An exponential function changes by a constant multiplicative factor over equal input intervals. This distinguishes it from a linear function, which changes by a constant additive amount.
1. The basic form
A common form is y=ab^x, where a is the value at x=0 and b is the multiplicative factor for an increase of 1 in x.
When a>0 and b>1, the function shows exponential growth. When a>0 and 0<b<1, it shows exponential decay.
2. Growth is multiplicative
For y=3(2^x), increasing x by 1 doubles the output. Values at x=0,1,2,3 are 3,6,12,24.
The differences are not constant, but the ratios are: each output is twice the previous one.
3. Decay uses a factor between zero and one
For y=80(0.75^x), each increase of 1 multiplies the output by 0.75. This corresponds to retaining 75% of the previous value, a decrease of 25% per step.
4. Percentage change to exponential factor
A 6% increase per period uses factor 1.06. A 6% decrease uses factor 0.94. The percentage applies to the current amount each time, not repeatedly to the original amount.
5. Graph behaviour
For y=ab^x with a>0 and b>0, b≠1, the output remains positive. The graph passes through (0,a). With no vertical translation, y=0 is a horizontal asymptote.
The graph approaches the asymptote but does not reach zero for any finite real x.
6. Vertical translations move the asymptote
For y=2^x+3, the horizontal asymptote is y=3. The whole basic exponential graph has moved upward by 3.
For y=5−2^x, the negative sign reflects the exponential output before the upward translation.
7. Worked model
A hypothetical quantity begins at 500 and increases by 8% each period. A model is Q=500(1.08^t).
After 3 periods, Q=500(1.08^3)≈629.86. The model assumes the same percentage rule continues over those periods; it does not prove that a real process must behave this way indefinitely.
8. Compare exponential and linear change
Starting from 100, adding 10 each period gives 100,110,120,130. Increasing by 10% gives 100,110,121,133.1. The first has constant differences; the second has constant ratios.
9. Common mistakes
- Using 0.08 instead of 1.08 for 8% growth.
- Treating repeated percentage change as constant addition.
- Calling y=0 an intercept when it is only an asymptote.
- Extending a model beyond its justified range without comment.
10. Practice
- State whether y=4(1.3^x) represents growth or decay.
- Write the factor for a 12% decrease per period.
- Find y when x=3 for y=5(2^x).
- State the horizontal asymptote of y=3^x−4.
- A quantity begins at 200 and grows by 5% per period. Write a model.
11. Answers
- Growth.
- 0.88.
- 40.
- y=−4.
- Q=200(1.05^t).
Continue through the Logarithms and Exponentials guide, Linear Law, or return to the Additional Mathematics Hub.