SAT Nonlinear functions Practice Question 20 (Hard) | SAT Owl

Question bank / Nonlinear functions

Math · Advanced MathHard

The quadratic function $f$ is defined by $f(x) = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants, $a < 0$ , and $c > 0$ . Which of the following must be true?

A

The graph of $y = f(x)$ has a $y$ -intercept below the $x$ -axis.

B

The vertex of the graph of $y = f(x)$ lies below the $x$ -axis.

C

The function $f$ has no real zeros.

D

The function $f$ has two real zeros, one positive and one negative.

Show answer and explanation

Answer: D

Since $a < 0$ , the parabola opens downward, and since $f(0) = c > 0$ , the graph is above the $x$ -axis at $x = 0$ . A downward parabola that is above the $x$ -axis somewhere must cross it once on each side of that point, so there is one negative zero and one positive zero. (Algebraically, the product of the zeros is $\frac{c}{a} < 0$ .)

Choice C ignores that a downward-opening parabola with a positive $y$ -intercept must eventually fall below the $x$ -axis on both sides.

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