SAT Nonlinear functions Practice Question 17 (Hard)

Math · Advanced MathHard

A quadratic function $f$ has a minimum value of $-6$ . The graph of $y = f(x)$ in the $xy$ -plane passes through the points $(-1, 21)$ and $(5, 21)$ . What is the value of $f(4)$ ?

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Answer: 6

Points with equal $y$ -values are symmetric about the axis of symmetry, so the vertex has $x = \frac{-1 + 5}{2} = 2$ , and the vertex is $(2, -6)$ . Write $f(x) = a(x - 2)^2 - 6$ . Using $(5, 21)$ : $21 = 9a - 6$ , so $a = 3$ . Then $f(4) = 3(2)^2 - 6 = 6$ .

Answering 21 assumes $f(4)$ matches $f(5)$ , but symmetry about $x = 2$ pairs $x = 4$ with $x = 0$ , not with $x = 5$ .

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