Chapter 9.1, Problem 49E

To determine

To find: The standard form of the equation of the parabola with vertex at origin and given graph.

  

Answer to Problem 49E

The equation of the parabola with vertex at origin and given graph is $x^2=\frac{3}{2}y$ .

Explanation of Solution

Given information:

The graph of the parabola is given below.

  

Calculation:

As shown in the figure, the parabola is vertically upwards parabola with vertex at origin and passing through $\left(3,6\right)$ .

The standard form of general equation of a vertically upwards parabola is ${\left(x-h\right)}^2=4p\left(y-k\right)$ .

Substitute $0$ , $0$ for $h$ , $k$ in the general equation.

  $\begin{array}{c}{\left(x-h\right)}^2=4p\left(y-k\right)\\ {\left(x-0\right)}^2=4p\left(y-0\right)\\ x^2=4py\end{array}$

Substitute $3$ , $6$ for $x$ , $y$ in the equation of parabola.

  $\begin{array}{c}{x}^2=4py\\ 3^2=4p\left(6\right)\\ 9=24p\\ p=\frac{3}{8}\end{array}$

Substitute $p=\frac{3}{8}$ in the general equation.

  $\begin{array}{c}{x}^2=4py\\ x^2=4\left(\frac{3}{8}\right)y\\ x^2=\frac{3}{2}y\end{array}$

Therefore, the equation of the parabola with vertex at origin and given graph is $x^2=\frac{3}{2}y$ .