# 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* - *x*^{2}. Which of the following represents a quadratic with no real zeros?

- A.
- B.
- C.
- D.

**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* - *x*^{2} 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 *x*^{2} 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 , and therefore, the vertex of the parabola is .

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.