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Advanced Math / Nonlinear functions Difficulty: Hard

$f (x) = (x - 10) (x + 13)$

The function $f$ is defined by the given equation. For what value of $x$ does $f (x)$ reach its minimum?

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Explanation

Choice D is correct. It's given that $f (x) = (x - 10) (x + 13)$ , which can be rewritten as $f (x) = x^{2} + 3 x - 130$ . Since the coefficient of the $x^{2}$ -term is positive, the graph of $y = f (x)$ in the xy-plane opens upward and reaches its minimum value at its vertex. The x-coordinate of the vertex is the value of $x$ such that $f (x)$ reaches its minimum. For an equation in the form $f (x) = a x^{2} + b x + c$ , where $a$ , $b$ , and $c$ are constants, the x-coordinate of the vertex is $- \frac{b}{2 a}$ . For the equation $f (x) = x^{2} + 3 x - 130$ , $a equals 1$ , $b equals 3$ , and $c = - 130$ . It follows that the x-coordinate of the vertex is $- \frac{3}{2 (1)}$ , or $- \frac{3}{2}$ . Therefore, $f (x)$ reaches its minimum when the value of $x$ is $- \frac{3}{2}$ .

Alternate approach: The value of $x$ for the vertex of a parabola is the x-value of the midpoint between the two x-intercepts of the parabola. Since it’s given that $f (x) = (x - 10) (x + 13)$ , it follows that the two x-intercepts of the graph of $y = f (x)$ in the xy-plane occur when $x equals 10$ and $x = - 13$ , or at the points $left parenthesis 10 comma 0 right parenthesis$ and $left parenthesis negative 13 comma 0 right parenthesis$ . The midpoint between two points, $left parenthesis x 1 comma y 1 right parenthesis$ and $left parenthesis x 2 comma y 2 right parenthesis$ , is $(\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})$ . Therefore, the midpoint between $left parenthesis 10 comma 0 right parenthesis$ and $left parenthesis negative 13 comma 0 right parenthesis$ is $(\frac{10 + (- 13)}{2}, \frac{0 + 0}{2})$ , or $left parenthesis negative three halves comma 0 right parenthesis$ . It follows that $f (x)$ reaches its minimum when the value of $x$ is $- \frac{3}{2}$ .

Choice A is incorrect. This is the y-coordinate of the y-intercept of the graph of $y = f (x)$ in the xy-plane.

Choice B is incorrect. This is one of the x-coordinates of the x-intercepts of the graph of $y = f (x)$ in the xy-plane.

Choice C is incorrect and may result from conceptual or calculation errors.