Cogito
AP Precalculus · Chapter 5 · Lesson 1
Exponential Functions
Constant ratios instead of constant differences.
12 problems · about 21 minutes · AP Precalculus 2.1, 2.2, 2.3
What this lesson teaches
The student identifies exponential functions from patterns and models exponential change.
- A linear function adds a constant; an exponential multiplies by one.
- Check differences for linear and ratios for exponential.
- The growth factor is 1 plus the rate, never the rate alone.
Warm Up
Straightforward practice. Get the method working first.
5 problems6, 12, 24, 48. Linear or exponential? 1 linear, 2 exponential.
Answer 2
Why Exponential.
What defines an exponential function?
Answer Equal input intervals give a constant ratio of outputs.
Why A constant ratio over equal intervals.
2, 6, 18, 54. Linear or exponential? 1 linear, 2 exponential.
Answer 2
Why Constant ratio of 3.
7, 12, 17, 22. Linear or exponential? 1 or 2?
Answer 1
Why Constant difference of 5.
f(x) = 100 · 2ˣ at x = 3. What is f?
Answer 800
Why 100 × 8.
Build It Up
The same ideas with more to keep track of.
3 problemsf(x) = 80 · 0.5ˣ at x = 3. What is f?
Answer 10
Why Halve three times.
A 40% annual fall. What is the decay factor?
Answer 0.6
Why 1 − 0.4.
f(x) = a · bˣ at x = 0. What is f, if a = 7?
Answer 7
Why b to the zero is 1.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsSort each pattern by the function type it indicates.
Answer Linear: Constant differences over equal intervals, A fixed amount added each step · Exponential: Constant ratios over equal intervals, A fixed percentage change each step
Why Adding against multiplying.
3, 9, 27, 81. What is the common ratio?
Answer 3
Why 9 ÷ 3.
The Ratio: 5, 10, 20, 40. What is the common ratio?
Answer 2
Why 2.
The Factor: A 25% annual rise. What is the growth factor?
Answer 1.25
Why 1.25.