SAT Math Multiple Choice Question 640: Answer and Explanation

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Question: 640

10. The function f is defined by the equation f(x) = x - x2. Which of the following represents a quadratic with no real zeros?

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

Correct Answer: B

Explanation:

B

Advanced Mathematics (quadratics) HARD

Perhaps the simplest way to begin this problem is to draw a quick sketch of the function in the xy-plane, and then compare this graph to the transformations of the original function given in the choices. Notice that the original function f(x) = x - x2 is easily factored as f(x) = x (1 - x). The Zero Product Property (Chapter 9, Lesson 5) tells us that this function must have zeros at x = 0 and x = 1. Notice, also, that since the coefficient of the x2 term in the original function is negative (-1), the graph of this quadratic is an "open-down" parabola. Also, the axis of symmetry is halfway between the zeros, at x = ½. Plugging x = ½ back into the function gives us images, and therefore, the vertex of the parabola is images.

images

The question asks us to find the function that has no real zeros. This means that the graph of this function must not intersect the x-axis at all. Each answer choice indicates a different transformation of the function f. Recall from Chapter 9, Lesson 3, that choice (A) f(x) + ½ is the graph of f shifted up ½ unit, choice (B) f(x) - ½ is the graph of f shifted down ½ unit, choice (C) f(x/2) is the graph of f stretched by a factor of 2 in the horizontal direction, and choice (D) f(x - ½) is the graph of f shifted right ½ unit. As the sketch above shows, only (B) yields a graph that does not intersect the x-axis.

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