Question

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

Focus is at$\left(-4,0\right)$and vertex is at$\left(0,0\right)$.

Short answer

The equation of a parabola is

$y^2=-16x$.

The latus rectum points are$\left(-4,8\right)$and$\left(-4,-8\right)$.

Step-by-step solution

Step 1. Given Information.

The given points of an equation are :

focus$\left(-4,0\right)$and vertex$\left(0,0\right)$.

Step 2. Equation of a parabola.

The distance from the vertex $\left(0,0\right)$and its focus $\left(-4,0\right)$is $a=-4$.

The equation of a parabola be :

$$\begin{gathered} y^2=4ax \\ {y}^2=-16x \end{gathered}$$.

Step 3. Calculation of a latus rectum.

The two points that determine the latus rectum by letting $x=-4$. Then,

$$\begin{gathered} y^2=-16x \\ {y}^2=-16\left(-4\right) \\ {y}^2=64 \\ y=\pm 8 \end{gathered}$$.

The points are$\left(-4,8\right)$and$\left(-4,-8\right)$.

Step 4. Graphing utility.

The graph of a parabola be: