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Expert-verified Found in: Page 643 ### Precalculus Enhanced with Graphing Utilities

Book edition 6th
Author(s) Sullivan
Pages 1200 pages
ISBN 9780321795465 # 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 $\left(2,3\right)$.

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 : See the 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

${3}^{2}=4a\left(2\right)\phantom{\rule{0ex}{0ex}}9=8a⇒a=\frac{9}{8}$.

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,

${y}^{2}=\frac{9}{2}x\phantom{\rule{0ex}{0ex}}{y}^{2}=\frac{9}{2}\left(\frac{9}{8}\right)\phantom{\rule{0ex}{0ex}}{y}^{2}=\frac{81}{16}\phantom{\rule{0ex}{0ex}}y=±\frac{9}{4}$.

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  ### Want to see more solutions like these? 