find a formula for the quadratic function depicted in the following graph.

Explanation:

Step1: Identify vertex form

The vertex - form of a quadratic function is $y = a(x - h)^2+k$, where $(h,k)$ is the vertex of the parabola. From the graph, the vertex is $(- 2,-3)$, so $h=-2$ and $k = - 3$. Then the function is $y=a(x + 2)^2-3$.

Step2: Find the value of $a$

The parabola passes through the point $(0,1)$. Substitute $x = 0$ and $y = 1$ into $y=a(x + 2)^2-3$. We get $1=a(0 + 2)^2-3$. [ \begin{align*} 1&=4a-3\ 4a&=4\ a&=1 \end{align*} ]

Step3: Write the quadratic function

Substitute $a = 1$ into $y=a(x + 2)^2-3$. The quadratic function is $y=(x + 2)^2-3=x^{2}+4x+4 - 3=x^{2}+4x + 1$.

Answer:

$y=x^{2}+4x + 1$