OG详解-OG6 数学1 Q21

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 = 2x² +bx-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 = 2x² +bx-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 bc 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 = 2x² +bx+ 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 x² term when the equation is written in standard form. It follows that a = 2. Substituting 2 for a, -1for 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(x² +2x+1)-8. Multiplying each term inside the parentheses on the right-hand side of this equation by 2 yields y = 2x² +4x+2-8, which is equivalent to y = 2x² +4x-6. From the given equation y = 2x² +bx+ c, it follows that the value of b is 4 and the value of c is -6. Therefore, the value of bc is (4)(-6), or -24.