write an equation (any form) for the quadratic graphed below: y = question help: video

Explanation:

Step1: Identify vertex

The vertex of the parabola is at $(-1, 4)$. The vertex - form of a quadratic equation is $y=a(x - h)^2 + k$, where $(h,k)$ is the vertex. So, $h=-1$ and $k = 4$, and the equation becomes $y=a(x + 1)^2+4$.

Step2: Find the value of $a$

The parabola passes through the point $(0,2)$. Substitute $x = 0$ and $y = 2$ into $y=a(x + 1)^2+4$: [ \begin{align*} 2&=a(0 + 1)^2+4\ 2&=a+4\ a&=2 - 4=-2 \end{align*} ]

Step3: Write the quadratic equation

Substitute $a=-2$ back into the vertex - form equation. The quadratic equation is $y=-2(x + 1)^2+4$. Expanding it gives $y=-2(x^{2}+2x + 1)+4=-2x^{2}-4x-2 + 4=-2x^{2}-4x + 2$.

Answer:

$y=-2(x + 1)^2+4$ (vertex - form) or $y=-2x^{2}-4x + 2$ (standard - form)