determine the equation of the parabola graphed below. note: when responding if the number is negative you…

Explanation:

Step1: Recall the vertex - form of a parabola

The vertex - form of a parabola is $y=a(x - h)^2+k$, where $(h,k)$ is the vertex of the parabola. From the graph, the vertex of the parabola is $(1,-2)$, so $h = 1$ and $k=-2$. Then the equation becomes $y=a(x - 1)^2-2$.

Step2: Find the value of $a$

The parabola passes through the point $(0,0)$. Substitute $x = 0$ and $y = 0$ into the equation $y=a(x - 1)^2-2$. We get $0=a(0 - 1)^2-2$, which simplifies to $0=a - 2$. Solving for $a$, we add 2 to both sides of the equation: $a=2$.

Step3: Write the final equation

Substitute $a = 2$ back into the vertex - form equation. The equation of the parabola is $y=2(x - 1)^2-2$. Expand it: $y=2(x^{2}-2x + 1)-2=2x^{2}-4x+2 - 2=2x^{2}-4x$.

Answer:

$y = 2x^{2}-4x$