Question

Finding the Equation of a Parabola

Find an equation of the parabola whose graph is shown

Short answer

The equation of parabola is $y^2=-3x$

Step-by-step solution

Step 1. Given information

Graph of parabola is given as-

Step 2. Concept used

The equation of a parabola having a vertex at origin is –

$y^2=4px$

Where, $p>0$ then parabola opens to the right

$p<0$ then parabola opens to the left.

The directrix is the right side of $y$-axis is so, which means the focus should be negative, the parabola opens to the left and the coordinates of focus are obtained. On substituting the value of $p$ in the equation $y^2=4px$ to obtain an equation of a parabola.

Step 3. Calculation

The focal diameter-

$\begin{array}{l}=\frac{3}{2}+\frac{3}{2}units\\ =3units.\end{array}$

The focal diameter of a parabola $y^2=4px$ is $4p$. Then, we get –

$\begin{array}{l}4p=3\\ \Rightarrow p=\frac{3}{4}\end{array}$

The focus is on the negative $x$ -axis. Therefore,

$p=-\frac{3}{4}$

Now, substituting the value of $p$ in equation $y^2=4px$ , we get the equation as.

$\begin{array}{l}{y}^2=4\left(-\frac{3}{4}\right)x\\ \Rightarrow y^2=-3x\end{array}$