AP Precalculus
AP Precalc Unit 2A (2.1–2.6): Conceptual FRQ
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Explain-and-justify FRQs for the Unit 2A AP Readiness test (2.1–2.6): unequal spacing, residual reasoning, equivalent forms, revealing a proportional pattern. Rubric and sample answer for every part.
11 questions across 4 sections. Write your answer, then check it against the rubric and score yourself.
Sample questions
Open an item to see its answer.
(a) A student says f cannot be exponential because the ratios of consecutive outputs are 1.5, 2.25, 1.5, 2.25. Explain why the student's reasoning is wrong.
Sample answer: The x-steps aren't the same length: 1, 2, 1, 2. For an exponential, only intervals of the same length must have the same ratio. The 1-unit intervals both give 1.5 and the 2-unit intervals both give 2.25.
(b) Show that f can be modeled by an exponential function and write the model.
Sample answer: Each 2-unit interval multiplies by 2.25 = 1.5², which matches two 1-unit steps of 1.5. So f(x) = 6(1.5)ˣ. Check: 6(1.5)⁶ = 68.34375.
(a) Which model is more appropriate? Justify your answer using the residuals.
Sample answer: The exponential model. The linear residuals go positive, then negative, then positive (a U shape), so the line misses the curve. The exponential residuals have no pattern and are small.
(b) For the exponential model, interpret the residual at x = 4.
Sample answer: The residual 0.5 means actual − predicted = 0.5, so the actual value is 0.5 above the model. The model underestimates at x = 4.
(c) Explain why a pattern in a residual plot is evidence that a model is not appropriate.
Sample answer: If the errors follow a pattern, the model is wrong in a predictable way, too low on the ends and too high in the middle, for example. That means it doesn't match the data's shape. A good model leaves only random error.