Exponential Functions
An exponential function has the variable in the exponent. Its form is f(x) = a \cdot b^x, where a is the starting value and b is the number you multiply by each time x goes up by 1. For example, f(x) = 3 \cdot 2^x doubles at every step.
Before this lesson: You should know function notation, the exponent rules, and zero and negative exponents.
What Makes a Function Exponential
Fold a sheet of paper in half. It is now 2 layers thick. Fold again and you have 4 layers, then 8, then 16. Each fold multiplies the layers by 2.
After x folds, there are 2^x layers. The number of folds is in the exponent. That is what makes the function exponential.
A linear function adds the same amount each step. An exponential function multiplies by the same amount each step. That number is the base, b.
The Exponential Function Form
f(x) = a \cdot b^x
| Part | Meaning | In f(x) = 3 · 2ˣ |
|---|---|---|
| a | The starting value, or y-intercept. It is the output when x = 0. It can’t be 0. | a = 3 |
| b | The base. Multiply by it each time x goes up by 1. It must be positive and not 1. | b = 2 |
| x | The input. It sits in the exponent. | x |
If b is greater than 1, the function grows. This is exponential growth. If b is between 0 and 1, the function shrinks. This is exponential decay.
How to Tell If a Table Is Exponential
- Check that the x-values go up by the same amount, such as 1.
- Divide each y-value by the one before it. This is the ratio.
- If every ratio is the same, the table is exponential. That ratio is b.
- Find a, the y-value when x = 0. Then write f(x) = a \cdot b^x.
Worked Examples
Example 1: Spot the Exponential Functions
Which are exponential functions? (a) y = 4 \cdot 3^x, (b) y = x^3, (c) y = 10(0.5)^x, (d) y = 5x + 2 and (e) y = 6 \cdot 1^x.
(a) Yes. The variable is in the exponent, a = 4 and b = 3.
(b) No. The variable is in the base, not the exponent.
(c) Yes. a = 10 and b = 0.5, so it is decay.
(d) No. This is linear. It adds 5 each step.
(e) No. 1^x = 1 for every x, so y = 6. That is a flat line.
Answer: (a) and (c)
Example 2: Evaluate an Exponential Function
Let f(x) = 4 \cdot 3^x. Find f(2), f(0) and f(-1).
Step 1: f(2) = 4 \cdot 3^2 = 4 \cdot 9 = 36. Do the exponent first, then multiply.
Step 2: f(0) = 4 \cdot 3^0 = 4 \cdot 1 = 4. This is the starting value, a.
Step 3: f(-1) = 4 \cdot 3^{-1} = 4 \cdot \dfrac{1}{3} = \dfrac{4}{3}. The output is still positive.
Answer: f(2) = 36, f(0) = 4, f(-1) = \dfrac{4}{3}
Example 3: Write a Function from a Table
Decide if each table is exponential. If it is, write its function.
| x | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| Table A | 5 | 10 | 20 | 40 |
| Table B | 81 | 27 | 9 | 3 |
| Table C | 2 | 5 | 8 | 11 |
Table A: The ratios are 10 \div 5 = 2, 20 \div 10 = 2 and 40 \div 20 = 2. They match, so b = 2. At x = 0, y = 5. So f(x) = 5 \cdot 2^x.
Table B: The ratios are 27 \div 81 = \tfrac{1}{3}, then \tfrac{1}{3} and \tfrac{1}{3}. So f(x) = 81\left(\tfrac{1}{3}\right)^x. This is decay.
Table C: The ratios are \tfrac{5}{2} and \tfrac{8}{5}. They don’t match, so it is not exponential. It adds 3 each time, so it is linear.
Answer: A is f(x) = 5 \cdot 2^x; B is f(x) = 81\left(\tfrac{1}{3}\right)^x; C is not exponential.
Example 4: Graph Growth and Decay
Graph f(x) = 2^x and g(x) = 8\left(\tfrac{1}{2}\right)^x. Give the domain and range.
Step 1: Make a table for each function.
| x | −1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|
| f(x) = 2^x | 1/2 | 1 | 2 | 4 | 8 |
| g(x) = 8\left(\tfrac{1}{2}\right)^x | 16 | 8 | 4 | 2 | 1 |
Step 2: Plot the points and join them with a smooth curve. f rises from left to right. g falls.
Step 3: Read the features. The y-intercepts are (0, 1) and (0, 8). Each curve gets closer and closer to the x-axis but never touches it. The line y = 0 is called a horizontal asymptote.
Answer: For both functions, the domain is all real numbers and the range is y \gt 0.
Example 5: Word Problem
A new video has 300 views. The number of views triples every day for the first week. Write a function for the views V after d days. Then find the views on day 4. On which day does it first pass 20,000 views?
Step 1: The start is 300, so a = 300. Tripling means b = 3. So V(d) = 300 \cdot 3^d.
Step 2: V(4) = 300 \cdot 3^4 = 300 \cdot 81 = 24{,}300.
Step 3: Check the day before: V(3) = 300 \cdot 27 = 8{,}100. That is less than 20,000, so day 4 is the first day over.
Answer: V(d) = 300 \cdot 3^d. Day 4 has 24,300 views, and day 4 is the first day over 20,000.
Common Mistakes
| Mistake | Example of the error | Fix |
|---|---|---|
| Multiplying a and b first | 3 \cdot 2^x = 6^x | Do the exponent first: 3 \cdot 2^2 = 3 \cdot 4 = 12, not 6^2 = 36 |
| Mixing up x² and 2ˣ | Calling y = x^2 exponential | Exponential means the variable is in the exponent |
| Checking differences instead of ratios | 2, 6, 18 has differences 4 and 12, so “not a pattern” | Divide: 6 \div 2 = 3 and 18 \div 6 = 3. It is exponential with b = 3 |
| Thinking the graph crosses the x-axis | “2^x = 0 when x is very negative” | 2^x gets close to 0 but is always positive |
| Using the first table value as a | Table starts at x = 1, y = 6, so a = 6 | a is the value at x = 0. Work back one step by dividing by b |
Growth vs. Decay at a Glance
| Feature | Growth (b \gt 1) | Decay (0 \lt b \lt 1) |
|---|---|---|
| Example | y = 2^x | y = \left(\tfrac{1}{2}\right)^x |
| Graph from left to right | Rises, faster and faster | Falls, slower and slower |
| Each step | Multiply by more than 1 | Multiply by a number between 0 and 1 |
| y-intercept | (0, a) | (0, a) |
| Horizontal asymptote | y = 0 | y = 0 |
| Domain and range (when a > 0) | All real numbers; y \gt 0 | All real numbers; y \gt 0 |
Real growth and decay are often given as a percent, like “grows 5% per year.” That is the next lesson, Exponential Growth and Decay. Later in this unit, Linear vs. Exponential Functions compares a constant ratio with a constant average rate of change. The domain and range ideas come from domain and range.
Practice Problems
- Is y = 7(1.5)^x an exponential function? If so, give a and b, and say if it is growth or decay.
- Let f(x) = 2 \cdot 5^x. Find f(3).
- Let f(x) = 6\left(\tfrac{1}{2}\right)^x. Find f(-2).
- Write a function for this table: x = 0, 1, 2, 3 and y = 3, 12, 48, 192.
- Is this table exponential? x = 1, 2, 3, 4 and y = 1, 4, 9, 16.
- Give the y-intercept of f(x) = 9\left(\tfrac{2}{3}\right)^x. Is it growth or decay?
- What is the range of f(x) = 5 \cdot 2^x?
- An exponential function passes through (0, 4) and (1, 12). Write its equation.
- Let h(x) = 160\left(\tfrac{1}{2}\right)^x. Find h(4).
- A lab starts with 25 cells. The number of cells doubles every hour. Write a function for the cells after h hours. How many cells are there after 5 hours?
Show answers
- Yes. a = 7 and b = 1.5. Since 1.5 \gt 1, it is growth.
- f(3) = 2 \cdot 5^3 = 2 \cdot 125 = 250
- f(-2) = 6\left(\tfrac{1}{2}\right)^{-2} = 6 \cdot 4 = 24
- The ratio is 12 \div 3 = 4 each time, and a = 3. So f(x) = 3 \cdot 4^x.
- No. The ratios are 4, \tfrac{9}{4} and \tfrac{16}{9}, which don’t match. This table is y = x^2.
- The y-intercept is (0, 9). Since 0 \lt \tfrac{2}{3} \lt 1, it is decay.
- y \gt 0. The outputs are always positive.
- a = 4. Then b = 12 \div 4 = 3. So f(x) = 4 \cdot 3^x.
- h(4) = 160 \cdot \tfrac{1}{16} = 10
- C(h) = 25 \cdot 2^h. After 5 hours: 25 \cdot 2^5 = 25 \cdot 32 = 800 cells.
FAQ
What makes a function exponential?
The variable is in the exponent, as in f(x) = a \cdot b^x. In a table, the outputs are multiplied by the same number each time x goes up by 1.
What is the horizontal asymptote of an exponential function?
For f(x) = a \cdot b^x, it is the line y = 0. The graph gets closer and closer to it but never touches it.
Why can’t the base be negative or 1?
With b = 1, every output is the same, so the graph is a flat line. With a negative base, the signs jump back and forth, as in (-2)^1 = -2 and (-2)^2 = 4, so there is no smooth curve.
How do you tell growth from decay?
Look at the base. If b \gt 1, it is growth. If b is between 0 and 1, it is decay.
Summary
- An exponential function has the form f(x) = a \cdot b^x, with the variable in the exponent.
- a is the starting value and the y-intercept. b is the number you multiply by each step.
- Growth when b \gt 1; decay when 0 \lt b \lt 1.
- In a table, the ratios between outputs are all the same.
- The graph never touches its horizontal asymptote, y = 0.
Previous lesson: Zero and Negative Exponents
Next lesson: Exponential Growth and Decay
Back to the Algebra 1 Index