Question

Write the quadratic function in form $f\left(x\right)=a{(x-h)}^2+k$whose graph is shown.

Short answer

The equation of quadratic function shown in the given graph is :

$f\left(x\right)={\left(x+3\right)}^2-1$

Step-by-step solution

Step 1. Given information.  

The vertex of the given graph is $\left(-3,-1\right)$.They-intercept is$\left(0,8)\right)$

Step 2. Substitute the vertex 

Since it is quadratic, we start with the

$f\left(x\right)=a{(x-h)}^2+k\_\_\_\_\left(1\right)$

And we know that $\left(h,k\right)$is the vertex.

So, substituting the $$\begin{gathered} h=-3 \\ k=-1 \end{gathered}$$in the equation (1), we get:

$$\begin{gathered} f\left(x\right)=a\left(x-(-3\right){)}^2-1 \\ f\left(x\right)=a{\left(x+3\right)}^2-1\_\_\_\_\left(2\right) \end{gathered}$$

Step 3. Find the value of a

The y-intercept is $\left(0,8\right)$

Substituting $$\begin{gathered} x=0 \\ f\left(x\right)=8 \end{gathered}$$ in equation (2), we get:

localid="1664210862662" $$\begin{gathered} f\left(x\right)=a{\left(x+3\right)}^2-1 \\ 8=a{(0+3)}^2-1 \\ 8+1=9a \\ 9a=9 \\ a=1 \end{gathered}$$

Step 4. Find the equation of the parabola 

Substituting the value $a=1$in the equation (2), we get:

$f\left(x\right)={\left(x+3\right)}^2-1$