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Algebra / Linear functions Difficulty: Medium

The graph of $y = f (x) - 11$ is shown.

Which equation defines the linear function $f$ ?

$f (x) = - 13 x - 11$

$f (x) = - 2 x + 7$

$f (x) = - 13 x + 7$

$f (x) = - 2 x - 11$

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Explanation

Choice B is correct. The graph of a line in the xy-plane can be represented by the equation $y = m x + b$ , where $m$ is the slope of the line and $left parenthesis 0 comma b right parenthesis$ is the y-intercept. The slope of a line that passes through the points $left parenthesis x 1 comma y 1 right parenthesis$ and $left parenthesis x 2 comma y 2 right parenthesis$ can be calculated using the formula $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ . The line shown passes through the points $(- 1, - 2)$ and $left parenthesis 0 comma negative 4 right parenthesis$ . Substituting $(- 1, - 2)$ and $left parenthesis 0 comma 4 right parenthesis$ for $left parenthesis x 1 comma y 1 right parenthesis$ and $left parenthesis x 2 comma y 2 right parenthesis$ , respectively, in the formula $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ yields $m = \frac{- 4 - (- 2)}{0 - (- 1)}$ , which is equivalent to $m = \frac{- 2}{1}$ , or $m = - 2$ . Since the line shown passes through the point $left parenthesis 0 comma negative 4 right parenthesis$ , it follows that $b = - 4$ . Substituting $negative 2$ for $m$ and $negative 4$ for $b$ in the equation $y = m x + b$ yields $y = - 2 x - 4$ . It’s given that the graph shown is the graph of $y = f (x) - 11$ . Substituting $- 2 x - 4$ for $y$ in the equation $y = f (x) - 11$ yields $- 2 x - 4 = f (x) - 11$ . Adding $11$ to both sides of this equation yields $- 2 x + 7 = f (x)$ . Therefore, the equation $f (x) = - 2 x + 7$ defines the linear function $f$ .

Choice A 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.