Chapter 7.5, Problem 21WE

To determine

The equation of the parabola in the form $y-k=a{\left(x-h\right)}^2$ having vertex $\left(0,0\right)$ and contains the point $\left(-3,3\right)$ .

Answer to Problem 21WE

The equation of the parabola is $y=\frac{1}{3}{x}^2$ .

Explanation of Solution

Given information: It is given that the vertex of the parabola is $\left(0,0\right)$ and it passes through the point $\left(-3,3\right)$ .

Calculation: Since the parabola has the vertex $\left(0,0\right)$ and it passes through the point $\left(-3,3\right)$ therefore satisfy the vertex $\left(0,0\right)$ and point $\left(-3,3\right)$ in the form $y-k=a{\left(x-h\right)}^2$ .

  $\begin{array}{c}y-\left(0\right)=a{\left(x-\left(0\right)\right)}^2\mathrm{....}\left\{\left(h,k\right)\to \left(0,0\right)\right\}\\ y=a{x}^2\mathrm{....}\left(1\right)\\ \left(3\right)=a{\left(-3\right)}^2\mathrm{....}\left\{\left(x,y\right)\to \left(-3,3\right)\right\}\\ 3=9\left(a\right)\mathrm{....}\left(\text{Simplified}\right)\\ \text{or, }a=\frac{1}{3}\end{array}$

Now substitute $a=\frac{1}{3}$ in (1) will give the equation of the parabola. Therefore from (1), equation of parabola is $y=\frac{1}{3}{x}^2$ .