9th grade · Math · Unit 13 of 22
Linear or Exponential?
Add each time or multiply each time
What your child practices
Telling linear growth (add the same amount) from exponential growth (multiply by the same factor), writing exponential functions, and reading growth and decay rates.
New York State Learning Standards learning standard: AI-F.LE.1; AI-F.LE.2; AI-F.LE.5 · telling linear from exponential situations and comparing their growth
How we explain it
Children can open this short lesson before they practice.
- Linear: add the same amount each step.
- Exponential: multiply by the same factor each step.
- In y = a(b)ˣ, a is the start and b is the growth factor.
Example: A population of 100 doubles each year. How many after 3 years? 100 × 2 × 2 × 2. 100 × 8 = 800. Answer: 800
Linear or Exponential? in New York
In the New York State Learning Standards, this is unit 13 of 22 in 9th grade Math, after Solving Systems and before Factoring & Quadratics.
Gradelings describes a child's practice on this unit as Beginning, Approaching, Meeting or Exceeding (Four practice steps). That reflects practice, not a report-card mark: the teacher decides the mark.
Sample questions
A taste of this unit. In the app, questions change every time and get easier or harder to match your child.
Question 1
In V(t) = 2000(1.2)ᵗ, what does 1.2 tell you?
- It starts at 1.2
- It grows by 1.2 each time period
- It grows by 20% each time period
- It shrinks by 20% each time period
It grows by 20% each time period
Hint: 1.2 = 1 + 0.2, which is 20%.
Question 2
The number of bacteria doubles every 20 minutes. Which model fits best?
- Linear
- Exponential
Exponential
Hint: Adding the same amount each time is linear. Multiplying by the same factor (a percent change) is exponential.
Question 3
f(t) = 4(2)ᵗ. What is f(3)?
- 512
- 12
- 32
- 24
32
Hint: 2³ = 8. Then × 4.
Question 4
A population of 50 doubles every hour. Which function gives the population after t hours?
- P(t) = 50t²
- P(t) = 50 + 2t
- P(t) = 2(50)ᵗ
- P(t) = 50(2)ᵗ
P(t) = 50(2)ᵗ
Hint: Exponential: start value × (growth factor) to the power t.
Question 5
Is the pattern in the table linear or exponential?
x y 0 3 1 6 2 12 3 24 - Neither: there is no pattern
- Exponential: it is multiplied by 2 each time
- Linear: it goes up by 3 each time
Exponential: it is multiplied by 2 each time
Hint: Equal differences mean linear. Equal ratios (same multiplier) mean exponential.
Question 6
f(t) = 4(3)ᵗ. What is f(2)?
- 13
- 36
- 24
- 144
36
Hint: 3² = 9. Then × 4.