Exponential Functions

Exponential functions model repeated multiplication, including percent growth, decay, doubling, and half-life.

Equal input steps create multiplicative output change. An increasing curve stays above a horizontal asymptote, grows slowly at first, and then rises rapidly. Three marked points show a constant multiplication factor rather than a constant difference.

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

The basic form

An exponential function can be written

$$ f(x)=ab^x, $$

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 $3x^2$.

Worked example: percent growth

Worked example: depreciation

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

$$ V(t)=1800(0.78)^t. $$

After $3$ years,

V(3)=1800(0.78)^3\approx853.93.

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

Transformations

The form

y=a b^{k(x-d)}+c

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.