Applying Exponential Models (College Board AP® Precalculus): Study Guide

Applying exponential models

What does the base of an exponential model tell us?

• For an exponential model in general form $f\left(x\right)=a{b}^x$

  • the base $b$ is the growth factor

    • it tells you by what factor the output is multiplied each time the input increases by 1 unit

• The growth factor is directly related to the percent change per unit

  • If $\mathit{b}\mathbf{>}\mathbf{1}$, the quantity is growing

    • and the percent increase per unit is $(b-1)\times 100\%$

    • E.g. $b=1.08$ means an 8% increase per unit input

  • If $\mathbf{0}\mathbf{<}\mathit{b}\mathbf{<}\mathbf{1}$, the quantity is decaying

    • and the percent decrease per unit is $(1-b)\times 100\%$

    • E.g. $b=0.95$ means a 5% decrease per unit input

• It is important to remember that the growth factor applies per unit of the input variable

  • E.g. if the model is $P\left(t\right)=500(1.03{)}^t$

  • and $t$ is in years

    • then the quantity grows by 3% per year

• If you need the growth rate per a different time period, you need to rewrite the model (see below)

How do I handle growth rates over non-unit intervals?

• A common exam scenario is when the growth rate is given per one time unit but the model uses a different time unit

  • The key is to express the exponent so that it counts the number of growth periods

• E.g. a population grows by 5% per month, and $t$ is measured in years

  • The monthly growth factor is $1.05$

    • and in $t$ years there are $12t$ months

  • So the model is $P\left(t\right)=P_0(1.05{)}^{12t}$

• E.g. a substance decays by 10% per decade, and $t$ is measured in years

  • The decay factor per decade is $0.90$

    • and in $t$ years there are $\frac{t}{10}$ decades

  • So the model is $A\left(t\right)=A_0(0.90{)}^{\frac{t}{10}}$

• A common error is confusing the exponent direction

  • E.g. if the rate is per quarter and $t$ is in years the exponent should be $4t$ (more growth periods per year), not $t/4$

    • $t/4$ would be correct if the rate were per year and $t$ were per quarter

How can equivalent forms of an exponential function reveal different information?

• The same exponential function can be written in equivalent forms that highlight different properties

  • The general principle is that ${\mathit{b}}^{\mathit{x}}\mathbf{=}\mathbf{(}{\mathit{b}}^{\mathit{k}}{\mathbf{)}}^{\mathit{x}\mathbf{/}\mathit{k}}$

    • for any nonzero constant $k$

• This means you can rewrite the base to show the growth factor over any time interval

• E.g. $f\left(d\right)=2^d$ (quantity doubles every day, where $d$ is in days)

  • Weekly form: $f\left(d\right)=(2^7{)}^{d/7}={128}^{d/7}$

    • the quantity grows by a factor of 128 every 7 days (every week)

  • Hourly form: $f\left(d\right)=(2^{1/24}{)}^{24d}$

    • the quantity grows by a factor of $2^{1/24}$ every hour

• E.g. A radioactive substance has a half-life of 8 days

  • This means that every 8 days the initial amount is multiplied by $\frac{1}{2}=0.5$

  • The decay model is $h\left(d\right)=A_0(0.5{)}^{d/8}$, where $d$ is in days

    • Every 8 days, the amount is multiplied by 0.5

  • To convert to $t$ measured in hours

    • There are 24 hours in a day

      • so $t=24d\Rightarrow d=t/24$

    • Therefore

      • $k\left(t\right)=A_0(0.5{)}^{(t/24)/8}=A_0(0.5{)}^{t/192}=A_0{\left(0.5^{1/192}\right)}^t$

    • This shows the decay factor per hour is $0.5^{1/192}$

  • To convert to $w$ measured in weeks

    • There are 7 days in a week

      • so $d=7w$

    • Therefore

      • $h\left(w\right)=A_0(0.5{)}^{7w/8}$

    • This shows the decay factor per week is $0.5^{7/8}$

• In general, converting between time units involves

  • substituting the relationship between the units into the exponent

    • then simplifying

How do I use an exponential model to make predictions?

• Once you have an exponential model, you can predict output values

  • by substituting input values into the function

• You can also predict input values

  • by substituting output values into the function

  • and solving to find the corresponding input value

• There are a number of things to keep in mind

  • Predictions are only meaningful within the contextual constraints of the domain

    • E.g. if $t$ represents time since an event, negative values of $t$ may not make sense

  • Models may not remain valid indefinitely

    • Real-world growth and decay often level off or change behavior over time

  • The model gives only approximate predictions

    • especially when based on a regression or limited data

  • The model may not remain valid beyond the range of data used to construct it

    • Just as with any model