Exponential Functions
The exponential function a·bˣ with positive a is the most disciplined curve in the course: increasing everywhere when b is above 1, decreasing everywhere when b is between 0 and 1, with no turns, no extremes, a domain of all real numbers, a range of positive outputs only, and a horizontal asymptote at y = 0 that the graph approaches from one side and never touches.
Most wrong answers here come from two slips: treating a negative exponent as a sign change instead of a reciprocal (so the graph is drawn dipping below zero or landing on the axis), and garbling the percent-to-factor translation (reading b = 0.85 as an 85% loss). The lesson drills both until the impossible graphs stop looking plausible.
The work
3 ways in · any order
Lesson
Exponential Functions
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The anatomy of a·bˣ: growth and decay bases, the always-positive outputs, the asymptote at y = 0, and negative exponents as reciprocals rather than sign changes. Ten scenarios close it out: evaluate, classify, find ranges and limits, and translate percent language into factors.
Diagnostic
10-item topic check
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Ten items spanning the two Topic 2.3 misconceptions: exponent arithmetic abuse, especially negative exponents read as negative outputs, and growth factors garbled into 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.