Exponential Functions

Linear change adds the same amount; exponential change multiplies by the same factor.

The basic form

An exponential function can be written

where $a\ne0$, $b>0$, and $b\ne1$. The value $a=f(0)$ is the initial value because $b^0=1$. The base $b$ is the factor applied whenever $x$ increases by one.

The variable is in the exponent. This distinguishes an exponential function such as $3(1.2)^x$ from a power function such as $3x^2$.

Growth and decay

If $b>1$, the magnitude grows as $x$ increases. If $0<b<1$, the function shows decay.

A rate form is

for growth rate $r$, and

for decay rate $r$. Write percentages as decimals: $6\%=0.06$, so a $6\%$ growth factor is $1.06$.

Graph features

For $f(x)=ab^x$ with $a>0$:

• domain: all real numbers;
• range: $y>0$;
• $y$-intercept: $(0,a)$;
• horizontal asymptote: $y=0$;
• no $x$-intercept.

The graph approaches the asymptote but never reaches it. Adding a vertical shift $k$ in $ab^{x-h}+k$ changes the horizontal asymptote to $y=k$.

Worked example: percent growth

Worked example: depreciation

A laptop worth $1800$ loses $22\%$ of its value each year. It retains $78\%$, giving factor $0.78$:

After $3$ years,

Subtracting $22\%$ of the original price each year would be linear and would not represent percentage depreciation of the current value.

Recognizing exponential data

For equally spaced inputs, exponential data have approximately constant ratios between consecutive outputs. Linear data instead have constant first differences.

For outputs $5,10,20,40$, each value is multiplied by $2$, so an exponential model with base $2$ is appropriate. For $5,10,15,20$, the constant difference $5$ indicates a linear model.

Real data rarely have perfectly constant ratios; context and residuals help judge whether the pattern is reasonable.

Doubling time and half-life

A doubling model can be written

where $d$ is the doubling time. A half-life model is

where $h$ is the half-life. At $t=h$, exactly half the initial amount remains; at $t=2h$, one quarter remains.

Transformations

The form

can reflect, stretch, compress, and translate the parent $y=b^x$. The horizontal asymptote becomes $y=c$. A negative $a$ reflects across the asymptote, while $d$ shifts notable points horizontally.

Transform the parent point $(0,1)$ to locate a useful anchor point on the new graph.

Interpreting models carefully

An exponential model assumes a constant multiplicative rate under similar conditions. Populations face limits, investment rates change, and technologies depreciate differently over time. State the time unit, reasonable domain, rounding rule, and whether interpolation or extrapolation is being used.

The formula can be mathematically defined beyond the interval where its assumptions remain credible.

Common mistakes

Using $r$ instead of $1+r$. A $7\%$ increase uses factor $1.07$, not $0.07$.

Using $1+r$ for decay. A $7\%$ decrease retains $0.93$.

Confusing constant difference with constant ratio. They signal different models.

Thinking the asymptote is an intercept. The graph approaches it without reaching it in the basic model.

Ignoring time units. A monthly rate and annual rate cannot share the same exponent without conversion.

Quick self-check

• What is the initial value?
• What factor is applied per time period?
• Does the base indicate growth or decay?
• Are the input and rate measured in matching units?
• Does the graph have the correct intercept, asymptote, domain, and range?
• Is the model reasonable over the stated interval?

Related topics

• Exponents
• Exponential Equations
• Geometric Sequences
• Compound Interest