Chapter 4.7, Problem 40PPS

To determine

To write: the standard form of quadratic equation in the vertex form as mentioned in the problem.

Answer to Problem 40PPS

The vertex form of quadratic equation is $y=\frac{3}{2}{\left(x+3\right)}^2+2$ .

Explanation of Solution

Given information:

The points on the graph is mentioned below.

  

Formula used:

The vertex form of the quadratic equation is $f\left(x\right)=a{\left(x-h\right)}^2+k$ where $\left(h,k\right)$ is the vertex.

Calculation:

Consider the points mentioned in the graph below.

  

Recall that the vertex form of the quadratic equation is $f\left(x\right)=a{\left(x-h\right)}^2+k\mathrm{......}\left(i\right)$ where $\left(h,k\right)$ is the vertex.

As the graph mentioned above we observe that vertex is $\left(-3,2\right)=\left(h,k\right)$ .

Therefore the equation $\left(i\right)$ becomes $y=a{\left(x+3\right)}^2+\mathrm{2.....}\left(ii\right)$ .

Now since the point $\left(-1,8\right)$ lies on the graph.

Therefore $x=-1\&y=8$ will be point which lie on $\left(ii\right)$ .

Therefore equation $\left(ii\right)$ becomes $8=a{\left(-1+3\right)}^2+2$ .

Therefore $8=a{\left(-1+3\right)}^2+2\Rightarrow 8=a*4+2\Rightarrow 6=a*4\Rightarrow a=\frac{3}{2}$ .

Therefore the vertex form of the quadratic equation is $y=\frac{3}{2}{\left(x+3\right)}^2+2$ .