SAT Math 5: Nonlinear Functions

Exponential function

A function of the form $f(x)=ab^x$, where $a\neq 0$ and $b>0$ with $b\neq 1$. The variable appears in the exponent, while the base is a positive constant.

What conditions on the base are required for $f(x)=ab^x$ to be an exponential function?

The base $b$ must be positive and must not equal $1$. A positive base ensures real outputs for every real exponent, and $b=1$ would produce a constant function.

Why is $f(x)=x^3$ not an exponential function?

The variable $x$ is the base and the exponent is constant. This is a power function, not an exponential function, because exponential functions have a constant base and a variable exponent.

Why is $f(x)=(-2)^x$ excluded from the usual definition of a real exponential function?

A negative base does not produce a real output for every real exponent; for example, $(-2)^{1/2}$ is not real. Exponential functions therefore use a positive base.

What role does $a$ play in $f(x)=ab^x$?

The nonzero constant $a$ is the initial value because $f(0)=a$. It also determines whether the outputs are positive or negative.

How does exponential growth differ from linear growth?

Exponential growth changes by a constant percentage or multiplicative factor over equal intervals, whereas linear growth changes by a constant amount or additive difference.

What is the multiplicative relationship between consecutive outputs of an exponential function?

For $f(x)=ab^x$, increasing $x$ by $1$ multiplies the output by $b$: $\frac{f(x+1)}{f(x)}=b$. This constant ratio distinguishes exponential growth from linear growth.

Exponential growth factor

The factor multiplying a quantity during each equal time interval. For a percentage growth rate $r$, the growth factor is $1+r$, where $r$ is written as a decimal.

Exponential decay factor

For a percentage decrease of $r$ per equal interval, the multiplicative factor is $1-r$, where $0<r<1$ when $r$ is expressed as a decimal.

How can you recognize exponential growth or decay from $f(x)=ab^x$?

If $b>1$, the function models exponential growth. If $0<b<1$, it models exponential decay.

How do you evaluate an exponential function such as $f(x)=30(2)^x$?

Substitute the given value for $x$, evaluate the power first, and then multiply by the coefficient. For example, $f(3)=30(2^3)=30(8)=240$.

How should $f(x)=5(3)^{x+1}$ be evaluated at $x=2$?

Substitute first and simplify the exponent: $f(2)=5(3)^{2+1}=5(3^3)=5(27)=135$.

What are the domain, range, intercept, and horizontal asymptote of $f(x)=ab^x$?

The domain is all real numbers, the $y$-intercept is $(0,a)$, and the horizontal asymptote is $y=0$. The range is $(0,\infty)$ if $a>0$ and $(-\infty,0)$ if $a<0$.

What happens to $f(x)=ab^x$ as $x\to -\infty$ when $b>1$?

Because $b^x\to 0$, $f(x)\to 0$. The graph approaches but never reaches the horizontal asymptote $y=0$.

What happens to $f(x)=ab^x$ as $x\to \infty$ when $b>1$ and $a>0$?

The function increases without bound: $f(x)\to\infty$. This is exponential growth.

How does the graph of $f(x)=ab^x$ behave when $0<b<1$ and $a>0$?

The graph decreases as $x$ increases, approaches $0$ as $x\to\infty$, and grows without bound as $x\to-\infty$.

How is a constant percentage growth rate represented in an exponential model?

If an initial amount is $a$ and the percentage growth rate per time period is $r$, then $f(t)=a(1+r)^t$, with $r$ written as a decimal.

How is a constant percentage decay rate represented in an exponential model?

If the initial amount is $a$ and the percentage decrease per time period is $r$, then $f(t)=a(1-r)^t$, with $r$ written as a decimal.

How do you choose the time variable in a real-world exponential model?

Define $t=0$ at the time corresponding to the known initial amount. Then the exponent is the elapsed number of time intervals since that reference time.

A population starts at $1.25$ billion and grows by $1.2\%$ per year. What model represents its population $t$ years later?

The model is $P(t)=1.25(1.012)^t$ billion. The factor $1.012$ comes from $1+0.012$.

How do you find an exponential model when one data point is the initial value?

Use $f(x)=ab^x$ and identify $a$ from the point $(0,a)$. Substitute the other point to solve for $b$; for a point $(x_1,y_1)$, $b=(y_1/a)^{1/x_1}$.

An exponential population changes from $80$ to $180$ over six years. What exact expression gives its yearly growth factor?

With $N(t)=80b^t$, use $180=80b^6$. Thus $b^6=\frac{180}{80}=\frac{9}{4}$, so $b=\left(\frac{9}{4}\right)^{1/6}$.

How do you find an exponential model from two points when neither point has $x=0$?

Substitute both points into $y=ab^x$, producing two equations in $a$ and $b$. Divide the equations to eliminate $a$, solve for $b$, and then substitute back to determine $a$.

What condition allows two points to determine a unique real exponential function of the form $f(x)=ab^x$?

The points must have different $x$-coordinates and their $y$-values must both be positive or both be negative. The model also must be known to be exponential.

How can you check an exponential model obtained from data?

Substitute each given point into the model and verify that the outputs match. A graph can also be checked to confirm that it passes through the data points and has the expected growth or decay behavior.

How can exponential regression be used to model data?

Enter the paired $x$- and $y$-values into a graphing calculator and use exponential regression, often labeled ExpReg. The calculator estimates parameters in $y=ab^x$.

Why can an exponential model become unreliable for long-term prediction?

An unrestricted exponential model assumes the same percentage change forever. Real populations and other systems typically face limits, so the model may eventually predict unrealistic values.

Compound interest

Interest that is added to an account and then earns additional interest in later periods. Thus, interest is earned on both the original principal and previously accumulated interest.

Compound-interest formula

The account value after $t$ years is $A(t)=P\left(1+\frac{r}{n}\right)^{nt}$, where $P$ is the principal, $r$ is the annual percentage rate as a decimal, and $n$ is the number of compounding periods per year.

In the compound-interest formula $A(t)=P\left(1+\frac{r}{n}\right)^{nt}$, what do $P$, $r$, $n$, and $t$ represent?

$P$ is the initial amount, $r$ is the nominal annual rate written as a decimal, $n$ is the number of compounding periods per year, and $t$ is time in years.

How does increasing the compounding frequency affect an account with a positive interest rate?

For the same principal, rate, and time, more frequent compounding generally produces a larger final balance because interest is reinvested sooner.

What is the difference between an APR and an effective annual rate?

The APR, or nominal rate, is the stated yearly rate before accounting for multiple compounding periods. With compounding more than once per year, the effective annual rate is greater because interest itself earns interest.

How do you solve the compound-interest formula for the principal $P$?

Rearrange the formula to obtain $P=\frac{A}{\left(1+\frac{r}{n}\right)^{nt}}$. This gives the amount that must be invested initially to reach a specified future value.

What values should be used for a $6\%$ interest rate compounded semiannually for $18$ years?

Use $r=0.06$, $n=2$, and $t=18$. The growth factor is $\left(1+\frac{0.06}{2}\right)^{2(18)}=(1.03)^{36}$.

What is the number $e$?

The irrational constant $e$ is defined by $e=\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n$, with approximate value $e\approx2.718282$.

What does the limit defining $e$ represent financially?

It is the limiting growth factor for one year when an investment earns a $100\%$ nominal annual rate and is compounded more and more frequently.

How can you evaluate a power of $e$, such as $e^{3.14}$?

Use the calculator's $e^x$ function rather than the 'Exp' key used for scientific notation. Numerically, $e^{3.14}\approx23.10387$.

What is the continuous growth or decay model?

A model of the form $A(t)=ae^{rt}$, where $a$ is the initial value, $r$ is the continuous rate per unit time, and $t$ is elapsed time. It is growth when $r>0$ and decay when $r<0$.

Continuous compounding formula

For an investment, continuous compounding is modeled by $A(t)=Pe^{rt}$, where $P$ is the principal, $r$ is the annual rate as a decimal, and $t$ is measured in years.

How does the sign of $r$ affect $A(t)=ae^{rt}$?

If $r>0$, then $e^{rt}$ increases with time and the model describes continuous growth. If $r<0$, then $e^{rt}$ decreases and the model describes continuous decay.

Why are models with base $e$ useful for continuous processes?

The base $e$ naturally describes change occurring continuously, rather than at separate compounding intervals. Such models are widely used for continuously changing quantities in science, finance, and other fields.