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
𝑓 𝑥 = 𝑥2
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?
2
𝑓 𝑥 = 𝑎(𝑥 − ℎ) +𝑘
Absolute Value and
Exponential Functions and
Their Transformations
Transformations
Parabolas Revisited: Vertex Form:
y = -a (x –
Reflection
across the
x-axis
Vertical Stretch
a>1
(makes it narrower)
OR
Vertical Compression
0<a<1
(makes it wider)
2
h)
+k
Vertical
Translation
Horizontal
Translation
(opposite of h)
*Remember that (h, k) is your vertex*
The Parent Graph of the
Absolute Value Function
9
8
7
6
5
4
3
2
1
-9 -8 -7 -6 -5 -4 -3 -2 -1 -1
-2
-3
-4
-5
-6
-7
-8
-9
y
x
1
2
3
4
5
6
7
8
9
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
y=x2
9
8
7
6
5
4
3
2
1
-9 -8 -7 -6 -5 -4 -3 -2 -1 -1
-2
-3
-4
-5
-6
-7
-8
-9
y
y=|x|
x
1
2
3
4
5
6
7
8
9
Vocabulary

The zeros of a function f(x) are the values of
x that make the value of f(x) zero.

f(x) = |x| - 3
On this graph where
x = -3 and x = 3 are
where the function
would equal 0.
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
Reflection
across the
x-axis
Vertical Stretch
a>1
(makes it narrower)
OR
Vertical Compression
0<a<1
(makes it wider)
Vertical
Translation
Horizontal
Translation
(opposite of h)
*Remember that (h, k) is your vertex*
Example 1:

Example 2:

Graph y = -2 |x + 3| + 4


What is your vertex?
What are the intercepts?
Absolute Value on your
calculator
Graphing example 2 on your
calculator
𝑦 = −2 |𝑥 + 3| + 4
You Try:

Graph 𝑦 =
1
−
2
𝑥– 1 − 2
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:
𝑓 𝑥 = 2𝑥
Exponential Parent Graph
y
4
Key Characteristics:
3
2
• There are no lines of symmetry
1
x
-4
-3
-2
-1
1
-1
2
3
4
• These functions will always
have an asymptote
-2
-3
-4
• There is no vertex point
Exponential Parent Graph
y
The ‘locater point’ for this
function is the asymptote.
4
3
2
1
x
-4
-3
-2
-1
1
-1
-2
-3
-4
2
3
4
Using this as our point
allows for quick
comparisons between the
parent and transformed
graphs.
Exponential Transformation
Example #1:
3
y
2
1
x
-4
-3
2
-2
-1
1
-1
-2
-3
2
3
4
Comparing the
asymptotes will
give the vertical
shift.
Exponential Transformation
Example #2:
y
4
3
2
1
x
-7
-6
-5
-4
-3
-2
-1
1
-1
-2
-3
-4
2
3
4
5
6
7
Horizontal translations
shift the point where the
graph would have
crossed the x-axis.
Exponential General Form
𝑓 𝑥 = −(a)2
Reflection across the x-axis
Vertical Stretch
a>1
(makes it narrower)
OR
Vertical Compression
0<a<1
(makes it wider)
(𝑥−ℎ)
+𝑘
Vertical Translation
(also the asymptote)
Horizontal Translation
(opposite of h)
You Try:
Identify the transformations:
1.
𝑓 𝑥 = −2(𝑥+6) + 10
2.
𝑓 𝑥 = 2(𝑥−9) − 8
3.
𝑓 𝑥 = 2(𝑥) − 89
4.
𝑓 𝑥 = −2(𝑥+12) + 48.55
Homework

Worksheet #4