Question

graph the given ellipse. begin by clicking on the ellipse graphing tool. then, click on the center. from the center, count the vertical and horizontal distances so that the vertices and co - vertices end up in the correct locations and click once. the ellipse will then be drawn for you.

$\frac{(x - 5)^2}{16}+\frac{(y - 6)^2}{4}=1$

Explanation:

Step1: Identify the center

The standard form of an ellipse is $\frac{(x - h)^2}{a^2}+\frac{(y - k)^2}{b^2}=1$, where $(h,k)$ is the center. For the given equation $\frac{(x - 5)^2}{16}+\frac{(y - 6)^2}{4}=1$, the center $(h,k)$ is $(5,6)$.

Step2: Determine $a$ and $b$

Compare with the standard form: $a^2 = 16$, so $a = 4$; $b^2=4$, so $b = 2$. Since $a>b$, the major axis is horizontal.

Step3: Find vertices and co - vertices

• Vertices: For a horizontal major axis, the vertices are at $(h\pm a,k)$. So, $(5\pm4,6)$, which are $(1,6)$ and $(9,6)$.
• Co - vertices: For a horizontal major axis, the co - vertices are at $(h,k\pm b)$. So, $(5,6\pm2)$, which are $(5,4)$ and $(5,8)$.

To graph the ellipse:

1. Plot the center $(5,6)$.
2. Plot the vertices $(1,6)$ and $(9,6)$.
3. Plot the co - vertices $(5,4)$ and $(5,8)$.
4. Use the ellipse graphing tool (as per the problem's instruction) to draw the ellipse passing through these vertices and co - vertices.

Answer:

To graph the ellipse $\frac{(x - 5)^2}{16}+\frac{(y - 6)^2}{4}=1$:

• Center: $(5,6)$
• Vertices: $(1,6)$ and $(9,6)$
• Co - vertices: $(5,4)$ and $(5,8)$

Then use the ellipse graphing tool to draw the ellipse by plotting these points and using the tool's functionality.