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Writing Exponential Functions Practice Problems

Writing an exponential function means finding two numbers, a and b. Two points are always enough: divide one equation by the other to get b, then substitute back for a. These twelve problems start from easy y-intercepts and finish with points that need roots and graphs with a moved asymptote.

  1. 1.

    Write the exponential function through and .

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    The y-intercept gives . One step in multiplies the output by , so and .

  2. 2.

    Write the exponential function through and .

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    and , so . A base below 1 means decay, which fits the output falling.

  3. 3.

    A table gives and . Write the exponential function.

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    Every ratio is , so . The value at is . So .

  4. 4.

    Write the exponential function through and .

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    . Then , so (the base must be positive). So .

  5. 5.

    Write the exponential function through and .

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    Divide: , so and . Substitute : , so . Answer: .

  6. 6.

    Write the exponential function through and .

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    Divide: , so and . Substitute : , so .

  7. 7.

    Write the exponential function through and .

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    , then , so and .

  8. 8.

    An exponential function passes through and . Find to four decimal places.

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    , so and . Keep the extra decimals: a rounded base gets magnified by the exponent.

  9. 9.

    Write the exponential function through and .

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    Divide: , so and . Substitute : , so .

  10. 10.

    Write the exponential function through and .

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    Divide: , so . Substitute : , so . Answer: .

  11. 11.

    A graph has horizontal asymptote and passes through and . Write its equation in the form .

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    . Measure heights above the asymptote: and . So and , giving .

  12. 12.

    A graph has horizontal asymptote and passes through and . Write its equation.

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    Heights above are and . So , , and .
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MPhil in Mathematics, Founder of Mathovia
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Writing Exponential Functions Practice Problems

The writing exponential functions lesson covers every method used here. Type your answers with the caret key for powers, such as 3*5^x; any equivalent form is marked correct.

The two-point method

Dividing cancels , which is what makes two points enough.

Quick checks

SituationShortcut
one point is is read directly
points one step apart is the ratio of the y-values
asymptote subtract from each y-value first

Always check the finished equation against both points before moving on.

Common mistakes to watch for

  • Using a percentage as the base. A 5% increase means a base of , not or .
  • Decay with the wrong base. A 20% decrease means , not .
  • Wrong starting value. In , is the value at .
  • Mixing time units. If growth is per year, must be in years.

Frequently Asked Questions

How do I find b from two points?+

Divide the two y-values. The ratio equals b raised to the gap between the x-values. If the points are one step apart the ratio is b itself; if they are n steps apart, take the nth root of the ratio.

What if neither point has x = 0?+

Find b first by dividing, which works for any pair of points. Then substitute b and either point into y = a·bˣ and solve for a.

Why do I reject a negative value of b?+

An exponential base must be positive. If b² = 4, both 2 and −2 square to 4, but only b = 2 gives a valid exponential function.

How should I type my answer?+

Type the function using ^ for powers, for example 3*5^x or 10*(1/2)^x. Equivalent forms such as 10*0.5^x are accepted, because answers are checked by value rather than by exact spelling.

What changes when the asymptote is not y = 0?+

The model becomes y = a·bˣ + k with k the asymptote. Subtract k from each y-value first, then find a and b exactly as usual.

More practice problems

Exponential Function Practice Problems Exponential Growth and Decay Practice Problems Graphing Exponential Functions Practice Problems