Warm-Up: you should be able to answer the following without the use of a calculator

1) State the domain, range and axis of symmetry for the following parent graph

2) Graph the following function and state the domain, range and axis of symmetry for the following function:

𝑓 (𝑥 )=−(𝑥+3)2+10

• How does the domain, range and axis of symmetry relate to the general vertex form?

Absolute Value and Exponential Functions and

Their Transformations

TransformationsParabolas Revisited: Vertex Form:

y = -a (x – h)2 + k

*Remember that (h, k) is your vertex*

Reflection across the

x-axis Vertical Stretcha > 1

(makes it narrower)OR

Vertical Compression

0 < a < 1 (makes it wider)

Horizontal Translation

(opposite of h)

Vertical Translation

The Parent Graph of the Absolute Value Function

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-9-8-7-6-5-4-3-2-1

123456789

x

y

Vocabulary

The function f(x) = |x| is an absolute value function.

The highest or lowest point on the graph of an absolute value function is called the vertex.

An axis of symmetry of the graph of a function is a vertical line that divides the graph into mirror images. An absolute value graph has one axis of

symmetry that passes through the vertex.

Absolute Value Function

Vertex

Axis of Symmetry

Quadratic and Absolute Value Functions

Quadratic and Absolute Value functions share some common characteristics:

Vertex

Line of Symmetry

Minimum/ Maximum point

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-9-8-7-6-5-4-3-2-1

123456789

x

yy=x2 y=|x|

Vocabulary

The zeros of a function f(x) are the values of x that make the value of f(x) zero.

On this graph where

x = -3 and x = 3 are

where the function

would equal 0.

f(x) = |x| - 3

Other Shared Characteristics

Review the vertex form of a parabola. Review how the changes in a , h and k

transform, reflect or translate the parent graph of a parabola.

Parent and general equation:

Given y=|x| how do you think the general equation of a an absolute value function looks like?

How do you think each component transforms, reflects or translates the parent graph?

Vocabulary

A transformation changes a graph’s size, shape, position, or orientation.

A translation is a transformation that shifts a graph horizontally and/or vertically, but does not change its size, shape, or orientation.

When a = -1, the graph y = a|x| is a reflection in the x-axis of the graph of y = |x|.

Transformations

y = -a |x – h| + k

*Remember that (h, k) is your vertex*

Reflection across the

x-axis Vertical Stretcha > 1

(makes it narrower)OR

Vertical Compression

0 < a < 1 (makes it wider)

Horizontal Translation

(opposite of h)

Vertical Translation

Example 1:  

Example 2: Graph y = -2 |x + 3| + 4

What is your vertex? What are the intercepts?

Absolute Value on your calculator

Where to locate the abs value

Graphing example 2 on your calculator

You Try: Graph

Compare the graph with the graph of y = |x|

(what are the transformations)

Example 3:

Write a function for the graph shown.

You Try:

Write a function for the graph shown.

Exponential Functions

The next family of functions we are going to look at are Exponential Functions

Our parent function being used

for comparing graphs will be:

Exponential Parent Graph

-4 -3 -2 -1 1 2 3 4

-4

-3

-2

-1

1

2

3

4

x

y

𝒇 (𝒙 )=𝟐𝒙

Key Characteristics:

• There are no lines of symmetry

• These functions will always have an asymptote

• There is no vertex point

Exponential Parent Graph

-4 -3 -2 -1 1 2 3 4

-4

-3

-2

-1

1

2

3

4

x

y

The ‘locater point’ for this function is the asymptote.

Using this as our point allows for quick comparisons between the parent and transformed graphs.

𝒇 (𝒙 )=𝟐𝒙

Exponential Transformation

-4 -3 -2 -1 1 2 3 4

-3

-2

-1

1

2

3

x

y

𝒇 (𝒙 )=𝟐𝒙

𝒇 (𝒙 )=𝟐𝒙−𝟐

Example #1:

2

Comparing the asymptotes will give the vertical shift.

Exponential Transformation

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

-4

-3

-2

-1

1

2

3

4

x

y

𝒇 (𝒙 )=𝟐𝒙

𝒇 (𝒙 )=𝟐(𝒙−𝟒)+𝟑

Example #2:

Horizontal translations shift the point where the graph would have crossed the x-axis.

Exponential General Form𝑓 (𝑥 )=−( a )2  (𝑥−h )+𝑘

Vertical Translation (also the asymptote)

Reflection across the x-axis

Vertical Stretcha > 1

(makes it narrower)OR

Vertical Compression 0 < a < 1

(makes it wider)

Horizontal Translation

(opposite of h)

You Try:

Identify the transformations:

Homework

Worksheet #4