Download Investigating Exponential Functions

Math 30-2
Exponential Functions
Name: _____________________
Investigating Exponential Functions
Your goal in this investigation is to be able to complete the summary statements on the last
page.
You will be using the Desmos online graphing calculator for this investigation.
- Website: www.desmos.com
- Click the “Launch Calculator” button.
- In box 1 at the top left of the screen, type the equation y  a  b x .
- It will ask you if you want to “add slider” – click “all”
- Start by setting a  1 and b  2 .
Use the Desmos graphing calculator to graph each exponential function. Draw a quick sketch of
each function, making sure the y-intercept, horizontal asymptote, and relative steepness of the
graph are accurate. Then state:
a) the y-intercept
b) the equation of the horizontal asymptote
c) the sign of the coefficient, a (positive or negative)
d) whether the graph is increasing or decreasing
e) whether the graph is above or below the x-axis
Also make sure you answer the questions that follow each table.
Investigating the Base, b: b > 1
Leave the a-value at set at 1. Play with the b slider, making it bigger than 1.
y  2x
f x   1.5x
y  5x
graph
y-intercept
equation of
asymptote
sign of a
increasing/
decreasing
above/below
x-axis

Explain how the b value changes the shape of the graph when b is bigger than 1.
Math 30-2
Exponential Functions
Investigating the Base, b: 0 < b < 1
Leave the a-value at set at 1. Play with the b slider, making it between 0 and 1.
1
f x    
2
x
f x   0.2 x
y  0.75 x
graph
y-intercept
equation of
asymptote
sign of a
increasing/
decreasing
above/below
x-axis

Explain how the graph of an exponential function is different when the base, b, is
between 0 and 1.
Investigating the Coefficient, a: a > 1
Set the b-value to 2. Play with the a slider, making it positive.
y  2  2x
graph
y-intercept
equation of
asymptote
sign of a
increasing/
decreasing
above/below
x-axis
y  42 
x
f x   0.75  2 x
Math 30-2
Exponential Functions

How does the y-intercept relate to the coefficient, a, of an exponential function?

Explain how changing the coefficient, a, changes the shape of the graph (when a is
positive).
Investigating the Coefficient, a: a < 0
Leave the b-value set at 2. Play with the a slider, making it negative.
f x   3 2 x
y  2 
x
f x   
1 x
2
2
graph
y-intercept
equation of
asymptote
sign of a
increasing/
decreasing
above/below
x-axis

Explain how the graph of an exponential function is different when the coefficient, a, is
negative.
Summary of Characteristics of Graphs of Exponential Functions
Use your observations about exponential functions to summarize what you have learned.

Every exponential function in the form y  a  b x has a horizontal asymptote at y = _____.

If the coefficient, a, is positive and the base, b, is greater than 1, the graph is _____________
(increasing/decreasing). If the base is between 0 and 1, the graph is _________________.

In the equation y  a  b x , the y-intercept = _______.

If the base, b, is greater than 1 and the coefficient, a, is positive, the graph is ____________
(above/below) the x-axis. If the coefficient is negative, the graph is ___________ the x-axis.