Exponential functions model situations where a quantity is repeatedly multiplied by the same factor. They appear in population growth, radioactive decay, compound interest, cooling, and the spread of information. Their graphs can rise or fall very quickly, which makes them important for understanding change that is not constant.
The basic form is f(x) = a b^x, where the base b controls the pattern of growth or decay.
When b is greater than 1, the function shows exponential growth because each step multiplies the output by a factor larger than 1. When 0 < b < 1, the function shows exponential decay because each step multiplies the output by a fraction. The y-intercept is often the starting value, and the horizontal asymptote shows the value the graph approaches but does not cross in the parent form.
Comparing the bases of exponential functions helps predict which graph grows faster or decays faster.
Key Facts
- General form: f(x) = a b^x, where a is the initial value and b is the constant multiplier.
- Exponential growth occurs when b > 1.
- Exponential decay occurs when 0 < b < 1.
- For f(x) = a b^x, the y-intercept is f(0) = a.
- The parent functions y = b^x have horizontal asymptote y = 0.
- For equal x-steps, exponential functions have equal output ratios, not equal output differences.
Vocabulary
- Exponential function
- A function in which the variable appears in the exponent, usually written as f(x) = a b^x.
- Base
- The number b in f(x) = a b^x that is repeatedly multiplied as x changes by 1.
- Growth factor
- A base greater than 1 that causes the output of an exponential function to increase as x increases.
- Decay factor
- A base between 0 and 1 that causes the output of an exponential function to decrease as x increases.
- Horizontal asymptote
- A horizontal line that a graph approaches closer and closer but may never reach.
Common Mistakes to Avoid
- Treating exponential growth like linear growth is wrong because exponential functions multiply by a constant factor instead of adding a constant amount.
- Using a negative base without caution is wrong because many exponential models require b > 0, and negative bases can create patterns that are not smooth exponential graphs over all real numbers.
- Forgetting that decay bases are between 0 and 1 is wrong because a base like 0.8 means the value keeps 80 percent each step, not that it decreases by 0.8 each step.
- Confusing the y-intercept with the asymptote is wrong because f(0) = a gives the starting output, while the horizontal asymptote is the value the graph approaches.
Practice Questions
- 1 For f(x) = 3(2)^x, find f(0), f(1), f(2), and f(3). State whether the function shows growth or decay.
- 2 A substance has mass M(t) = 80(0.5)^t grams after t hours. Find the mass after 0, 1, 2, and 3 hours.
- 3 Compare y = 4(1.5)^x and y = 4(0.7)^x. Explain which graph shows growth, which shows decay, and how their shapes differ as x increases.