find the equation of the quadratic function g whose graph is shown below.

Explanation:

Step1: Recall quadratic - vertex form

The vertex - form of a quadratic function is $g(x)=a(x - h)^2 + k$, where $(h,k)$ is the vertex of the parabola. From the graph, the vertex is $(-2,-5)$, so $h=-2$ and $k = - 5$. Then $g(x)=a(x + 2)^2-5$.

Step2: Find the value of $a$

We know that the graph passes through the point $(-1,-4)$. Substitute $x=-1$ and $g(x)=-4$ into $g(x)=a(x + 2)^2-5$. [ \begin{align*} -4&=a(-1 + 2)^2-5\ -4&=a(1)^2-5\ -4&=a-5\ a&=1 \end{align*} ]

Step3: Write the quadratic function

Substitute $a = 1$ into $g(x)=a(x + 2)^2-5$. So $g(x)=(x + 2)^2-5$. Expand it: $g(x)=x^{2}+4x+4 - 5=x^{2}+4x - 1$.

Answer:

$g(x)=x^{2}+4x - 1$