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Advanced Math / Nonlinear functions Difficulty: Hard

The graph of $y = 2 x^{2} + b x + c$ is shown, where $b$ and $c$ are constants. What is the value of $b c$ ?

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Explanation

The correct answer is $negative 24$ . Since the graph passes through the point $left parenthesis 0 comma negative 6 right parenthesis$ , it follows that when the value of $x$ is $0$ , the value of $y$ is $negative 6$ . Substituting $0$ for $x$ and $negative 6$ for $y$ in the given equation yields $- 6 = 2 {(0)}^{2} + b (0) + c$ , or $- 6 = c$ . Therefore, the value of $c$ is $negative 6$ . Substituting $negative 6$ for $c$ in the given equation yields $y = 2 x^{2} + b x - 6$ . Since the graph passes through the point $(- 1, - 8)$ , it follows that when the value of $x$ is $negative 1$ , the value of $y$ is $negative 8$ . Substituting $negative 1$ for $x$ and $negative 8$ for $y$ in the equation $y = 2 x^{2} + b x - 6$ yields $- 8 = 2 {(- 1)}^{2} + b (- 1) - 6$ , or $- 8 = 2 - b - 6$ , which is equivalent to $- 8 = - 4 - b$ . Adding $4$ to each side of this equation yields $- 4 = - b$ . Dividing each side of this equation by $negative 1$ yields $4 equals b$ . Since the value of $b$ is $4$ and the value of $c$ is $negative 6$ , it follows that the value of $b c$ is $left parenthesis 4 right parenthesis left parenthesis negative 6 right parenthesis$ , or $negative 24$ .

Alternate approach: The given equation represents a parabola in the xy-plane with a vertex at $(- 1, - 8)$ . Therefore, the given equation, $y = 2 x^{2} + b x + c$ , which is written in standard form, can be written in vertex form, $y = a {(x - h)}^{2} + k$ , where $(h, k)$ is the vertex of the parabola and $a$ is the value of the coefficient on the $x^{2}$ term when the equation is written in standard form. It follows that $a equals 2$ . Substituting $2$ for $a$ , $negative 1$ for $h$ , and $negative 8$ for $k$ in this equation yields $y = 2 {(x - (- 1))}^{2} + (- 8)$ , or $y = 2 {(x + 1)}^{2} - 8$ . Squaring the binomial on the right-hand side of this equation yields $y = 2 (x^{2} + 2 x + 1) - 8$ . Multiplying each term inside the parentheses on the right-hand side of this equation by $2$ yields $y = 2 x^{2} + 4 x + 2 - 8$ , which is equivalent to $y = 2 x^{2} + 4 x - 6$ . From the given equation $y = 2 x^{2} + b x + c$ , it follows that the value of $b$ is $4$ and the value of $c$ is $negative 6$ . Therefore, the value of $b c$ is $left parenthesis 4 right parenthesis left parenthesis negative 6 right parenthesis$ , or $negative 24$ .