Exponential graphs
Objective. Interpret growth, decay, intercepts and asymptotic behaviour.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Exponential graphs show curves of the form y=ab^x, where growth occurs if b>1 and decay occurs if 0<b<1, and they always pass through the point (0,a) since any base raised to the power zero is 1. As x increases, growth curves rise ever more steeply while decay curves approach the x-axis but never actually touch it, making y=0 a horizontal asymptote. The value of a determines the y-intercept, and changing b alters how quickly the curve grows or shrinks. For example, y=5×2^x has y-intercept (0,5) and doubles in value every time x increases by 1.
Worked example
The graph of y=3×2^x is sketched. State its y-intercept and its horizontal asymptote.
- 1Find the y-intercept by substituting x=0: y=3×2⁰=3×1=3
- 2The y-intercept is therefore (0, 3)
- 3As x→−∞, 2^x→0, so y→3×0=0
- 4The horizontal asymptote is the line y=0
Answer: y-intercept (0, 3); asymptote y=0
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Practice
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Start practisingCommon mistake. Pupils sometimes think an exponential decay curve eventually reaches y=0 and crosses the x-axis, but it only ever approaches the asymptote y=0 without ever touching or crossing it, however large x becomes.
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