k is not grouped with x, so
k affects the output, or y-values. The
value of k is less than 0, so the graph of
f(x) = x is translated |−2| units down, or
2 units down.

g(x) = x − 2 is the translation of the
graph of the parent function 2 units
down.

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Example 1
Vertical and Horizontal Translations
of Linear Functions

b. g(x) = (x + 5)

Graph the parent graph for
linear functions.

Because f(x) = x, g(x) = f(x − h)
where h = −5.

x

x + 5

f(x + 5)

(x, g(x))

−2

3

3

(−2, 3)

0

5

5

(0, 5)

1

6

6

(1, 6)

g(x) = (x + 5) → g(x) = f(x − (−5))

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Example 1
Vertical and Horizontal Translations
of Linear Functions

The constant h is grouped with x,
so h affects the input, or x-values.
The value of h is less than 0, so
the graph of f(x) = x is translated
|−5| units left, or 5 units left.

g(x) = (x + 5) is the translation of
the graph of the parent function 5
units left.

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Check

Describe the translation in g(x) = (x + 12) as it relates to
the graph of the parent function.

Example 1 (We will do together)
Vertical and Horizontal Translations
of Linear Functions

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Check

Describe the translation in g(x) = (x + 12) as it relates to
the graph of the parent function.

The graph of g(x) = (x + 12) is a translation of the graph
of the parent function 12 units left.

Example 1
Vertical and Horizontal Translations
of Linear Functions

​Group Question

​

​

​

​

Multiple Choice

Individual Question:

Describe the translation in the function.

• g(x) is a translation of the parent function 11 units up.

• g(x) is a translation of the parent function 11 units down.

• g(x) is a translation of the parent function 9 units up.

• g(x) is a translation of the parent function 9 units down.

​Group Question

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Example 2 (Watch as I do)
Multiple Translations of Linear Functions

Describe the translation in g(x) = (x − 6) + 3 as it
relates to the graph of the parent function.

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Example 2
Multiple Translations of Linear Functions

Graph the parent graph
for linear functions.

Because f(x) = x,
g(x) = f(x − h) + k where
h = 6 and k = 3.

x

x − 5 f(x − 6) f(x − 6) + 3 (x, g(x))

−2

−8

−8

−5

(−2, −5)

0

−6

−6

−3

(0, −3)

1

−5

−5

−2

(1, −2)

g(x) = (x − 6) + 3 → g(x) = f(x − 6) + 3

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Example 2
Multiple Translations of Linear Functions

The value of h is grouped with x and
is greater than 0, so the graph of
f(x) = x is translated 6 units right.

The value of k is not grouped with x
and is greater than 0, so the graph
of f(x) = x is translated 3 units up.

g(x) = (x − 6) + 3 is the translation of
the graph of the parent function 6
units right and 3 units up.

​We will complete as a class

​

​

Group Question

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Example 3 (Watch as I do)
Translations of Linear Functions

TICKETS A website sells tickets to
concerts and sporting events. The
total price of the tickets to a certain
game can be modeled by f(t) = 12t,
where t represents the number of
tickets purchased. The website then
charges a standard service fee of $4
per order. The total price of an order
can be modeled by g(t) = 12t + 4.
Describe the translation of g(t) as it
relates to f(t).

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Example 3
Translations of Linear Functions

Because f(t) = 12t, g(t) = f(t) + k, where k = 4.

g(t) = 12t + 4 → f(t) + 4

The constant k is added to f(t) after the total
price of the tickets has been evaluated and is
greater than 0, so the graph will be shifted 4
units up.

g(t) = 12t + 4 is the translation of the graph of
f(t) = 4 units up.

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Check

RETAIL Jerome is buying paint for a mural. The total
cost of the paint can be modeled by the function
f(p) = 6.99p. He has a coupon for $5.95 off his purchase
at the art supply store, so the final cost of his purchase
can be modeled by g(p) = 6.99p − 5.95. Describe the
translation in g(p) as it relates to f(p).

Example 3 (We will complete as a class)
Translations of Linear Functions

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Check

RETAIL Jerome is buying paint for a mural. The total
cost of the paint can be modeled by the function
f(p) = 6.99p. He has a coupon for $5.95 off his purchase
at the art supply store, so the final cost of his purchase
can be modeled by g(p) = 6.99p − 5.95. Describe the
translation in g(p) as it relates to f(p).

The graph of g(p) = 6.99p − 5.95 is the translation of the
graph of f(p) = 5.95 units down.

Example 3
Translations of Linear Functions

​Group Question

​

​

Multiple Choice

Individual Question:

• g(x)=4x; g(x) is the translation of f(x) 3.5 units up.

• g(x)=4x; g(x) is the translation of f(x) 3.5 units to the left.

• g(x)=4x; g(x) is the translation of f(x) 3.5 units down.

• g(x)=4x; g(x) is the translation of f(x) 3.5 units to the right.

Classwork 2.27.2023

-On a separate sheet of paper or on the workbook page, complete the following pages.

***Page. 173-174 #'s 3-6 & 8-9

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A dilation stretches or compresses the graph of a function.

When a linear function f(x) is multiplied by a positive constant a,
the result a · f(x) is a vertical dilation.

Learn
Dilations of Linear Functions

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Translations of Linear Functions

If |a| > 1, the graph of f(x) is stretched
vertically away from the x-axis

The slope of the graph of a · f(x) is
steeper than that of the graph of f(x).

If 0 < |a| < 1, the graph of f (x) is
compressed vertically toward the x-axis.

The slope of the graph of a · f(x) is less
steep than that of the graph of f(x).

Key Concept: Vertical Dilations of Linear Functions

g(x) = ax

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Dilations of Linear Functions

When x is multiplied by a positive constant a before a linear
function f(x) is evaluated, the result f(a · x) is a horizontal
dilation.

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Translations of Linear Functions

If |a| > 1, the graph of f(x) is
compressed horizontally toward the
x-axis

The slope of the graph of a · f(x) is
steeper than that of the graph of f(x).

If 0 < |a| < 1, the graph of f(x) is
stretched horizontally away from the
x-axis.

The slope of the graph of a · f(x) is less
steep than that of the graph of f(x).

Key Concept: Horizontal Dilations of Linear Functions

g(x) = (a • x)

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Example 4
Vertical and Horizontal Dilations of Linear
Functions

Describe the dilation in each function as it
relates to the graph of the parent function.

a. 𝒈(𝒙) = 𝟐(𝒙)

b. 𝒈(𝒙) =

𝟏
𝟒𝒙

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Example 4
Vertical and Horizontal Translations
of Linear Functions

a. 𝒈(𝒙) = 𝟐(𝒙)

Graph the parent graph for
linear functions.

Because f(x) = x, g(x) = a ∙ f(x)
where a = 2.

g(x) = 2(x) → g(x) = 2f(x)

x

f(x)

2f(x)

(x, g(x))

−2

−2

−4

(−2, −4)

0

0

0

(0, 0)

1

1

2

(1, 2)

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Example 4
Vertical and Horizontal Translations
of Linear Functions

The positive constant a is not grouped
with x, and |a| is greater than 1, so the
graph of f(x) = x is stretched vertically
by a factor of a, or 2.

g(x) = 2(x) is a vertical stretch of the
graph of the parent function. The slope
of the graph of g(x) is steeper than that
of f(x).

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Example 4
Vertical and Horizontal Translations
of Linear Functions

b. 𝒈(𝒙) =

𝟏
𝟒𝒙

Graph the parent graph for
linear functions.

Because f(x) = x, g(x) = f(a ∙ x)
where 𝑎 =

1
4.

𝑔 𝑥 =

1
4𝑥

→ 𝑔 𝑥 = 𝑓

1
4𝑥

x

𝟏
𝟒𝒙

𝒇

𝟏
𝟒𝒙
(x, g(x))

−4

−1

−1

(−4, −1)

0

0

0

(0, 0)

4

1

1

(4, 1)

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Example 4
Vertical and Horizontal Translations
of Linear Functions

The positive constant a is grouped with
x, and |a| is between 0 and 1, so the
graph of f(x) = x is stretched horizontally
by a factor of

1
𝑎, or 4.

𝑔 𝑥 =

is a horizontal stretch of

the graph of the parent function. The
slope of the graph of g(x) is less steep
than that of f(x).

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Check

Describe the transformation in g(x) = 6(x) as it relates to
the graph of the parent function.

Example 4 (Group Question)
Vertical and Horizontal Translations
of Linear Functions

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Check

Describe the transformation in g(x) = 6(x) as it relates to
the graph of the parent function.

The graph of g(x) = 6(x) is a vertical stretch of the graph
of the parent function.

The slope of the graph g(x) is steeper than that of the
parent function.

Example 4
Vertical and Horizontal Translations
of Linear Functions

Group Question

Group Question

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Reflections of Linear Functions

A reflection is a transformation in which a figure, line, or curve is
flipped across a line. When a linear function f(x) is multiplied by
−1 before or after the function has been evaluated, the result is a
reflection across the x- or y-axis. Every x- or y-coordinate of f(x)
is multiplied by −1.

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Reflections of Linear Functions

The graph of −f(x) is the reflection of the
graph of f(x) = x across the x-axis.

Every y-coordinate of −f(x) is the
corresponding y-coordinate of f(x)
multiplied by −1.

The graph of f(−x) is the reflection of the
graph of f(x) = x across the y-axis.

Every x-coordinate of f(−x) is the
corresponding x-coordinate of f(x)
multiplied by −1.

Key Concept: Reflections of Linear Functions

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Example 5
Reflections of Linear Functions Across the
x- or y-Axis

Describe how the graph of each function is
related to the graph of the parent function.

a. 𝒈(𝒙) = −

𝟏
𝟐(𝒙)

b. 𝒈(𝒙) = (−𝟑𝒙)

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Example 5
Reflections of Linear Functions Across the
x- or y-Axis

a. 𝒈(𝒙) = −

𝟏
𝟐(𝒙)

Graph the parent graph for
linear functions.

Because f(x) = x,
g(x) = −1 ∙ a ∙ f(x) where 𝑎 =

1
2.

x
𝒇(𝒙)

−

𝟏
𝟐𝒇 𝒙
(x, g(x))

−2

−2

1

(−2, 1)

0

0

0

(0, 0)

4

4

−2

(4, −2)

𝑔(𝑥) = −

1
2(𝑥) → 𝑔(𝑥) = −

1
2𝑓(𝑥)

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Example 5
Reflections of Linear Functions Across the
x- or y-Axis

The constant a is not grouped with x, and
|a| is less than 1, so the graph of f(x) = x is
vertically compressed.

The negative is not grouped with x, so the
graph is also reflected across the
x-axis.

The graph of 𝑔(𝑥) = −

1
2(𝑥) is the graph of

the parent function vertically compressed
and reflected across the x-axis.

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Example 5
Reflections of Linear Functions Across the
x- or y-Axis

b. g(x) = (−3x)

Graph the parent graph for
linear functions.

Because f(x) = x,
g(x) = f(−1 ∙ a ∙ x) where a = 3.

g(x) = −3x → g(x) = f(−3x)

x

−3x

f(−3x)

(x, g(x))

−1

3

3

(−1, 3)

0

0

0

(0, 0)

1

−3

−3

(1, −3)

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Example 5
Reflections of Linear Functions Across the
x- or y-Axis

The constant a is grouped with x, and |a|
is greater than 1, so the graph of f(x) = x
is horizontally compressed.

The negative is grouped with x, so the
graph is also reflected across the y-axis.

The graph of g(x) = (−3x) is the graph of
the parent function horizontally
compressed and reflected across the y-
axis.

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Check
How can you tell whether multiplying −1 by the parent
function will result in a reflection across the x-axis?
A. If the constant is not grouped with x, the result will

be a reflection across the x-axis.

B. If the constant is grouped with x, the result will be a

reflection across the x-axis.

C. If the constant is greater than 0, the result will be a

reflection across the x-axis.

D. If the constant is less than 0, the result will be a

reflection across the x-axis.

Example 5 (Group Question)
Reflections of Linear Functions Across the
x- or y-Axis

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Check
How can you tell whether multiplying −1 by the parent
function will result in a reflection across the x-axis? A
A. If the constant is not grouped with x, the result will

be a reflection across the x-axis.

B. If the constant is grouped with x, the result will be a

reflection across the x-axis.

C. If the constant is greater than 0, the result will be a

reflection across the x-axis.

D. If the constant is less than 0, the result will be a

reflection across the x-axis.

Example 5
Reflections of Linear Functions Across the
x- or y-Axis

Group Question

Group Question

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Classwork 3.2.2023

​On a separate sheet of paper or on the workbook page complete the following pages & numbers.

**Workbook pg.174-175 #'s 12-15, & 18-25

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Exit Ticket

1. Describe each pair of lines written in slope-intercept form.

a. Two lines have the same value for m, but they have

different values for b.

b. Two lines have different values for m, but they have the

same value for b.

2. Which graph is steepest: y = 3x, y = −4x − 7, or 𝑦 =

1
2𝑥 + 4?

Explain.

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Exit Ticket

3. How can knowing about the effects of m and b help you

sketch the graph of an equation?

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Exit Ticket

1. Describe each pair of lines written in slope-intercept form.

a. Two lines have the same value for m, but they have

different values for b. The lines are parallel.

b. Two lines have different values for m, but they have the

same value for b. The lines intersect at (0, b).

2. Which graph is steepest: y = 3x, y = −4x − 7, or 𝑦 =

1
2𝑥 + 4?

Explain.

The graph of y = −4x − 7 is steepest because |m| is greatest.

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Exit Ticket

3. How can knowing about the effects of m and b help you

sketch the graph of an equation?

The value of m tells you how steep the graph should be
compared to the graph of y = x. The value of b tells you how
many units higher or lower than y = x the graph should be.