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Advanced Math / Nonlinear functions Difficulty: Hard

Function $f$ is a quadratic function where $f (- 20) = 0$ and $f (- 4) = 0$ . The graph of $y = f (x)$ in the xy-plane has a vertex at $left parenthesis r comma negative 64 right parenthesis$ . What is the value of $r$ ?

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Explanation

The correct answer is $negative 12$ . It’s given that function $f$ is a quadratic function where $f (- 20) = 0$ and $f (- 4) = 0$ . It follows that the graph of $y = f (x)$ in the xy-plane passes through the points $left parenthesis negative 20 comma 0 right parenthesis$ and $left parenthesis negative 4 comma 0 right parenthesis$ . When the graph of a quadratic function contains two points $left parenthesis a comma 0 right parenthesis$ and $left parenthesis b comma 0 right parenthesis$ , the x-coordinate of the vertex of the graph is the average of $a$ and $b$ . Therefore, the x-coordinate of the vertex of the graph of $y = f (x)$ is $\frac{- 20 + (- 4)}{2}$ , or $negative 12$ . It's given that the graph of $y = f (x)$ in the xy-plane has a vertex at $left parenthesis r comma negative 64 right parenthesis$ . It follows that the value of $r$ is $negative 12$ .