SAT Math Multiple Choice Question 660: Answer and Explanation

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Question: 660

15. Which of the following can represent the graph in the xy-plane of y = a(x - b)(x + c)2, where a, b, and c are all positive constants?

  • A. Images
  • B. Images
  • C. Images
  • D. Images

Correct Answer: D



Advanced Mathematics (analyzing polynomial graphs) HARD

By the Zero Product Property (Chapter 9, Lesson 5), the graph of y = a(x - b)(x + c)2 has zeroes at x = b and a "double root" at x = -c (because this expression has two factors of (x + c)). Since b and c are both positive, this means that the graph must have one single positive root and a "double" negative root. That is, the graph passes through the x-axis at a positive value of x and "bounces" off of the x-axis at a negative value of x. Notice that this eliminates choices (B) and (C). We also know that a, the "leading coefficient" of the polynomial, is positive. If the leading coefficient of the polynomial is positive, the polynomial must eventually "shoot up" toward positive infinity; that is, it must go up as we move to the right. This rules out choice (A) and leaves only choice (D) as correct.

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