Study Guide

Learning Objectives

• Identify key characteristics of $f(x)=ab^x$ (2.3.A).
• Classify exponential growth ($a>0$, $b>1$) vs decay ($a>0$, $0<b<1$) (2.3.A.1).
• Explain natural-number inputs as repeated multiplication; state domain all reals (2.3.A.2).
• Describe monotonicity, concavity, lack of extrema/inflection on $\mathbb{R}$ (2.3.A.3).
• Recognize that proportional outputs over equal intervals imply exponential structure (2.3.A.4).
• State end behavior with limits as $x\to\pm\infty$ (2.3.A.5).

Suggested Practice Sequence

1. Read $a$ and $b$ from $ab^x$; classify growth/decay. 2. Evaluate $f(0), f(1), f(2)$; verify constant ratio. 3. Sketch qualitative graphs (increasing/decreasing, concave up/down). 4. State domain, range tendencies, and limits. 5. Answer "extrema/inflection?" on $\mathbb{R}$ vs closed interval. 6. Connect to geometric sequences at integer inputs.

Classroom / Check-In Moves

• Parameter table: $a$, $b$, growth/decay, $f(0)$.
• Ratio check every time.
• Limit sentences for both directions.

Misconception Watch-List

• $b<0$ as a standard base.
• Parabola-shaped "U" with a vertex on the full domain.
• Inflection point because the curve bends.
• $f(x)=2^x$ has $a=0$.

Vocabulary to Own

• initial value $a$; base $b$; growth factor
• exponential growth; exponential decay
• domain; end behavior; limit
• monotonic; concave up/down
• proportional change over equal intervals

Materials Map

• Practice: identify parameters and features
• Word: population, decay, interest-style contexts (feature focus)
• Mixed / multi-*: link to 2.1/2.2 sequences
• Review & practice test: timed characteristic fluency
• Interactive: quick checks