3.7 Part 2 NOTES ­ Reflections and Shrinks and Stretches

3 . 7 LESSON 3.7 - Transformations• Today we will learn how the graphs of functions change based on how their equations change. 

• These changes include:  Translations, Reflections, Shrinks, and Stretches.

• A REFLECTION is a transformation that flips a graph over a line called the line of reflection.

REFLECTIONS

EXAMPLE: The function f(x) = 2x + 3 was reflected over the y‑axis to create the red line.

The equation of the red line is:

y = ‑2x + 3

• A REFLECTION is a transformation that flips a graph over a line called the line of reflection.

REFLECTIONS

EXAMPLE: The function f(x) = x + 4 was reflected over the x‑axis to create the red line.

The equation of the red line is:

y = ‑x ‑ 4

REFLECTIONS

REFLECTIONS IN THE x-AXIS: •The graph of g(x) = ‑f(x) is the reflection of f(x) in the x‑axis.

EXAMPLE:  f(x) = |x + 2|       g(x) = ‑|x + 2|

REFLECTIONS IN THE y-AXIS: •The graph of g(x) = f(‑x) is the reflection of f(x) in the y‑axis.

EXAMPLE: f(x) = |x ‑ 9|        g(x) = |‑x ‑ 9|

Match each graph to its function.

f(x) = |x + 3| g(x) = ‑|x + 3|

BA

h(x) = |‑x + 3| B

A

3.7 Part 2 NOTES ­ Reflections and Shrinks and Stretches

Match each graph to its function.

f(x) = |x ‑ 2| + 5 g(x) = ‑|x ‑ 2| ‑ 5

A B

h(x) = |‑x ‑ 2| + 5

B

Ag(x) = |‑x|

Describe the transformation(s) from the parent function f(x) to the function below.

The parent function f(x) was reflected in the y-axis to arrive at

the transformed function g(x).

g(x) = ‑|x|

Describe the transformation(s) from the parent function f(x) to the function below.

The parent function f(x) was reflected in the x-axis to arrive at

the transformed function g(x).

REFLECTIONS EXAMPLES

f(x) = |x + 7| and g(x) = f(‑x)

A) Describe the reflection that was performed

 

B) Write the equation for g(x)

C) Graph both functions.

Reflection in the y-axis

g(x) = |-x + 7|

REFLECTIONS EXAMPLES

f(x) = |x| + 8 and g(x) = ‑f(x) 

A) Describe the reflection that was performed

 

B) Write the equation for g(x)

C) Graph both functions.

Reflection in the x-axis

g(x) = -(|x| + 8)g(x) = -|x| - 8

REFLECTION EXAMPLES

f(x) = ‑|x + 5| ‑ 4  and  g(x) = ‑f(x) 

A) Describe the reflection that was performed

 

B) Write the equation for g(x)

C) Graph both functions.

Reflection in the x-axis

g(x) = -(-|x + 5| - 4)g(x) = |x + 5| + 4

3.7 Part 2 NOTES ­ Reflections and Shrinks and Stretches

• A VERTICAL SHRINKS & STRETCHES are transformations that change the shape of a graph.  This transformation is achieved by multiplying all the y‑coordinates by a number a.

• The transformation is a VERTICAL STRETCH if |a| > 1 because the graph stretches away from the x‑axis.

• The transformation is a VERTICAL SHRINK if |a| < 1 because the graph shrinks towards the x‑axis.

VERTICAL SHRINKS AND STRETCHES• A VERTICAL SHRINKS & STRETCHES are transformations that change the shape of a graph.

EXAMPLE: The red function is a vertical stretch of the parent function.

The equation of the red function is:

y = 2|x|

VERTICAL SHRINKS AND STRETCHES

• A VERTICAL SHRINKS & STRETCHES are transformations that change the shape of a graph.

EXAMPLE: The red function is a vertical shrink of the parent function.

The equation of the red function is:

y =   |x|

VERTICAL SHRINKS AND STRETCHES

13

VERTICAL STRETCHES: •The graph of g(x) = a(f(x)) is the vertical stretch of f(x) if |a| > 1.

EXAMPLE:  f(x) = x       g(x) = 4|x|

VERTICAL SHRINKS: •The graph of g(x) = a(f(x)) is the vertical shrink of f(x) if |a| < 1

EXAMPLE: f(x) = |x|        g(x) =    |x|

VERTICAL SHRINKS AND STRETCHES

Match each graph to its function.

f(x) =    |x|

f(x) = 3|x|

f(x) = ‑    |x|

f(x) = ‑3|x|

A B

C D

C

B

D

A

g(x) = 2|x|

Describe the transformation(s) from the parent function f(x) to the function below.

The parent function f(x) was vertically stretched with a = 2 to arrive at the transformed function

g(x).

3.7 Part 2 NOTES ­ Reflections and Shrinks and Stretches

g(x) =    |x|

Describe the transformation(s) from the parent function f(x) to the function below.

The parent function f(x) was vertically shrunk with a = to

arrive at the transformed function g(x).

14

g(x) =    |x|

Describe the transformation(s) from the parent function f(x) to the function below.

The parent function f(x) was vertically stretched with a = to arrive at the transformed function

g(x).

53

g(x) = 7|x|

Describe the transformation(s) from the parent function f(x) to the function below.

The parent function f(x) was vertically stretched with a = 7 to arrive at the transformed function

g(x).

SHRINKS & STRETCHES EXAMPLES

f(x) = |x| and g(x) = 5 f(x)

A) Describe the shrink or stretch that was performed

 

B) Write the equation for g(x)

C) Graph both functions.

Vertical stretch of a = 5

g(x) = 5|x|

SHRINKS & STRETCHES EXAMPLES

f(x) = |x| and g(x) =    f(x)

A) Describe the shrink or stretch that was performed

 

B) Write the equation for g(x)

C) Graph both functions.

Vertical shrink of a =

g(x) = |x|

1

16

6

SHRINKS & STRETCHES EXAMPLES

f(x) = |x + 7| ‑ 2  and  g(x) = 3 f(x)

A) Describe the shrink or stretch that was performed

 

B) Write the equation for g(x)

C) Graph both functions.

Vertical stretch of a = 3

g(x) = 3(|x + 7| - 2)

g(x) = 3|x + 7| - 6

3.7 Part 2 NOTES ­ Reflections and Shrinks and Stretches

SHRINKS & STRETCHES EXAMPLES

f(x) = ‑|x + 4| + 10  and  g(x) =     f(x)

A) Describe the shrink or stretch that was performed

 

B) Write the equation for g(x)

C) Graph both functions.

Vertical shrink of a = 2

25

5

g(x) = (-|x + 4| + 10)

g(x) = - |x + 4| + 825

SHRINKS & STRETCHES EXAMPLES

f(x) = 3|x| ‑ 1  and  g(x) = ‑3 f(x)

A) Describe the shrink or stretch that was performed

 

B) Write the equation for g(x)

C) Graph both functions.

Vertical stretch of a = 3

g(x) = -3(3|x| - 1)

g(x) = -9|x| + 9

Reflection in the x-axis

HOMEWORK:3.7 Part 2 Worksheet ‑ Reflections and Shrinks & Stretches