which equation represents a graph with a vertex at (-3, 2)?\n$y = 4x^{2}+24x + 38$\n$y = 4x^{2}-24x +…

Explanation:

Step1: Recall vertex - form of quadratic function

The vertex - form of a quadratic function is $y=a(x - h)^2+k$, where $(h,k)$ is the vertex of the parabola. Here, $h=-3$ and $k = 2$, so the equation is $y=a(x + 3)^2+2$. Expand it: $y=a(x^{2}+6x + 9)+2=ax^{2}+6ax+9a + 2$. We can also use the formula for the x - coordinate of the vertex of a quadratic function $y = ax^{2}+bx + c$, which is $x=-\frac{b}{2a}$.

Step2: Check option A

For $y = 4x^{2}+24x + 38$, where $a = 4$, $b = 24$. Using the formula $x=-\frac{b}{2a}$, we have $x=-\frac{24}{2\times4}=-\frac{24}{8}=-3$. Substitute $x=-3$ into $y = 4x^{2}+24x + 38$: $y=4\times(-3)^{2}+24\times(-3)+38=4\times9-72 + 38=36-72 + 38=2$.

Step3: Check option B

For $y = 4x^{2}-24x + 38$, $a = 4$, $b=-24$. Then $x=-\frac{-24}{2\times4}=\frac{24}{8}=3\neq - 3$.

Step4: Check option C

For $y = 4x^{2}+12x + 2$, $a = 4$, $b = 12$. Then $x=-\frac{12}{2\times4}=-\frac{12}{8}=-\frac{3}{2}\neq - 3$.

Step5: Check option D

For $y = 4x^{2}+16x + 13$, $a = 4$, $b = 16$. Then $x=-\frac{16}{2\times4}=-\frac{16}{8}=-2\neq - 3$.

Answer:

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