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Advanced Math Difficulty: Hard

$x$ $x$	$y$ $y$
$21$ $21$	$- 8$ $negative 8$
$23$ $23$	$8$ $8$
$25$ $25$	$negative 8$

The table shows three values of $x$ and their corresponding values of $y$ , where $y = f (x) + 4$ and $f$ is a quadratic function. What is the y-coordinate of the y-intercept of the graph of $y = f (x)$ in the xy-plane?

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Explanation

The correct answer is $negative 2,112$ . It's given that $f$ is a quadratic function. It follows that $f$ can be defined by an equation of the form $f (x) = a {(x - h)}^{2} + k$ , where $a$ , $h$ , and $k$ are constants. It's also given that the table shows three values of $x$ and their corresponding values of $y$ , where $y = f (x) + 4$ . Substituting $a {(x - h)}^{2} + k$ for $f (x)$ in this equation yields $y = a {(x - h)}^{2} + k + 4$ . This equation represents a quadratic relationship between $x$ and $y$ , where $k plus 4$ is either the maximum or the minimum value of $y$ , which occurs when $x = h$ . For quadratic relationships between $x$ and $y$ , the maximum or minimum value of $y$ occurs at the value of $x$ halfway between any two values of $x$ that have the same corresponding value of $y$ . The table shows that x-values of $21$ and $25$ correspond to the same y-value, $negative 8$ . Since $23$ is halfway between $21$ and $25$ , the maximum or minimum value of $y$ occurs at an x-value of $23$ . The table shows that when $x equals 23$ , $y equals 8$ . It follows that $h equals 23$ and $k + 4 = 8$ . Subtracting $4$ from both sides of the equation $k + 4 = 8$ yields $k equals 4$ . Substituting $23$ for $h$ and $4$ for $k$ in the equation $y = a {(x - h)}^{2} + k + 4$ yields $y = a {(x - 23)}^{2} + 4 + 4$ , or $y = a {(x - 23)}^{2} + 8$ . The value of $a$ can be found by substituting any x-value and its corresponding y-value for $x$ and $y$ , respectively, in this equation. Substituting $25$ for $x$ and $negative 8$ for $y$ in this equation yields $- 8 = a {(25 - 23)}^{2} + 8$ , or $- 8 = a {(2)}^{2} + 8$ . Subtracting $8$ from both sides of this equation yields $- 16 = a {(2)}^{2}$ , or $- 16 = 4 a$ . Dividing both sides of this equation by $4$ yields $- 4 = a$ . Substituting $negative 4$ for $a$ , $23$ for $h$ , and $4$ for $k$ in the equation $f (x) = a {(x - h)}^{2} + k$ yields $f (x) = - 4 {(x - 23)}^{2} + 4$ . The y-intercept of the graph of $y = f (x)$ in the xy-plane is the point on the graph where $x equals 0$ . Substituting $0$ for $x$ in the equation $f (x) = - 4 {(x - 23)}^{2} + 4$ yields $f (0) = - 4 {(0 - 23)}^{2} + 4$ , or $f (0) = - 4 {(- 23)}^{2} + 4$ . This is equivalent to $f (0) = - 2,112$ , so the y-intercept of the graph of $y = f (x)$ in the xy-plane is $left parenthesis 0 comma negative 2,112 right parenthesis$ . Thus, the y-coordinate of the y-intercept of the graph of $y = f (x)$ in the xy-plane is $negative 2,112$ .