Precalculus
Exponential Functions
Graphing Basic Exponentials
Name:__________________________________
Graphing Basic Exponential Functions
Your goal today is to determine the characteristics of the graphs of exponential functions. In an exponential
function, the variable is an exponent. The parent function is 𝑓(𝑥) = 𝑏 𝑥 where 𝑏 is any real number greater than
0, except 1.
Graph 𝒇(𝒙) = 𝟐𝒙 . Complete the table below. Graph the points and connect with a smooth curve.
1. What happens to f(x) as x increases? What happens to f(x) as x decreases?
2. Does the graph intersect the x-axis? Explain how you know.
3. What are the domain and range of f(x)?
Worksheet and Images are adapted from the Holt McDougal Activity Generator version 1.1 copyright 2009 Houghton Mifflin Harcourt Publishing Company.
𝟏 𝒙
Graph 𝒇(𝒙) = (𝟐) . Complete the table below. Graph the points and connect with a smooth curve.
4. What happens to f(x) as x increases? What happens to f(x) as x decreases?
1 𝑥
5. How does the domain and range of 𝑓(𝑥) = (2) compare to the domain and range of 𝑓(𝑥) = 2𝑥 ?
6.
1 𝑥
What do you notice about the y-intercepts of the graphs of 𝑓(𝑥) = (2) and f(x) = 2x? Why does this
make sense?
A function of the form f(x) = bx is an exponential growth function if b > 1 and an
exponential decay function if 0 < b < 1.
7. Describe the end behavior of an exponential growth function.
8. Describe the end behavior of an exponential decay function.
9. Explain why the point (1, b) is always on the graph of f(x) = bx.
10. Explain why the point (0, 1) is always on the graph of f(x) = bx.
11. Are f(x) = 3x and g(x) = 5x both exponential growth functions or both exponential decay functions?
Although they have the same end behavior, how you do think their graphs differ? Explain your
reasoning.
1 𝑥
1 𝑥
12. Are 𝑓(𝑥) = (3) and 𝑔(𝑥) = (5) both exponential growth functions or both exponential decay
functions? Although they have the same end behavior, how you do think their graphs differ? Explain
your reasoning.
Compare each of the functions f(x) = x + 3 and g(x) = x3 to the exponential function h(x) = 3x for x ≥ 0.
The graph of ℎ(𝑥) = 3𝑥 is already graphed below.
Graph 𝒇(𝒙) = 𝒙 + 𝟑 here:
Graph 𝒈(𝒙) = 𝒙𝟑 here:
13. How do the values of h(x) compare to those of f(x) and g(x) as x increases without bound?
Complete the tables below for the two functions as indicated and sketch a graph of each function. You can
graph by hand or you can use your graphing calculator or a different graphing tool, such as DESMOS.
𝒇(𝒙) = 𝟑−𝒙
x
y
-4
-3
-2
-1
0
1
2
3
𝟏
𝒇(𝒙) = ( )𝒙
𝟑
x
y
-4
-3
-2
-1
0
1
2
3
1. What do you notice about these two graphs?
2. Using your knowledge of the rules of exponents, show how this is occurring.