which equation represents a graph with a vertex at (-3, 2)?\no y = 4x² + 24x + 38\no y = 4x² - 24x + 38\no y…

Explanation:

Step1: Recall vertex - form of quadratic equation

The vertex - form of a quadratic equation is $y=a(x - h)^2+k$, where $(h,k)$ is the vertex of the parabola. Given the vertex $(-3,2)$, the equation is $y=a(x + 3)^2+2$. Expand it: $y=a(x^{2}+6x + 9)+2=ax^{2}+6ax+9a + 2$. Here $a = 4$. So $y=4x^{2}+24x+36 + 2=4x^{2}+24x+38$.

Step2: Check each option

We expand and compare. For option 1: [ \begin{align*} y&=4x^{2}+24x + 38 \end{align*} ] For option 2: [ \begin{align*} y&=4x^{2}-24x + 38 \end{align*} ] For option 3: [ \begin{align*} y&=4x^{2}+12x + 2 \end{align*} ] For option 4: [ \begin{align*} y&=4x^{2}+16x + 13 \end{align*} ]

Answer:

$y = 4x^{2}+24x + 38$ (First option)