Exponential Functions

Graphing Exponential Functions

100

Does the function 

  f(x) = -2/9(5/4)^x

represent exponential growth or decay?  

Exponential growth

200

Identify the y-intercept of the function:

f(x) = -6(1/4)^x+3

(0, -0.09)

300

What is the end behavior of the graph of an exponential decay function?

As x approaches infinity, f(x) approaches the asymptote.

As x approaches negative infinity, f(x) approaches infinity.  

400

Graph y = 4(1/4)^x

On paper.  

500

Look at the base 2 graph.  Write the equation of the function.  

f(x) = (2)^x+3 -5

Transformations of Exponential Functions

100

How is the function y = 2^x transformed to create the graph of y = -2^(x+3) - 4?

Translated left 3 units, reflected over the x-axis, and translated down 4 units.

200

Write an example of a function that is translated right 3 units and up 5 units.  

Answers vary.

300

Write an example of a function that shows a reflection in x and another example that shows a reflection in y.  

Answers vary.  

400

Describe all the transformations applied to 

f(x) = 4^x  to obtain the function   f(x) = -2(4)^x

How do these transformations affect the range of the function? 

There are two transformations:

1. A vertical stretch by a factor of 2. 
2. A reflection across the x-axis.

Effect on the Range:  The original range was y> 0.  The new range is y<0.  

500

What are two ways that a vertical stretch is different from a horizontal stretch?

For a vertical stretch, the number that is multiplied to the base is NOT in the exponent.

For a vertical stretch, the number that is multiplied in between 0 and 1 (a proper fraction).  

Logarithms

100

Rewrite the equation in exponential form:

log₄64=3

4³=64

200

Write the equation  in logarithmic form:

2^-3= 1/8

log₂(1/8) =  -3

300

Evaluate using your knowledge of exponents (no calculators):

log₂₅5

1/2

400

Write the expression that is equivalent to:  log₇50 using the change of base formula and common logarithms.  Do not evaluate.  

log 50 / log 7

500

Solve the equation for x:    ln(x-4)=2

Show your work!

Approximately   11.389

e

100

Clue: The exponential form of the equation  lnx=5

e⁵=x

200

What are the horizontal asymptote and the 

y-intercept of the function f(x)=e^x+1-4

The asymptote is    y=-4

The y-intercept is approx. -1.28

300

A $1,000 investment in a Vanguard index fund grows at 8.1%.  What is it's value in 5 years?

Approximately $1,500.  

400

Students at B-P started coming down with a highly contagious virus. After t days, the number of people, P, who have been infected by the virus is modeled by the equation on your template. How many people have the virus after 7.95 years? 

500 people

500

Write the exponential function and fill in the blank.

x: 1, 2, 3, 4

y: 1536, 384, 96, ___

y=6144*0.25^x, y = 24

Exponential Growth & Decay

100

Write a simplified exponential function for the scenario: an investment of $20,000 earns 2.5% interest per year.

y = 20000(1.025)^x

200

A town starts with a population of 1,000 people, and it grows at a constant annual rate of 8%. This equation, f(t)=1000(1.08)^t models the population over time in years.  How much time has passed when the town has 1,469 people?

5 years

300

Tim's red F-150 was worth $20,000 when he bought it in 2022. It depreciates in value by 15% each year.  How much is it worth this year?

$12,285.42

400

Carbon-14 has a half-life of 5,730 years. Brennan found an ancient bone that initially contained 100 milligrams of Carbon-14 and determined it was 11,460 years old.  How many milligrams did it have when she found it? 25 milligrams.

25 milligrams

100(0.5)^t/5730

500

A bank account earns 4% interest compounded continuously. If you invest $500, how long would it take to double your money?

17.3286 years

Use this equation:  1000=500e^0.04t