Problem
The keyword “write an equation for the ellipse graphed in standard form” requires reading key points from a graph and writing the ellipse equation in standard form. Since no specific graph is provided, the following representative graph description is used:
The ellipse is centered at the point halfway between its left and right vertices. The vertices are at \((-3,-1)\) and \((7,-1)\), and the co-vertices are at \((2,2)\) and \((2,-4)\).
Standard forms for an ellipse
Horizontal major axis: \[ \frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1 \] where the vertices are \((h\pm a,\,k)\) and co-vertices are \((h,\,k\pm b)\).
Vertical major axis: \[ \frac{(x-h)^2}{b^2}+\frac{(y-k)^2}{a^2}=1 \] where the vertices are \((h,\,k\pm a)\) and co-vertices are \((h\pm b,\,k)\).
Step-by-step: write the ellipse equation in standard form
1) Identify the center \((h,k)\)
The center is the midpoint of the two opposite vertices on the major axis. Using \((-3,-1)\) and \((7,-1)\):
\[ h=\frac{-3+7}{2}=2,\qquad k=\frac{-1+(-1)}{2}=-1 \]
So the center is \((h,k)=(2,-1)\).
2) Determine whether the major axis is horizontal or vertical
The vertices \((-3,-1)\) and \((7,-1)\) share the same \(y\)-coordinate, so they lie on a horizontal line. This indicates a horizontal major axis.
3) Find the semi-axis lengths \(a\) and \(b\)
The distance from the center to a vertex is \(a\). From \((2,-1)\) to \((7,-1)\):
\[ a=|7-2|=5 \]
The distance from the center to a co-vertex is \(b\). From \((2,-1)\) to \((2,2)\):
\[ b=|2-(-1)|=3 \]
4) Substitute into the standard form
Since the major axis is horizontal, use \(\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1\). Substitute \(h=2\), \(k=-1\), \(a=5\), \(b=3\):
\[ \frac{(x-2)^2}{25}+\frac{(y+1)^2}{9}=1 \]
Key points check (quick verification)
Plugging a vertex such as \((7,-1)\) into the equation should satisfy it:
\[ \frac{(7-2)^2}{25}+\frac{(-1+1)^2}{9} =\frac{25}{25}+0 =1 \]
The same check works for \((-3,-1)\), \((2,2)\), and \((2,-4)\).
Summary table (read from the graph)
| Feature | From the graph | Value used in standard form |
|---|---|---|
| Center | Midpoint of \((-3,-1)\) and \((7,-1)\) | \((h,k)=(2,-1)\) |
| Major axis direction | Vertices share the same \(y\) | Horizontal |
| Semi-major axis | Distance center → vertex | \(a=5\) |
| Semi-minor axis | Distance center → co-vertex | \(b=3\) |
| Ellipse equation | Standard form substitution | \(\frac{(x-2)^2}{25}+\frac{(y+1)^2}{9}=1\) |
Visualization: ellipse with center, vertices, and co-vertices
Final answer
The ellipse equation in standard form is \[ \frac{(x-2)^2}{25}+\frac{(y+1)^2}{9}=1. \]