The parabola opens upward. The vertex is at point (negative 1 comma negative 8). The parabola passes through the following

Question

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

Accepted Answer

The correct answer is - 24 . Since the graph passes through the point 0 , - 6 , it follows that when the value of x is 0 , the value of y is - 6 . Substituting 0 for x and - 6 for y in the given equation yields - 6 = 2 0 2 + b 0 + c , or - 6 = c . Therefore, the value of c is - 6 . Substituting - 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 - 1 , the value of y is - 8 . Substituting - 1 for x and - 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 - 1 yields 4 = b . Since the value of b is 4 and the value of c is - 6 , it follows that the value of b c is 4 - 6 , or - 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 = 2 . Substituting 2 for a , - 1 for h , and - 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 - 6 . Therefore, the value of b c is 4 - 6 , or - 24 .

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