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Algebra Difficulty: Easy

For the linear function $f$ , the graph of $y = f (x)$ in the xy-plane has a slope of $2$ and has a y-intercept at $left parenthesis 0 comma negative 5 right parenthesis$ . Which equation defines $f$ ?

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Explanation

Choice D is correct. An equation defining the linear function $f$ can be written in the form $f (x) = m x + b$ , where $m$ is the slope and $left parenthesis 0 comma b right parenthesis$ is the y-intercept of the graph of $y = f (x)$ in the xy-plane. It’s given that the graph of $y = f (x)$ has a slope of $2$ . Therefore, $m equals 2$ . It’s also given that the graph of $y = f (x)$ has a y-intercept at $left parenthesis 0 comma negative 5 right parenthesis$ . Therefore, $b = - 5$ . Substituting $2$ for $m$ and $negative 5$ for $b$ in the equation $f (x) = m x + b$ yields $f (x) = 2 x - 5$ . Thus, the equation that defines $f$ is $f (x) = 2 x - 5$ .

Choice A is incorrect. For this function, the graph of $y = f (x)$ in the xy-plane has a slope of $\frac{1}{2}$ , not $2$ .

Choice B is incorrect. For this function, the graph of $y = f (x)$ in the xy-plane has a slope of $- \frac{1}{2}$ , not $2$ .

Choice C is incorrect. For this function, the graph of $y = f (x)$ in the xy-plane has a slope of $negative 2$ , not $2$ .