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Advanced Math / Nonlinear functions Difficulty: Hard

$f (x) = (x - 1) (x + 3) (x - 2)$

In the xy-plane, when the graph of the function $f$ , where $y = f (x)$ , is shifted up $6$ units, the resulting graph is defined by the function $g$ . If the graph of $y = g (x)$ crosses through the point $left parenthesis 4 comma b right parenthesis$ , where $b$ is a constant, what is the value of $b$ ?

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Explanation

The correct answer is $48$ . It's given that in the xy-plane, when the graph of the function $f$ , where $y = f (x)$ , is shifted up $6$ units, the resulting graph is defined by the function $g$ . Therefore, function $g$ can be defined by the equation $g (x) = f (x) + 6$ . It's given that $f (x) = (x - 1) (x + 3) (x - 2)$ . Substituting $(x - 1) (x + 3) (x - 2)$ for $f (x)$ in the equation $g (x) = f (x) + 6$ yields $g (x) = (x - 1) (x + 3) (x - 2) + 6$ . For the point $left parenthesis 4 comma b right parenthesis$ , the value of $x$ is $4$ . Substituting $4$ for $x$ in the equation $g (x) = (x - 1) (x + 3) (x - 2) + 6$ yields $g (4) = (4 - 1) (4 + 3) (4 - 2) + 6$ , or $g left parenthesis 4 right parenthesis equals 48$ . It follows that the graph of $y = g (x)$ crosses through the point $left parenthesis 4 comma 48 right parenthesis$ . Therefore, the value of $b$ is $48$ .