PSAT Math Multiple-Choice Question 439: Answer and Explanation

Question: 439

Which of the following equations could represent the above graph?

• A. y = -x² + 18x - 32
• B. y = -x² + 14x - 32
• C. y = x² - 14x - 32
• D. y = x² + 18x + 32

Correct Answer: A

Explanation:

A

Difficulty: Hard

Category: Graphs of Quadratics

Strategic Advice: Use the visual information in the graph to eliminate answers quickly. A negative coefficient of the squared term means that the graph opens downward. The roots of a quadratic are where the graph crosses the x-axis.

Getting to the Answer: Because the quadratic opens downward, the x²-coefficient must be negative, so you can eliminate (C) and (D). According to the graph, the roots are x = 2 and x = 16. That means that the factored form of the quadratic will be either (-x + 2)(x - 16) or (x - 2)(-x + 16). These factored forms are actually equivalent because (-x + 2) = (-1)(x - 2) and (x - 16) = (-1)(-x + 16), which means:

(-x + 2)(x - 16) = (-1 × -1)(x - 2)(-x + 16)

(-x + 2)(x - 16) = (x - 2)(-x + 16)

Use FOIL on one of them to see if it matches the expanded form of the quadratic in (A) or (B):

(-x + 2)(x - 16) = -x² + 16x + 2x - 32

= -x² + 18x - 32

Thus, (A) is correct.

Another approach would be to use Picking Numbers. After eliminating (C) and (D) for having upward parabolas and calculating the roots, simply pick the more manageable of the two x-intercepts, x = 2, and plug it into the equations in (A) and (B) to see which one results in y = 0:

(A): y = -(2)² + 18(2) - 32

y = -(4) + 36 - 32 = 0, keep.

(B): y = -(2)² + 14(2) - 32

y = -(4) + 28 - 32 = -8 ≠ 0, eliminate.

(A) is indeed correct.