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Advanced Math / Nonlinear functions Difficulty: Medium

The function $f$ is defined by $f left parenthesis x right parenthesis equals 7 x cubed$ . In the xy-plane, the graph of $y = g (x)$ is the result of shifting the graph of $y = f (x)$ down $2$ units. Which equation defines function $g$ ?

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Explanation

Choice D is correct. If the graph of $y = g (x)$ is the result of shifting the graph of $y = f (x)$ down $k$ units in the xy-plane, the function $g$ can be defined by an equation of the form $g (x) = f (x) - k$ . It’s given that $f left parenthesis x right parenthesis equals 7 x cubed$ and the graph of $y = g (x)$ is the result of shifting the graph of $y = f (x)$ down $2$ units. Substituting $7 x cubed$ for $f (x)$ and $2$ for $k$ in the equation $g (x) = f (x) - k$ yields $g (x) = 7 x^{3} - 2$ .

Choice A is incorrect and may result from conceptual errors.

Choice B is incorrect and may result from conceptual errors.

Choice C is incorrect. This equation defines a function $g$ for which the graph of $y = g (x)$ is the result of shifting the graph of $y = f (x)$ up, not down, $2$ units.