Family Overview

In Plain Language

An exponential function has the form $f(x)=ab^x$ with initial value $a\neq 0$ and base $b>0$, $b\neq 1$.

• Growth: $a>0$ and $b>1$ (outputs multiply by more than 1 each unit).
• Decay: $a>0$ and $0<b<1$ (outputs shrink toward 0).
• Domain: all real numbers.
• Behavior: always increasing or always decreasing; graph always concave up or always concave down.
• No global extrema (except on a closed interval) and no inflection points.
• End behavior: as $x\to\infty$ or $x\to-\infty$, outputs may grow without bound or approach 0.

For whole-number inputs, the exponent tells how many factors of $b$ are applied to $a$.

Why It Matters

Topic 2.3 is the core definition of exponential functions before manipulation (2.4) and modeling (2.5). AP questions ask students to identify growth/decay, describe concavity and limits, and read key features from equations and graphs.

How Families Can Help

• Ask: "What is $a$? What is $b$? Is $b>1$ or between 0 and 1?"
• Practice: compute $f(0), f(1), f(2)$ and check constant ratios.
• Emphasize: exponential graphs are smooth curves, not connected dots (unlike sequences).

What Not to Do

• Do not allow $b\le 0$ or $b=1$ in standard exponential form.
• Do not claim a turning point on the full real domain.
• Do not confuse $ab^x$ with a linear function.
• Do not forget end-behavior limits.