Study Guide
Learning Objectives
- Construct linear functions comparable to arithmetic sequences: $f(x)=b+mx$ and $f(x)=y_i+m(x-x_i)$ (2.2.A.1-A.2).
- Construct exponential functions comparable to geometric sequences: $f(x)=ab^x$ and $f(x)=y_i r^{x-x_i}$ (2.2.A.3-A.4).
- Explain that sequences and corresponding functions may have different domains (2.2.A.5).
- Distinguish linear vs exponential change over equal-length intervals (2.2.B.1).
- Compare additive vs multiplicative structure and shared "initial value + change constant" form (2.2.B.2).
- Recognize that arith/linear/geo/exponential models can be determined by two distinct values (2.2.B.3).
Suggested Practice Sequence
1. Map $a_0,d$ to $b,m$ and write $f(x)$. 2. Map $g_0,r$ to $a,b$ and write $f(x)=ab^x$. 3. Point-slope / point-ratio forms from one known point. 4. Table diagnosis: constant differences vs constant ratios. 5. Domain contrast: sequence vs function graphs. 6. Two-point determination drills for both families. 7. Context: money/population (multiplicative) vs steady add (additive).
Classroom / Check-In Moves
- Side-by-side: sequence list vs continuous function with same parameters.
- Cold-call: "addition or multiplication?"
- Require domain statement whenever switching sequence <-> function.
Misconception Watch-List
- Calling any increasing curve linear.
- Using $f(x)=a+bx$ for multiplicative contexts.
- Ignoring domain differences (2.2.A.5).
- Thinking you need many points when two suffice (in ideal exact data).
Vocabulary to Own
- linear function; exponential function
- initial value; rate of change; growth/decay factor
- proportional change; equal-length input intervals
- comparable sequence; domain difference
- point-slope form; point-ratio form
Materials Map
- Practice: build linear/exponential from sequence parameters and points
- Word: additive vs multiplicative contexts
- Mixed / multi-*: connect to 2.1 sequences
- Review & practice test: timed recognition + construction
- Interactive: quick checks