Exponential functions
Objective. Interpret exponential graphs, parameters and asymptotes.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
An exponential function has the form f(x) = a×bˣ, where the variable sits in the power rather than the base, so equal increases in x multiply the output by a constant factor b. When b > 1 the graph grows ever more steeply and when 0 < b < 1 it decays towards zero, always staying above the x-axis, which is a horizontal asymptote. The parameter a gives the y-intercept since f(0) = a. For example f(x) = 3×2ˣ starts at (0,3) and doubles every time x increases by 1, giving f(1) = 6 and f(2) = 12. Changing a stretches the curve vertically while b controls how fast it grows or decays.
Worked example
The exponential function f(x) = 5×2ˣ models a population in thousands after x years. Find f(3) and state the equation of the horizontal asymptote.
- 1Substitute x = 3 into f(x) = 5×2ˣ.
- 2Compute 2³ = 8.
- 3Multiply: f(3) = 5×8 = 40.
- 4As x → −∞, f(x) → 0, so the asymptote is y = 0.
Answer: f(3) = 40 (thousand); asymptote y = 0
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Practice
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Start practisingCommon mistake. Students often think the asymptote is at x = 0 rather than y = 0, confusing the vertical intercept with the horizontal asymptote. Remember y = bˣ never reaches zero however negative x becomes, so the asymptote is always the line y = 0.
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