Introduction to Exponential Functions

Show your work for each problem.

1.
Which function is exponential?
1. a)
2. b)
3. c)
4. d)
Answer:
is exponential because the variable is in the exponent. The others are polynomial (, ) or linear ().
2.
For , what is the y-intercept?
1. a)
2. b)
3. c)
4. d)
Answer:
. The y-intercept of is always .
3.
Evaluate
Answer: 16
4.
If , find
Answer: 9
5.
Which describes exponential growth?
1. a)
2. b)
3. c)
4. d)
Answer:
shows growth because . When (like ), it's decay. When , it's constant. Negative bases aren't used for exponential functions.
6.
Evaluate . Give your answer as a decimal.
Answer: 0.125
. This is also .
7.
What is the horizontal asymptote of ?
1. a)
2. b)
3. c)
4. d)
Answer:
For any exponential function with and , the horizontal asymptote is always . The function approaches but never reaches zero.
8.
A population doubles every year. If you start with 100, find the population after 4 years.
Answer: 1600
1. What is the growth factor (base) if the population doubles? 2
2. Write the exponential model: . What goes in the blank? 2
3. What is ? 16
4. Calculate 1600
9.
A radioactive substance has a half-life of 1 hour. If you start with 80 grams, how much remains after 3 hours?
Answer: 10
1. What is the decay factor if half remains each hour? 0.5
2. How many half-lives is 3 hours? 3
3. What is ? 0.125
4. Calculate 10
10.
If , find . Give your answer as a fraction or decimal.
Answer: 4/3
11.
You invest 500 dollars at 10% annual compound interest. How much will you have after 2 years?
Answer: 605
1. What is the growth factor where ? 1.10
2. What is ? 1.21
3. Calculate 605
12.
Write the exponential function that passes through and .
Answer: f(x) = 4 * 3^x
1. From the point , what is the value of in ? 4
2. Using : . Solve for . 9
3. What is if and ? 3
4. Write the final function 4 * 3^x
13.
Which statement about is TRUE?
1. a)The horizontal asymptote is
2. b)The y-intercept is 0.8
3. c)The function shows exponential growth
4. d)The function shows exponential decay
Answer: The function shows exponential decay
Since , this is exponential decay. The y-intercept is , and the horizontal asymptote is always (not ).
14.
A car depreciates by 15% each year. If it's worth 20000 dollars now, what will it be worth in 3 years? Round to the nearest dollar.
Answer: 12283
dollars.
15.
Compare: Which is larger, or ?
Answer: 2^10
1. Calculate 100
2. Calculate 1024
3. Which is larger? 2^10