SAT Question 09d21d79: Nonlinear Functions answer and explanation with Desmos solution

Since the graph passes through the point $\left(0,-6\right)$, 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 $-6=2{\left(0\right)}^2+b\left(0\right)+c$, or $-6=c$. Therefore, the value of $c$ is $-6$. Substituting $-6$ for $c$ in the given equation yields $y=2x^2+bx-6$. Since the graph passes through the point $\left(-1,-8\right)$, 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 $y=2x^2+bx-6$ yields $-8=2{\left(-1\right)}^2+b\left(-1\right)-6$, 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 $\left(4\right)\left(-6\right)$, or $-24$.

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