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Advanced Math / Nonlinear functions Difficulty: Hard

The function $f$ is defined by $f (x) = (x - 6) (x - 2) (x + 6)$ . In the xy-plane, the graph of $y = g (x)$ is the result of translating the graph of $y = f (x)$ up $4$ units. What is the value of $g left parenthesis 0 right parenthesis$ ?

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Explanation

The correct answer is $76$ . It's given that the graph of $y = g (x)$ is the result of translating the graph of $y = f (x)$ up $4$ units in the xy-plane. It follows that the graph of $y = g (x)$ is the same as the graph of $y = f (x) + 4$ . Substituting $g (x)$ for $y$ in the equation $y = f (x) + 4$ yields $g (x) = f (x) + 4$ . It's given that $f (x) = (x - 6) (x - 2) (x + 6)$ . Substituting $(x - 6) (x - 2) (x + 6)$ for $f (x)$ in the equation $g (x) = f (x) + 4$ yields $g (x) = (x - 6) (x - 2) (x + 6) + 4$ . Substituting $0$ for $x$ in this equation yields $g (0) = (0 - 6) (0 - 2) (0 + 6) + 4$ , or $g left parenthesis 0 right parenthesis equals 76$ . Thus, the value of $g left parenthesis 0 right parenthesis$ is $76$ .