6.3 Graphing Exponential Functions

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35 questions

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OBJECTIVES & STANDARDS
Math Objectives
Identify and graph asymptotes for exponential functions
Describe end behavior from a graph, function, or table
Understand how changing coefficients changes the graph of an exponential function
Understand how changing the base changes the graph of an exponential function
Understand how adding constants changes the graph of an exponential function
Graph exponential functions from equations and tables
Common Core Math Standards
Link to all CCSS Math
CCSS.HSF.IF.A.2
CCSS.HSF.IF.C.7
CCSS.HSF.LE.A.1
CCSS.HSF.LE.A.2
CCSS.HSF.LE.A.3
CCSS.HSF.LE.B.5
Personal Finance Objectives
Compare the growth of savings and investment accounts by looking at their graphs
Predict anticipated compound growth from a graph
Understand the concept of annualized rate of return on investments
National Standards for Personal Financial Education
Investing
3b: Investigate the long-run average rates of returns on small-company stocks, large-company stocks, corporate bonds, and Treasury bonds
7b: Compare the expense ratios for several mutual funds
DISTRIBUTION & PLANNING
Distribute to students
Student Activity Packet
Application Problems

Instructions

OBJECTIVES & STANDARDS
Math Objectives
Identify and graph asymptotes for exponential functions
Describe end behavior from a graph, function, or table
Understand how changing coefficients changes the graph of an exponential function
Understand how changing the base changes the graph of an exponential function
Understand how adding constants changes the graph of an exponential function
Graph exponential functions from equations and tables
Common Core Math Standards
Link to all CCSS Math
CCSS.HSF.IF.A.2
CCSS.HSF.IF.C.7
CCSS.HSF.LE.A.1
CCSS.HSF.LE.A.2
CCSS.HSF.LE.A.3
CCSS.HSF.LE.B.5
Personal Finance Objectives
Compare the growth of savings and investment accounts by looking at their graphs
Predict anticipated compound growth from a graph
Understand the concept of annualized rate of return on investments
National Standards for Personal Financial Education
Investing
3b: Investigate the long-run average rates of returns on small-company stocks, large-company stocks, corporate bonds, and Treasury bonds
7b: Compare the expense ratios for several mutual funds
DISTRIBUTION & PLANNING
Distribute to students
Student Activity Packet
Application Problems

Intro/Warm-Up:

Predicting the graph
Use the data table to answer the questions.

Plot the points on the graph. Use Desmos and Screen shot your answer in the show your work here and in question 3.

Would a linear function be a good fit for these points? Why or why not?

Sketch a smooth curve that best fits the data.(screen shot question 1 desmos here, then sketch over plotted points).

Describe the shape of the sketch and how it is different from a straight line.

What do you predict the function value to be when x=-100? When x=-1000? Why do you say this?

Explore It

DESMOS: Marbleslides: Exponentials
An exponential function has the general form y = abx +c Manipulating these variables will alter the function graph. Follow your teacher’s instructions to complete this exploratory Desmos activity. Then, answer the summary questions.

How does the graph change as the base, b, increases above 1?

How does the graph change when the base, b, decreases below 1 but above 0?

How does the graph change when you add or subtract a constant in the exponent, x?

Learn It

Asymptotes
You likely have noticed some similarities in all of the exponential graphs you looked at in that activity. One unique feature is that for every exponential graph, one end will go on toward positive or negative infinity and the other end will approach, but not cross a certain, specific y value. This value that the function approaches is called an asymptote. In the case of exponential functions, they will have a horizontal asymptote (y = some value).
Example 1
Let’s take a look back at the problem from the intro to explain some key features. The data was from the function y = 2x. You can get this information from either the table or the graph of an exponential function. Oftentimes you can use the information about the asymptote to sketch a more accurate graph!



Coefficient and the constant terms
Changing the constant term affects the y-intercept of an exponential function, just as it does a linear function. You may have noticed that changing the coefficient changes the y-intercept as well. This is because b0 is always 1, so the function at x = 0 becomes y = a + c.
The coefficient is important in another way though! You can tell the value of the asymptote because it is the same value as the constant term that is added. For example, in the equation y = 3x - 4, the asymptote would be at y = -4. Take a look at the previous three examples and their equations to see how the constant term matches your predicted asymptotic values.

a._______ 
b. _______ 
c. _______ 

Practice It

5-Point Practice: Writing an Exponential Equation From a Table
Complete any number of problems that add up to 5 points. Identify how the graph behaves when x is very large and when x is very small: does it approach ∞, -∞, or a specific value. Then give the equation of the asymptote.

APPLICATION: Exponential Change in Investments and Cars

Level 1
Jerry bought a brand new car at the dealership one year ago for $35,000. The value of his car depreciates and is worth 20% less every year.

Write the exponential equation that models the value of his car._______ 

Complete this table of Jerry’s car’s value over the first 10 years of its life.*use commas in your answer*
x= Car Age (years)    F(x)=Car Value
x=0    f(x)=$35,000
x=1     f(x)=$_______ 
x=2     f(x)=$_______ 
x=3     f(x)=$_______ 
x=4     f(x)=$_______ 
x=5     f(x)=$_______ 
x=6     f(x)=$_______ 
x=7     f(x)=$_______ 
x=8     f(x)=$_______ 
x=9     f(x)=$_______ 
x=10   f(x)=$_______ 

How long do you predict it will take for the car’s value to drop to $1000?

In Desmos, plot the value of Jerry’s car under this model over the first 20 years of its life. Screen shot your answer in the drawing. windows+shift+s

How long does it take for the value to drop to about $1000? About ______ years.

Approximately how long does it take for the value to drop to 50% of its original value?_______  To 10%?_______ 

Jerry wants to know what would have a bigger impact on the value of his car long-term, an increased purchase price or decreased depreciation. He decides to compare a purchase price of $50,000 and 20% depreciation (blue graph) with a $35,000 purchase price and 17% depreciation (red graph). He graphs both of the equations. Use this graph to answer the following questions and help him figure it out.

Approximately what is the value of the $50,000 car after 9.5 years with 20% depreciation?_______ 
Approximately what is the value of the $35,000 car after 9.5 years with 17% depreciation?_______ 
How much value did each vehicle lose over 9.5 years?
50,000 car cost: _______ 
35,000 car cost: _______ 
4.  Given your answer to part c, which vehicle would you rather purchase? Explain your answer._______ 
5.  What is the domain and range of this situation?
Domain:_______ 
Range: _______ 

Level 2:
Although the prices of stocks and bonds fluctuate constantly, you can model the growth by using an exponential function that approximates the growth as if it were the same year after year. This growth is called the annualized rate of return, and you can use it as the base in an exponential function model.
Let’s say Jenny and Sarah are both gifted $5,000 when they were born and they are 16 years old now. Jenny’s money is invested in an S&P Index Fund with 7.5% average annualized returns while Sarah’s is invested in a bond fund with 3.5% average annualized returns.

Write an exponential equation to model each of their investment returns.
Jenny j(x)=_______ 
Sarah s(x)=_______ 

How much more does Jenny have in her account than Sarah on their 16th birthday?
Jenny:_______ 
Sarah: _______ 

In Desmos or below, make a graph of both of their investments from birth until age 30. Label who’s function is whose.

What part of the function caused Jennie's graph to go up faster?

Approximately how long did it take for Jenny to have $2000 more than Sarah? about _______ years

If they both save until retirement at age 65, what will be their total balances (to the nearest dollar)? 
Jenny:_______ 
Sarah:_______ 
How much more will Jenny have than Sarah at that time?_______ 

Let’s say Jenny’s account had a management fee which brought down her returns by just 1%.
Write the new function j(x) for Jenny’s account value_______ 
How would the shape of the graph change based on this change?_______ 
What would not change about the graph?_______ 
Plot and label this new function on the same graph. Were your predictions correct?_______ 

How much did fees cost Jenny over the lifetime of her investment until age 65?_______ 

Bonus: Level 3 Application

The Power of Time
Shandra took a finance class in high school , and her twin brother Abdul did not. So when they both started working full-time jobs at the age of 25, Shandra invested in a retirement account right away. She put in $5,000 the first year in an account that has an average annual rate of return of 6%. Abdul waited 5 years until he was 30 before doing the same. In reality, they would both continue investing, but we will look at the growth of just this FIRST $5,000.

If x represents the number of years since age 25, write the equation for Shandra’s investment growth as a function of x._______ 

How would Abdul’s number of years invested compare to Shandra’s?  Abdul would have _______ years _______ (less/more) than Shandra. 

Using your answer to question 3, modify Shandra’s equation to write the equation for Abdul’s investment growth as a function of x._______ 

Graph and label both equations. What did your change in the exponent do to the graph? Screen shot your desmos graph in the show your work.

How much less does Abdul have in his account when they are both 40 years old?_______  How about when they’re both 60?_______ 

How much did waiting 5 years cost Abdul by the time he was 60?_______ 

If we took inflation into account, what would have happened to the $5000 if it was uninvested?_______  By what equation? _______ 

If inflation averages 2% annualized, the real value of this money is the value if the inflation is subtracted.
What parameter of the function would you need to adjust to account for inflation?_______ 
What would this do to the graph?_______ 
Write the new equations for Shandra_______  and Abdul _______ account for inflation