Activity
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1. Each of the following functions, , , , and , represents the amount of money in a bank account, in dollars, as a function of time , in years. They are each written in the form .
a. Use graphing technology to graph each function on the same coordinate plane. It may be helpful to use a different color for each function.
Compare your answers:
b. Explain how changing the value of changes the graph.
Compare your answer:
When is greater than 1, larger values of mean that the function grows more quickly as increases. A positive value of that is less than 1 means the function is decreasing as increases.
2. Here are equations defining functions , , and . They are also written in the form .
a. Use graphing technology to graph each function on the same coordinate plane. It may be helpful to use a different color for each function.
Compare your answers:
b. Explain how changing the value of changes the graph.
Compare your answer:
It changes the -intercept to .
3. Here are equations defining functions , , and written in the form .
a. Use graphing technology to graph each function on the same coordinate plane. It may be helpful to use a different color for each function.
Compare your answers:
a. Explain how changing the value of changes the graph.
Compare your answers:
b. Discuss with a partner how changing the value of changes the asymptote of the graph.
Compare your answer:
Changing the value of changes the horizontal asymptote from to . So function has a horizontal asymptote of , and has a horizontal asymptote of .
Are you ready for more?
Extending Your Thinking
As before, consider bank accounts whose balances are given by the following functions:
Which function would you choose? Does your choice depend on ?
Compare your answer:
We can rewrite each of the functions as multiples of to more easily compare them:
Since , we see that has the most money for any value of .
Video: Exponential Equations and How They Affect the Graphs
Watch the following video to learn more about exponential functions and how they affect the graphs:
Self Check
Additional Resources
Vertical Shifts with Exponential Functions
The first transformation occurs when we add a constant to the parent function , giving us a vertical shift units in the same direction as the sign. For example, if we begin by graphing a parent function, , we can then graph two vertical shifts alongside it, using : the upward shift, , and the downward shift, . Both vertical shifts are shown in the graph below.
Observe the results of shifting vertically:
- The domain, , remains unchanged.
- When the function is shifted up 3 units to :
- The -intercept shifts up 3 units to .
- The asymptote shifts up 3 units to .
- The range becomes .
- When the function is shifted down 3 units to :
- The -intercept shifts down 3 units to .
- The asymptote also shifts down 3 units to .
- The range becomes .
Try it
Try It: Vertical Shifts with Exponential Functions
Where would the horizontal asymptote be located for the graph of ?
Compare your answer:
Here is how to find the equation of the horizontal asymptote:
In the function , the vertical shift is represented by –6.
The parent function, , has a horizontal asymptote at . For the function , the horizontal asymptote will be shifted down 6 to .