Question

Find the equation of the parabola described. Find the two points that define the latus rectum, and graph the equation by hand.

Vertex at $\left(0,0\right)$, axis of symmetry is the x-axis; containing the point$\left(2,3\right)$.

Short answer

The equation of a parabola is $y^2=\frac{9}{2}x$. The points are $\left(\frac{9}{8},\frac{9}{4}\right)$and $\left(\frac{9}{8},-\frac{9}{4}\right)$.

The graph of a parabola is :

Step-by-step solution

Step 1. Given Information.

The given vertex is at the point $\left(0,0\right)$and the axis of symmetry is the x-axis and containing the point$\left(2,3\right)$.

Step 2. Equation of a parabola.

The vertex is at the origin, the axis of symmetry is the x-axis and the graph contains a point in the first quadrant. The general form of the equation is

$y^2=4ax$

Because the point $\left(2,3\right)$is on the parabola, the coordinates $x=2,y=3$must satisfy the equation of the parabola. Substitute the values, we get

$$\begin{gathered} 3^2=4a\left(2\right) \\ 9=8a\Rightarrow a=\frac{9}{8} \end{gathered}$$.

The equation will be$y^2=\frac{9}{2}x$.

Step 3. Latus Rectum.

The focus is at the point $\left(\frac{9}{8},0\right)$. The two points that determines the latus rectum by letting $x=\frac{9}{8}$. Then,

$$\begin{gathered} y^2=\frac{9}{2}x \\ {y}^2=\frac{9}{2}\left(\frac{9}{8}\right) \\ {y}^2=\frac{81}{16} \\ y=\pm \frac{9}{4} \end{gathered}$$.

The points are$\left(\frac{9}{8},\frac{9}{4}\right)$and$\left(\frac{9}{8},-\frac{9}{4}\right)$.

Step 4. Graphing Utility.

The graph of a parabola is