# Reflecting Graphs ```
1.
2.
Reflections in the coordinate axes of the
graph of y = f(x) are represented by:
Reflection in the x-axis: h(x) = -f(x)
Reflection in the y-axis: h(x) = f(-x)

g is related to one of the parent functions.
Identify the parent function, describe the
sequence of transformation from f to g.
1. 𝑔 𝑥 = 2 − 𝑥 + 5 2
1
−
4
𝑥+2 2−2
3. 𝑔 𝑥 = 𝑥 + 3 + 9
1
4. 𝑔 𝑥 = −
𝑥+3−1
2
5. 𝑔 𝑥 = − 𝑥 + 10 2 + 5
2. 𝑔 𝑥 =

Use a graphing utility to graph the three
functions in the same viewing window.
Describe the graphs of g and h relative the
graph of f.
1. 𝑓 𝑥 = 𝑥 3 − 3𝑥 2
2. 𝑔 𝑥 =
1
− 𝑓(𝑥)
3
3. ℎ 𝑥 = 𝑓(−𝑥)

Compare the graph of each function with the
graph of f(x) = 𝑥 alegebraically
1. 𝑔 𝑥 = − 𝑥
2. ℎ 𝑥 = −𝑥
3. 𝑘 𝑥 = − 𝑥 + 2

Horizontal, vertical, and reflection shifts are
all call rigid transformations. These
transformations only change the position of
the graph in the coordinate plane

Non-rigid transformations are those that
cause distortion of the graph.

A non-rigid transformation of the graph
y= f(x) is represented by y = cf(x), where the
transformation is a vertical stretch if c &gt; 1 and
a vertical shrink if 0 &lt; c&lt; 1.

Another non-rigid transformation of the
graph y = f(x) is represented by h(x) = f(cx),
where the transformation is a horizontal
shrink if c &gt; 1 and a horizontal stretch if 0&lt;c&lt;1
𝑓 𝑥 = 𝑥
ℎ 𝑥 =3𝑥
1
𝑔 𝑥 = 𝑥
3
Write a few sentences comparing the graphs
shown above.

You may form into groups of 2-3 to complete
the following quiz.

Each member of your group must show all
work in order to receive credit.

After you have finished quiz, please be sure
own paper)
Given the three following functions:
𝑓 𝑥 = 𝑥 3 − 3𝑥 2 + 2
𝑔 𝑥 = −𝑓 𝑥
ℎ 𝑥 = 𝑓(2𝑥)
1. Identify the parent function of f.
2. Describe the graphs of g and h relative to the
graphs of each functions. Label the graph
appropriatly.

Just as real numbers can be combined by the
multiplication, and division to form other real
numbers, two functions can be combined to
create a new function.

This is known as an arithmetic combination
of functions.

The domain of an arithmetic combination of
functions f and g consists of all real numbers
that are common to the domains of f and g.

State the domain of the following
combination:
𝑓 𝑥
𝑔(𝑥)

Sum: f + 𝑔 (𝑥) = 𝑓 𝑥 + 𝑓 𝑔

Difference: 𝑓 − 𝑔 (𝑥) = 𝑓 𝑥 − 𝑔 𝑥

Product: 𝑓𝑔 𝑥 = 𝑓 𝑥 𝑔 𝑥

Quotient:
𝑓
𝑔
𝑥 =
𝑓 𝑥
, 𝑔(𝑥)
𝑔(𝑥)
≠0

Given the two functions find (f + g)(x):
𝑓 𝑥 = 2𝑥 + 1
𝑔 𝑥 = 𝑥 2 + 2𝑥 − 1

Given:
𝑓 𝑥 = 2𝑥 + 1
𝑔 𝑥 = 𝑥 2 + 2𝑥 − 1
Evaluate the difference of the two functions.
Then evaluate the difference when x = 2

Given:
𝑓 𝑥 = 𝑥2
𝑔 𝑥 =𝑥−3
Find the product of the two functions then
evaluate the product when x = 4.

Given
𝑓 𝑥 = 𝑥
𝑔 𝑥 = 4 − 𝑥2
Find the quotient of the functions
𝑓
𝑔
𝑥 𝑎𝑛𝑑
function.
𝑔
𝑓
𝑥 . Find the domain of each

Pg. 58 # 5 – 26
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