Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Looking for algebra 1 exponential functions? Then this Algebra 1 lesson on Exponential Growth and Decay Models explains each idea step by step, with worked examples.
Quick answer: An exponential function has the form y = a · b^x, where a is the starting value and b is the factor applied each time x increases by 1. When b is greater than 1 the quantity grows, and when b is between 0 and 1 it decays. Linear functions add equal amounts, and exponential functions multiply by equal factors.
Standards: F-LE.A.1a, F-LE.A.1c, F-LE.A.2, F-LE.A.3, F-IF.C.8b, F-LE.B.5 · Level: Grade 9 Algebra 1
What makes a function exponential?
A linear function adds the same amount each time x goes up by 1. An exponential function multiplies by the same factor each time. Compare a savings jar that gains $8 per week with a viral video whose views double every day.
A function of the form y = a · bx with a ≠ 0, b > 0 and b ≠ 1. The number a is the initial value (the y-value at x = 0). The number b is the growth or decay factor.
How do you tell linear from exponential in a table?
Look at equal steps in x. Equal differences in y mean linear. Equal ratios (each y divided by the one before it) mean exponential.
| Type | Check | Equation |
|---|---|---|
| Linear | y-differences are equal | y = mx + c |
| Exponential | y-ratios are equal | y = a · bx |
What do growth and decay look like?
If b > 1 the graph rises; if 0 < b < 1 it falls toward the x-axis without touching it. Write b = 1 + r for a growth rate r, and b = 1 − r for a decay rate r.
With interest added once per year, A = P(1 + r)t. Here P is the starting amount, r is the yearly rate as a decimal and t is the number of years.
How do you write an exponential model?
- Find the initial value a (the value when x = 0).
- Divide any y-value by the previous one to get b.
- Write y = a · bx and test it on a second point.
A trail camera uploads 6, 9, 13.5 photos on days 0, 1, 2. Then a = 6 and b = 9 ÷ 6 = 1.5, so y = 6(1.5)x. Check: 6(1.5)2 = 13.5.
A 20% decay is b = 0.8, not 0.2. Subtract the rate from 1.
Why does an exponential eventually beat a line?
A line gains a fixed amount, while a curve with b > 1 gains more each step. However steep the line, some value of x exists where the curve is higher for good.
Ask “plus or times?” Same amount added means linear. Same multiplier means exponential.
- Linear: equal differences in y. Exponential: equal ratios in y.
- In y = a · b^x, a is the initial value and b is the factor per step.
- Growth rate r gives b = 1 + r; decay rate r gives b = 1 − r.
- Compound interest once a year: A = P(1 + r)^t.
- Any exponential with b above 1 eventually exceeds any linear function.
Exponential Growth and Decay Models: frequently asked questions
How can you tell if a table is exponential?
Divide each y-value by the one before it for equal steps in x. If every ratio is the same number, the table is exponential and that number is the factor b.
What does the number a mean in y = a · b^x?
It is the initial value, the y-value when x = 0. On a graph it is the y-intercept, the point where the curve crosses the y-axis.
How do you turn a percent rate into the factor b?
For growth, add the rate (as a decimal) to 1, so 7% growth gives b = 1.07. For decay, subtract it from 1, so 7% decay gives b = 0.93.
Does an exponential graph ever touch the x-axis?
No. When a is positive, the values of a · b^x stay positive for every x. A decay curve gets closer to the x-axis forever but never reaches it.
Why does exponential growth beat linear growth?
A linear function adds the same amount every step. An exponential function adds a bigger amount every step, so it eventually passes any line, no matter how steep.
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