Change in Linear and Exponential Functions
Linear and exponential functions are the grown-up versions of arithmetic and geometric sequences. Over equal input intervals a linear function changes by equal amounts; an exponential function changes by equal factors. In f(x) = a·bˣ, the a is the starting output and the b is the per-step factor, with percent change b − 1, and across an interval of length k the output is multiplied by b to the power k.
The mistakes are exactly the sequence mistakes wearing function clothes: extending a constant-factor pattern by adding, reading the factor 1.04 as 104% growth, and computing a slope where a ratio was needed. The lesson drills the diagnosis, the parameter reading, and the long-run comparison where the exponential always wins.
The work
3 ways in · any order
Lesson
Change in Linear and Exponential Functions
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Equal amounts or equal factors: the lesson runs the difference test and the ratio test side by side, reads a and b in f(x) = a·bˣ without garbling either, and closes with ten scenarios on classifying tables, building models from two points, and the long run.
Diagnostic
10-item topic check
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Ten items spanning the two Topic 2.2 misconceptions: additive reasoning applied to multiplicative growth, and growth factors confused with growth rates. The bank ships with the Unit 2 data drop.
Targeted Practice
Drill a single misconception
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Pick one of the failure modes you missed and drill it on its own. The round is adaptive: two correct in a row clears the misconception and moves you to the next.