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Algebra / Linear functions Difficulty: Medium

In the xy-plane, the graph of the linear function $f$ contains the points $left parenthesis 0 comma 3 right parenthesis$ and $left parenthesis 7 comma 31 right parenthesis$ . Which equation defines $f$ , where $y = f (x)$ ?

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Explanation

Choice C is correct. In the xy-plane, an equation of the graph of a linear function can be written in the form $f (x) = m x + b$ , where $m$ represents the slope and $left parenthesis 0 comma b right parenthesis$ represents the y-intercept of the graph of $y = f (x)$ . It’s given that the graph of the linear function $f$ , where $y = f (x)$ , in the xy-plane contains the point $left parenthesis 0 comma 3 right parenthesis$ . Thus, $b equals 3$ . The slope of the graph of a line containing any two points $left parenthesis x 1 comma y 1 right parenthesis$ and $left parenthesis x 2 comma y 2 right parenthesis$ can be found using the slope formula, $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ . Since it’s given that the graph of the linear function $f$ contains the points $left parenthesis 0 comma 3 right parenthesis$ and $left parenthesis 7 comma 31 right parenthesis$ , it follows that the slope of the graph of the line containing these points is $m = \frac{31 - 3}{7 - 0}$ , or $m equals 4$ . Substituting $4$ for $m$ and $3$ for $b$ in $f (x) = m x + b$ yields $f (x) = 4 x + 3$ .

Choice A is incorrect. This function represents a graph with a slope of $28$ and a y-intercept of $left parenthesis 0 comma 34 right parenthesis$ .

Choice B is incorrect. This function represents a graph with a slope of $3$ and a y-intercept of $left parenthesis 0 comma 38 right parenthesis$ .

Choice D is incorrect. This function represents a graph with a slope of $7$ and a y-intercept of $left parenthesis 0 comma 3 right parenthesis$ .