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Advanced Math / Nonlinear functions Difficulty: Hard

$f (x) = 4 x^{2} + 64 x + 262$

The function $g$ is defined by $g (x) = f (x + 5)$ . For what value of $x$ does $g (x)$ reach its minimum?

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Explanation

Choice A is correct. It's given that $g (x) = f (x + 5)$ . Since $f (x) = 4 x^{2} + 64 x + 262$ , it follows that $f (x + 5) = 4 {(x + 5)}^{2} + 64 (x + 5) + 262$ . Expanding the quantity $left parenthesis x plus 5 right parenthesis squared$ in this equation yields $f (x + 5) = 4 (x^{2} + 10 x + 25) + 64 (x + 5) + 262$ . Distributing the $4$ and the $64$ yields $f (x + 5) = 4 x^{2} + 40 x + 100 + 64 x + 320 + 262$ . Combining like terms yields $f (x + 5) = 4 x^{2} + 104 x + 682$ . Therefore, $g (x) = 4 x^{2} + 104 x + 682$ . For a quadratic function defined by an equation of the form $g (x) = a {(x - h)}^{2} + k$ , where $a$ , $h$ , and $k$ are constants and $a$ is positive, $g (x)$ reaches its minimum, $k$ , when the value of $x$ is $h$ . The equation $g (x) = 4 x^{2} + 104 x + 682$ can be rewritten in this form by completing the square. This equation is equivalent to $g (x) = 4 (x^{2} + 26) + 682$ , or $g (x) = 4 (x^{2} + 26 x + 169 - 169) + 682$ . This equation can be rewritten as $g (x) = 4 ({(x + 13)}^{2} - 169) + 682$ , or $g (x) = 4 {(x + 13)}^{2} - 4 (169) + 682$ , which is equivalent to $g (x) = 4 {(x + 13)}^{2} + 6$ . This equation is in the form $g (x) = a {(x - h)}^{2} + k$ , where $a equals 4$ , $h = - 13$ , and $k equals 6$ . Therefore, $g (x)$ reaches its minimum when the value of $x$ is $negative 13$ .

Choice B is incorrect. This is the value of $x$ for which $f (x)$ , rather than $g (x)$ , reaches its minimum.

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

Choice D is incorrect. This is the value of $x$ for which $f left parenthesis x minus 5 right parenthesis$ , rather than $f left parenthesis x plus 5 right parenthesis$ , reaches its minimum.