AP® Precalculus review sheet from Aim for Five (aimforfive.com/precalc/units/2/2-2)
Unit 2 · Topic 2.2
2.2 Change in Linear and Exponential Functions
Linear functions add a constant amount over equal input steps, and exponential functions multiply by a constant ratio. They are the continuous versions of arithmetic and geometric sequences. You'll write each from two points and tell them apart from a table.
Key terms
- linear function
- exponential function
- constant ratio
- proportional change
- point-slope form
Linear functions are like arithmetic sequences
A linear function f(x) = b + mx starts at the initial value b and adds the slope m for every 1-unit increase in x, just like aₙ = a₀ + dn adds d each step.
Point-slope form matches aₙ = aₖ + d(n − k): if you know a point (xᵢ, yᵢ) and the slope m, then f(x) = yᵢ + m(x − xᵢ).
Exponential functions are like geometric sequences
An exponential function f(x) = a · bˣ starts at the initial value a and multiplies by b for every 1-unit increase in x, just like gₙ = g₀ · rⁿ.
The matching point form: if you know a point (xᵢ, yᵢ) and the ratio b, then f(x) = yᵢ · b^(x − xᵢ).
The big difference is the domain. A sequence uses only whole numbers, but a linear or exponential function accepts every real number, so its graph is a connected line or curve.
Telling them apart from data
Look at outputs over equal-length input intervals:
- If the outputs change by a constant amount (constant differences), the function is linear.
- If the outputs change by a constant ratio (each is the same multiple of the one before), the function is exponential. This is called proportional change.
- For an exponential function, the ratio can't be 0 or 1. A ratio of 1 would just be a constant function.
Two points are enough
Arithmetic sequences, geometric sequences, linear functions and exponential functions are each determined by just two values. That's why so many exam questions give you two points and ask for the function.
For a line, divide the change in output by the change in input to get m. For an exponential function, divide the outputs to get the total ratio, then take a root to get the ratio per 1 unit of input.
Linear is built on repeated addition; exponential is built on repeated multiplication. If you remember only one thing, remember that.
Same start, different growth
Two savings plans both start at 100 dollars. Plan A adds 20 dollars each year, so A(t) = 100 + 20t. Plan B grows 15% each year, so B(t) = 100(1.15)ᵗ. At first the linear plan is ahead, because 15% of 100 dollars is only 15 dollars. But B's yearly gain keeps growing while A's stays at 20 dollars.
B passes A between years 4 and 5. An increasing exponential function eventually passes any linear function, no matter how steep the line is.
| Year t | Plan A: 100 + 20t | Plan B: 100(1.15)ᵗ |
|---|---|---|
| 0 | 100 | 100 |
| 1 | 120 | 115 |
| 4 | 180 | 174.90 |
| 5 | 200 | 201.14 |
| 10 | 300 | 404.56 |
Worked examples
Try each one yourself first, then open the solution.
- Example 1
Linear and exponential through the same points
Write (a) a linear function and (b) an exponential function through (2, 18) and (5, 486).
Show the solutionHide the solution
- Step 1: (a) Slope m = (486 − 18)/(5 − 2) = 468/3 = 156. Point-slope form: f(x) = 18 + 156(x − 2).
- Step 2: (b) From x = 2 to x = 5 is 3 units, so 18 · b³ = 486. Then b³ = 27 and b = 3.
- Step 3: Point-ratio form: g(x) = 18 · 3^(x − 2).
- Step 4: To get a · bˣ form, find the initial value: g(0) = 18 · 3⁻² = 18/9 = 2. So g(x) = 2 · 3ˣ. Check: 2 · 3⁵ = 486.
Answer: (a) f(x) = 18 + 156(x − 2); (b) g(x) = 18 · 3^(x − 2) = 2 · 3ˣ.
- Example 2
Deciding from a table
A function h has h(0) = 5, h(2) = 20, h(4) = 80 and h(6) = 320. Is h linear or exponential? Write a formula.
Show the solutionHide the solution
- Step 1: The inputs go up by 2 each time, so the intervals are equal.
- Step 2: Differences: 15, 60, 240. Not constant, so not linear.
- Step 3: Ratios: 20/5 = 4, 80/20 = 4, 320/80 = 4. Constant, so exponential.
- Step 4: The ratio 4 is for 2 units of input. Per 1 unit, b² = 4, so b = 2. The initial value is h(0) = 5.
Answer: Exponential: h(x) = 5 · 2ˣ.
- Example 3
Trap: unequal steps hide the pattern
A table shows k(1) = 6, k(2) = 12, k(4) = 48, k(5) = 96. A student says the ratios 2, 4, 2 aren't constant, so k isn't exponential. Is that right?
Show the solutionHide the solution
- Step 1: The input steps are 1, 2 and 1. Ratios can only be compared directly over equal-length intervals.
- Step 2: Convert to a ratio per 1 unit. Over the 2-unit step, the ratio is 4 = 2², which means 2 per unit. Every interval gives a ratio of 2 per unit.
- Step 3: So k is exponential with b = 2. Using k(1) = 6: k(x) = 6 · 2^(x − 1) = 3 · 2ˣ.
Answer: No. k(x) = 3 · 2ˣ fits every value.
Common mistakes
- Dividing the total ratio by the number of steps instead of taking a root. If the output is multiplied by 27 over 3 units, b is 3, not 9.
- Comparing differences or ratios when the input steps aren't equal.
- Mixing up the forms: the linear point form adds m(x − xᵢ), while the exponential point form multiplies by b^(x − xᵢ).
On the exam
- Expect to write a linear or exponential function from two points, or to justify which type fits a table: “over equal input intervals the outputs change by a constant ratio, so exponential.”
- On the non-periodic modeling free-response question, you may need to find the constants of an exponential model from two given values, by hand or with a calculator.
Connected topics
Videos
Check yourself
4 questions on 2.2 Change in Linear and Exponential Functions. Pick an answer to see if you got it, and why.
| x | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| f(x) | 5 | 8 | 11 | 14 |
| g(x) | 5 | 7.5 | 11.25 | 16.875 |
Table of values
Which of the following statements is consistent with the table?
Which of the following is an expression for g(x)?
An exponential function f satisfies f(2) = 18 and f(5) = 486. Which of the following is an expression for f(x)?
A linear function f and an exponential function g both pass through the points (0, 3) and (2, 12). What is the value of g(4) − f(4)?
0 of 4 answered