AP® Precalculus · Free response

A rational function with a hole and two asymptotes

write it first, then score yourself point by point

Q1: Function Concepts · Part A: graphing calculator allowed · 6 points

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The question

Name: ______________________   Date: ____________   A rational function with a hole and two asymptotes (6 points)

Graph

The graph of ff

The function ff is given by f(x)f(x)=2x2+2x−4x2+3x+2= \frac{2x^2 + 2x - 4}{x^2 + 3x + 2}. Part of its graph is shown, with a hole at (−2,6)(-2, 6), a vertical asymptote x=−1x = -1 and a horizontal asymptote y=2y = 2. The function gg is given by g(x)=0.5x2−1g(x) = 0.5x^2 - 1.
−7−6−5−4−3−2−11234567−8−6−4−224681012xyf

  1. (A(i)) Find every real zero of ff. Give a reason why x=−2x = -2 is not a zero of ff. [1 point]

  2. (A(ii)) Find limx→−2f(x)\lim_{x \to -2} f(x). Show the work that leads to your answer. [1 point]

  3. (B(i)) Write the behavior of ff on each side of x=−1x = -1 using limit notation. [1 point]

  4. (B(ii)) Write the end behavior of ff using limit notation, and give a reason for it. [1 point]

  5. (C(i)) Solve f(x)=g(x)f(x) = g(x). Give every solution correct to three decimal places. [1 point]

  6. (C(ii)) Is the rate of change of ff increasing or decreasing on the interval [2,5][2, 5]? Give a reason using the average rates of change of ff on [2,3.5][2, 3.5] and [3.5,5][3.5, 5]. [1 point]