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Advanced Math / Nonlinear functions Difficulty: Hard

$x$ $x$	$g (x)$ $g (x)$
$- 27$ $negative 27$	$3$ $3$
$- 9$ $negative 9$	$0$ $0$
$21$ $21$	$5$ $5$

The table shows three values of $x$ and their corresponding values of $g (x)$ , where $g (x) = \frac{f (x)}{x + 3}$ and $f$ is a linear function. What is the y-intercept of the graph of $y = f (x)$ in the xy-plane?

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Explanation

Choice A is correct. It's given that the table shows values of $x$ and their corresponding values of $g (x)$ , where $g (x) = \frac{f (x)}{x + 3}$ . It's also given that $f$ is a linear function. It follows that an equation that defines $f$ can be written in the form $f (x) = m x + b$ , where $m$ represents the slope and $b$ represents the y-coordinate of the y-intercept $left parenthesis 0 comma b right parenthesis$ of the graph of $y = f (x)$ in the xy-plane. The slope of the graph of $y = f (x)$ can be found using two points, $left parenthesis x 1 comma y 1 right parenthesis$ and $left parenthesis x 2 comma y 2 right parenthesis$ , that are on the graph of $y = f (x)$ , and the formula $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ . Since the table shows values of $x$ and their corresponding values of $g (x)$ , substituting values of $x$ and $g (x)$ in the equation $g (x) = \frac{f (x)}{x + 3}$ can be used to define function $f$ . Using the first pair of values from the table, $x = - 27$ and $g left parenthesis x right parenthesis equals 3$ , yields $3 = \frac{f (- 27)}{- 27 + 3}$ , or $3 = \frac{f (- 27)}{- 24}$ . Multiplying each side of this equation by $negative 24$ yields $- 72 = f (- 27)$ , so the point $(- 27, - 72)$ is on the graph of $y = f (x)$ . Using the second pair of values from the table, $x = - 9$ and $g left parenthesis x right parenthesis equals 0$ , yields $0 = \frac{f (- 9)}{- 9 + 3}$ , or $0 = \frac{f (- 9)}{- 6}$ . Multiplying each side of this equation by $negative 6$ yields $0 = f (- 9)$ , so the point $left parenthesis negative 9 comma 0 right parenthesis$ is on the graph of $y = f (x)$ . Substituting $(- 27, - 72)$ and $left parenthesis negative 9 comma 0 right parenthesis$ 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 $m = \frac{0 - (- 72)}{- 9 - (- 27)}$ , or $m equals 4$ . Substituting $4$ for $m$ in the equation $f (x) = m x + b$ yields $f (x) = 4 x + b$ . Since $0 = f (- 9)$ , substituting $negative 9$ for $x$ and $0$ for $f (x)$ in the equation $f (x) = 4 x + b$ yields $0 = 4 (- 9) + b$ , or $0 = - 36 + b$ . Adding $36$ to both sides of this equation yields $36 equals b$ . It follows that $36$ is the y-coordinate of the y-intercept $left parenthesis 0 comma b right parenthesis$ of the graph of $y = f (x)$ . Therefore, the y-intercept of the graph of $y = f (x)$ is $left parenthesis 0 comma 36 right parenthesis$ .

Choice B is incorrect. $12$ is the y-coordinate of the y-intercept of the graph of $y = g (x)$ .

Choice C is incorrect. $4$ is the slope of the graph of $y = f (x)$ .

Choice D is incorrect. $negative 9$ is the x-coordinate of the x-intercept of the graph of $y = f (x)$ .