3.2, 3.3: Graphs and Properties of Functions
Vertical Line Test
What defining feature must the graph of a function always have?
1.
Use the vertical line test to identify graphs in which y is a function
of x.
Domain and Range:
Recall: the domain of a function or relation is the set of all its inputs (x
values on the graph) and the range is the set of all its outputs (y values
on the graph).
2.
Identify the domain and range of each graph.
The graph of a function
Given the graph of a function, you should be able to:
– Determine whether or not it is a function.
– Identify all intercepts.
– Identify any local maxima and minima, as well as absolute
maxima and minima.
– Identify all intervals where the function is increasing,
decreasing, or constant.
– Determine if a function is even (symmetric w.r.t. the y-axis)
or odd (symmetric w.r.t. the origin).
– Determine the domain and range.
– Find any output values that correspond with an input value.
– Find the input value that corresponds with an output value.
Before we begin, let’s get some definitions out of the way.
Parity of a function
 A function is even if it is symmetric with respect to the y-axis. i.e.
𝑓(−𝑥 ) = 𝑓(𝑥) for every x in the domain.
 A function is odd if it is symmetric with respect to the origin. i.e.
𝑓(−𝑥 ) = −𝑓(𝑥) for every x in the domain.
Increasing, decreasing, and constant intervals
Maxima and minima
3.
Given the function to the right.
a. Is it a function?
b. Identify all intercepts
c. Where is the function increasing?
d. Where is the function decreasing?
e. Where is the function constant?
f. Where is the function positive?
g. Where is the function negative?
h. Identify any local maxima/minima.
i. Identify any absolute maxima/minima.
j. What is the function’s domain?
k. What is the function’s range?
l. What is 𝑓(3)
m. Where is 𝑓(𝑥 ) = 3?
4.
Given the function to the right
a. Is it a function?
b. Identify all intercepts
c. Where is the function increasing?
d. Where is the function decreasing?
e. Where is the function constant?
f. Where is the function positive?
g. Where is the function negative?
h. Identify any local maxima/minima.
i. Identify any absolute maxima/minima.
n. What is the function’s domain?
o. What is the function’s range?
p. What is 𝑓(3)
q. Where is 𝑓(𝑥 ) = 3?
Extreme Value Theorem
5.
Given ℎ(𝑥) below
Assume the function continues indefinitely in both directions
a. State the domain and range of ℎ(𝑥)
b. Solve ℎ(𝑥 ) < 0
6.
Determine whether each function is even, odd, or neither.
a. 𝑓(𝑥 ) = −3𝑥 4 − 2𝑥 2
b. 𝑔(𝑥 ) = 𝑥 − 𝑥 2
c. ℎ(𝑥 ) =
𝑥3
𝑥 2 −4
Average rate of change
7.
Find the average rate of change of 𝑓(𝑥 ) = 𝑥 3 − 2𝑥
a. from 1 to 2
b. from -1 to 4
8.
𝑔(𝑥 ) = −𝑥 + 3𝑥 2
a. Find the average rate of change from 0 to 3
b. What is the slope of the secant line containing (0, 𝑔(0)) and
(3, 𝑔(3))?
Calculator Stuff
9.
Given the function 𝑓(𝑥 ) = 0.15𝑥 3 − 𝑥 2 + 𝑥 + 4
a. Identify all intercepts.
b. Identify all local maxima/minima