Selecting & Constructing Exponential Models (College Board AP® Precalculus): Revision Note

Selecting & constructing exponential models

When is an exponential model appropriate?

• An exponential function models situations where successive output values over equal-length input-value intervals are proportional

  • i.e. they have a constant ratio

• When the input values are whole numbers

  • this corresponds to repeated multiplication of a constant factor applied to an initial value

• To determine whether data should be modeled by an exponential function

  • check whether the ratios of consecutive output values

  • for equally spaced input values

  • are approximately constant

• E.g. given the data $t=0,1,2,3$ with $f\left(t\right)=20,30,45,67.5$:

  • $\frac{f\left(1\right)}{f\left(0\right)}=\frac{30}{20}=1.5$, $\frac{f\left(2\right)}{f\left(1\right)}=\frac{45}{30}=1.5$, $\frac{f\left(3\right)}{f\left(2\right)}=\frac{67.5}{45}=1.5$

  • The ratio is constant at $1.5$, so an exponential model is appropriate

• Compare this with the tests for other function types

  • Linear: successive output values have a constant difference (1st differences are constant)

  • Quadratic: the 2nd differences in output values are constant

  • Exponential: successive output values have a constant ratio

What if the ratios aren't quite constant?

• Sometimes a constant needs to be added to (or subtracted from) the output values before the proportional pattern becomes visible

• E.g. the data $(1,7),(2,9),(3,13),(4,21)$ does not have constant ratios as is

  • But subtracting 5 from each output gives $(1,2),(2,4),(3,8),(4,16)$

  • for which the output values have a constant ratio of 2

    • I.e. $\frac{4}{2}=\frac{8}{4}=\frac{16}{8}=2$

• This means the original data can be modeled by a vertically shifted exponential function

  • I.e. an additive transformation of an exponential

• In general, if the output values of an additive transformation of a function are proportional over equal-length input-value intervals

  • then the function can be modeled by a transformation of an exponential function

How do I construct an exponential model from data?

• Method 1: From the initial value and ratio

  • If you can identify the initial value $a=f\left(0\right)$

    • and the constant ratio $b$ between consecutive outputs (for unit input intervals)

  • then the model is $f\left(x\right)=a{b}^x$

  • E.g. for the data above with $f\left(0\right)=20$ and ratio $b=1.5$

    • $f\left(t\right)=20{\left(\frac{3}{2}\right)}^t$

• Method 2: From two input-output pairs

  • If you have two data points $(x_1,y_1)$ and $(x_2,y_2)$,

    • you can construct the model $y=a{b}^x$ by solving a system of two equations

  • Substituting the two points gives

    • $y_1=a{b}^{x_1}$

    • $y_2=a{b}^{x_2}$

  • Dividing the second equation by the first eliminates $a$

    • $\frac{y_2}{y_1}=\frac{a{b}^{x_2}}{a{b}^{x_1}}=\frac{b^{x_2}}{b^{x_1}}=b^{(x_2-x_1)}$

  • You can then solve for $b$, and substitute back into $y=a{b}^x$ to find $a$

  • E.g. given the data points $(0,5)$ and $(3,40)$:

    • $\frac{40}{5}=\frac{a{b}^3}{a{b}^0}=\frac{b^3}{b^0}=b^3$

      • so $b^3=\frac{40}{5}=8\Rightarrow b=2$

    • Using $b=2$ and the point $\left(0,5\right)$ in $y=a{b}^x$

    • $5=a·b^0\Rightarrow a=5$

    • Model: $f\left(x\right)=5·2^x$

• Method 3: Applying transformations to $f\left(x\right)=a{b}^x$

  • An exponential model can be built by applying transformations based on the characteristics of the data or context

  • E.g. if data shows exponential growth starting from a value of 10 at $t=2$ with a growth factor of 3

    • the model might be $f\left(t\right)=10·3^{(t-2)}$

    • which is a horizontal shift of $y=10·3^t$

• Method 4: Using technology (exponential regression)

  • When data is not perfectly exponential

    • a graphing calculator can fit an exponential regression model of the form $y=a{b}^x$ to the data

  • This is done using the ExpReg function on most graphing calculators

    • The calculator determines the values of $a$ and $b$ that best fit the data using a least-squares method

    • The resulting model can then be used to predict output values at input values not in the original data set (including non-integer values)

  • See the first Worked Example for an example of this

What role does the natural base e play?

• The natural base $e\approx 2.718$ is commonly used in exponential models for real-world situations

• Any exponential function $f\left(x\right)=a{b}^x$ can be rewritten using base $e$

  • $f\left(x\right)=a{e}^{\left(\ln b\right)x}$

• Models using base $e$ appear frequently in science, economics, and engineering

  • e.g. continuous growth/decay, compound interest

How do I identify the correct model type on the exam?

• In exam questions, you are typically given a table of values and asked which function type best models the data

  • e.g. linear, quadratic, exponential, or logarithmic

• Check the 1st differences of the output values (for equally spaced inputs)

  • If these are constant, the model should be linear

• Check the 2nd differences of the output values (for equally spaced inputs)

  • If these are constant, the model should be quadratic

• Check the ratios of successive output values (for equally spaced inputs)

  • If these are constant, the model should be exponential

• If none of the above are constant, consider a logarithmic model

  • Here the input values change proportionally

    • while output values change additively

• To earn full credit you must give a reason that references specific values from the table

  • E.g. "The ratios of successive output values are constant at 0.5 over equal-length input-value intervals of 1, so an exponential model is best."