Study Guide
Learning Objectives
- Construct exponential models for proportional output change over equal input intervals (2.5.A.1).
- Adjust data with a constant offset when needed to reveal proportional growth (2.5.A.2).
- Find $a$ and $b$ from a ratio and initial value or from two input-output pairs (2.5.A.3).
- Build models using transformations of $ab^x$ from scenario features (2.5.A.4).
- Use exponential regression with technology when appropriate (2.5.A.5).
- Recognize and use the natural base $e$ in contextual models (2.5.A.6).
- Apply models; interpret $b$ as growth factor and percent change (2.5.B.1).
- Rewrite equivalent forms to compare rates across time units (2.5.B.2).
- Make predictions with attention to domain constraints (2.5.B.3).
Suggested Practice Sequence
1. Read a context; write $f(t)=ab^t$ with correct units. 2. Build ratio tables; verify constant multiplicative change. 3. Solve two-point systems for $a$ and $b$. 4. Shift/offset data when a baseline is present. 5. Rewrite daily model to weekly (or vice versa). 6. Predict a future value; state domain limits.
Classroom / Check-In Moves
- "What is $a$ in context?" "What does $b$ mean per unit $t$?"
- Percent link: $b=1.05$ means 5% growth per unit.
- Regression sanity check: parameters vs story.
Misconception Watch-List
- Using $b=r$ instead of $b=1+r$ for percent growth.
- Treating additive patterns as exponential.
- Extrapolating without context limits.
- Wrong time-unit conversion when rewriting bases.
Vocabulary to Own
- initial value; growth factor; decay factor
- proportional change; ratio table
- regression; natural base $e$
- percent change; equivalent form
- domain constraint; prediction; extrapolation
Materials Map
- Practice: parameter identification and construction
- Word: finance, population, decay, science contexts
- Mixed / multi-*: link to 2.3–2.4 manipulation
- Review & practice test: timed modeling fluency
- Interactive: quick interpretation checks