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Algebra Difficulty: Hard

The graph of $y = f (x) + 14$ is shown. Which equation defines function $f$ ?

$f (x) = - \frac{1}{4} x - 12$

$f (x) = - \frac{1}{4} x + 16$

$f (x) = - \frac{1}{4} x + 2$

$f (x) = - \frac{1}{4} x - 14$

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Explanation

Choice A is correct. An equation for the graph shown can be written in slope-intercept form $y = m x + b$ , where $m$ is the slope of the graph and its y-intercept is $left parenthesis 0 comma b right parenthesis$ . Since the y-intercept of the graph shown is $left parenthesis 0 comma 2 right parenthesis$ , the value of $b$ is $2$ . Since the graph also passes through the point $left parenthesis 4 comma 1 right parenthesis$ , the slope can be calculated as $\frac{1 - 2}{4 - 0}$ , or $- \frac{1}{4}$ . Therefore, the value of $m$ is $- \frac{1}{4}$ . Substituting $- \frac{1}{4}$ for $m$ and $2$ for $b$ in the equation $y = m x + b$ yields $y = - \frac{1}{4} x + 2$ . It’s given that an equation for the graph shown is $y = f (x) + 14$ . Substituting $f left parenthesis x right parenthesis plus 14$ for $y$ in the equation $y = - \frac{1}{4} x + 2$ yields $f (x) + 14 = - \frac{1}{4} x + 2$ . Subtracting $14$ from both sides of this equation yields $f (x) = - \frac{1}{4} x - 12$ .

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

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

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