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Algebra / Linear functions Difficulty: Easy

For the function $f$ , the graph of $y = f (x)$ in the xy-plane has a slope of $3$ and passes through the point $left parenthesis 0 comma negative 8 right parenthesis$ . Which equation defines $f$ ?

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Explanation

Choice B is correct. An equation defining a linear function can be written in the form $f (x) = m x + b$ , where $m$ and $b$ are constants, $m$ is the slope of the graph of $y = f (x)$ in the xy-plane, and $left parenthesis 0 comma b right parenthesis$ is the y-intercept of the graph. It's given that for the function $f$ , the graph of $y = f (x)$ in the xy-plane has a slope of $3$ . Therefore, $m equals 3$ . It's also given that this graph passes through the point $left parenthesis 0 comma negative 8 right parenthesis$ . Therefore, the y-intercept of the graph is $left parenthesis 0 comma negative 8 right parenthesis$ , and it follows that $b = - 8$ . Substituting $3$ for $m$ and $negative 8$ for $b$ in the equation $f (x) = m x + b$ yields $f (x) = 3 x - 8$ . Thus, the equation that defines $f$ is $f (x) = 3 x - 8$ .

Choice A is incorrect. For this function, the graph of $y = f (x)$ in the xy-plane passes through the point $left parenthesis 0 comma 0 right parenthesis$ , not $left parenthesis 0 comma negative 8 right parenthesis$ .

Choice C is incorrect. For this function, the graph of $y = f (x)$ in the xy-plane passes through the point $left parenthesis 0 comma 5 right parenthesis$ , not $left parenthesis 0 comma negative 8 right parenthesis$ .

Choice D is incorrect. For this function, the graph of $y = f (x)$ in the xy-plane passes through the point $left parenthesis 0 comma 11 right parenthesis$ , not $left parenthesis 0 comma negative 8 right parenthesis$ .