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Algebra / Linear functions Difficulty: Medium

In the linear function $h$ , $h left parenthesis 28 right parenthesis equals 15$ and $h left parenthesis 26 right parenthesis equals 22$ . Which equation defines $h$ ?

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Explanation

Choice D is correct. An equation defining $h$ can be written in the form $y = m x + b$ , where $y = h (x)$ , $m$ represents the slope of the graph of $y = h (x)$ in the xy-plane, and $b$ represents the y-coordinate of the y-intercept of the graph. It’s given that $h left parenthesis 28 right parenthesis equals 15$ and $h left parenthesis 26 right parenthesis equals 22$ . It follows that the points $left parenthesis 28 comma 15 right parenthesis$ and $left parenthesis 26 comma 22 right parenthesis$ are on the graph of $y = h (x)$ in the xy-plane. The slope can be found by using any two points, $left parenthesis x 1 comma y 1 right parenthesis$ and $left parenthesis x 2 comma y 2 right parenthesis$ , and the formula $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ . Substituting $left parenthesis 28 comma 15 right parenthesis$ and $left parenthesis 26 comma 22 right parenthesis$ for $left parenthesis x 1 comma y 1 right parenthesis$ and $left parenthesis x 2 comma y 2 right parenthesis$ , respectively, in the slope formula yields $m = \frac{22 - 15}{26 - 28}$ , or $m = - \frac{7}{2}$ . Substituting $- \frac{7}{2}$ for $m$ and $left parenthesis 28 comma 15 right parenthesis$ for $(x, y)$ in the equation $y = m x + b$ yields $15 = (- \frac{7}{2}) (28) + b$ , or $15 = - 98 + b$ . Adding $98$ to both sides of this equation yields $113 equals b$ . Substituting $- \frac{7}{2}$ for $m$ and $113$ for $b$ in the equation $y = m x + b$ yields $y = - \frac{7}{2} x + 113$ . Since $y = h (x)$ , it follows that the equation that defines $h$ is $h (x) = - \frac{7}{2} x + 113$ .

Choice A is incorrect. For this function, $h (26) = \frac{109}{7}$ , not $h left parenthesis 26 right parenthesis equals 22$ .

Choice B is incorrect. For this function, $h left parenthesis 28 right parenthesis equals 105$ , not $h left parenthesis 28 right parenthesis equals 15$ , and $h (26) = \frac{739}{7}$ , not $h left parenthesis 26 right parenthesis equals 22$ .

Choice C is incorrect. For this function, $h (28) = - 75$ , not $h left parenthesis 28 right parenthesis equals 15$ , and $h (26) = - 68$ , not $h left parenthesis 26 right parenthesis equals 22$ .