Exponential Functions Review and Extension

By the end of this lesson, you will be able to graph exponential functions with transformations, identify growth vs. decay, determine the asymptote and intercepts, and connect exponential models to real-world phenomena.

The Big Idea

Exponential functions model any process where growth (or decay) is proportional to the current size — populations, interest, radioactive decay, viral spread, battery discharge. The rate of change at any moment is proportional to the current value. No other function family has this property, which is why exponentials appear throughout science and economics.

Exponential Function Form

$$f(x) = a \cdot b^x$$

• $b > 0$, $b \neq 1$ (the base)
• $a > 0$ (the initial value, $y$-intercept is $a$)
• $b > 1$: exponential growth
• $0 < b < 1$: exponential decay

Natural exponential: $f(x) = e^x$ where $e \approx 2.718$

Key Features of $f(x) = a \cdot b^x$

• Domain: $(-\infty, \infty)$
• Range: $(0, \infty)$ for $a > 0$; $(-\infty, 0)$ for $a < 0$
• $y$-intercept: $(0, a)$
• Horizontal asymptote: $y = 0$ (the $x$-axis)
• No $x$-intercept (never crosses the $x$-axis for $a \neq 0$)

Transformations

For $g(x) = a \cdot b^{x - h} + k$:

Change	Effect
$+k$ (outside)	Shift up $k$; asymptote becomes $y = k$
$-h$ (inside)	Shift right $h$
$a < 0$	Reflect over $x$-axis
$0 < a < 1$	Vertical compression

Example 1 — Identifying Growth vs. Decay

(a) $f(x) = 3(1.05)^x$: $b = 1.05 > 1$ → growth; 5% per period

(b) $g(x) = 100(0.82)^x$: $b = 0.82 < 1$ → decay; 18% decrease per period

(c) $h(x) = 5e^{-0.3x} = 5(e^{-0.3})^x$: $e^{-0.3} \approx 0.74 < 1$ → decay

Example 2 — Graphing with Transformation