Ellipse Worksheet 1 - Sketch the graph of each given ellipse equation on the coordinate plane.

Problem: Sketch the graphs of the given ellipses.

The general equation of an ellipse is:

\[
\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1
\]

where:
- \((h, k)\) is the center of the ellipse,
- \(a\) is the semi-major axis length (horizontal if \(a^2 > b^2\)),
- \(b\) is the semi-minor axis length (vertical if \(a^2 < b^2\)).

We will analyze each equation step by step and sketch the corresponding ellipse.

---

1. \(\frac{(x - 2)^2}{25} + \frac{(y - 2)^2}{9} = 1\)

#### Step 1: Identify the center \((h, k)\)
The equation is in the form \(\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1\). Here:
- \(h = 2\),
- \(k = 2\).

So, the center of the ellipse is \((2, 2)\).

#### Step 2: Identify \(a\) and \(b\)
- \(a^2 = 25 \implies a = \sqrt{25} = 5\),
- \(b^2 = 9 \implies b = \sqrt{9} = 3\).

Since \(a > b\), the major axis is horizontal.

#### Step 3: Determine the vertices and co-vertices
- The vertices are located at \((h \pm a, k)\):
\[
(2 \pm 5, 2) = (7, 2) \text{ and } (-3, 2).
\]
- The co-vertices are located at \((h, k \pm b)\):
\[
(2, 2 \pm 3) = (2, 5) \text{ and } (2, -1).
\]

#### Step 4: Sketch the ellipse
- Plot the center \((2, 2)\).
- Mark the vertices \((7, 2)\) and \((-3, 2)\).
- Mark the co-vertices \((2, 5)\) and \((2, -1)\).
- Draw the ellipse through these points, ensuring it is horizontally elongated.

---

2. \((x - 5)^2 + \frac{(y + 2)^2}{25} = 1\)

#### Step 1: Identify the center \((h, k)\)
Here:
- \(h = 5\),
- \(k = -2\).

So, the center of the ellipse is \((5, -2)\).

#### Step 2: Identify \(a\) and \(b\)
Rewrite the equation in standard form:
\[
\frac{(x - 5)^2}{1} + \frac{(y + 2)^2}{25} = 1.
\]
- \(a^2 = 1 \implies a = \sqrt{1} = 1\),
- \(b^2 = 25 \implies b = \sqrt{25} = 5\).

Since \(b > a\), the major axis is vertical.

#### Step 3: Determine the vertices and co-vertices
- The vertices are located at \((h, k \pm b)\):
\[
(5, -2 \pm 5) = (5, 3) \text{ and } (5, -7).
\]
- The co-vertices are located at \((h \pm a, k)\):
\[
(5 \pm 1, -2) = (6, -2) \text{ and } (4, -2).
\]

#### Step 4: Sketch the ellipse
- Plot the center \((5, -2)\).
- Mark the vertices \((5, 3)\) and \((5, -7)\).
- Mark the co-vertices \((6, -2)\) and \((4, -2)\).
- Draw the ellipse through these points, ensuring it is vertically elongated.

---

3. \(\frac{(x - 1)^2}{25} + y^2 = 1\)

#### Step 1: Identify the center \((h, k)\)
Here:
- \(h = 1\),
- \(k = 0\).

So, the center of the ellipse is \((1, 0)\).

#### Step 2: Identify \(a\) and \(b\)
Rewrite the equation in standard form:
\[
\frac{(x - 1)^2}{25} + \frac{y^2}{1} = 1.
\]
- \(a^2 = 25 \implies a = \sqrt{25} = 5\),
- \(b^2 = 1 \implies b = \sqrt{1} = 1\).

Since \(a > b\), the major axis is horizontal.

#### Step 3: Determine the vertices and co-vertices
- The vertices are located at \((h \pm a, k)\):
\[
(1 \pm 5, 0) = (6, 0) \text{ and } (-4, 0).
\]
- The co-vertices are located at \((h, k \pm b)\):
\[
(1, 0 \pm 1) = (1, 1) \text{ and } (1, -1).
\]

#### Step 4: Sketch the ellipse
- Plot the center \((1, 0)\).
- Mark the vertices \((6, 0)\) and \((-4, 0)\).
- Mark the co-vertices \((1, 1)\) and \((1, -1)\).
- Draw the ellipse through these points, ensuring it is horizontally elongated.

---

4. \(\frac{x^2}{49} + (y + 2)^2 = 1\)

#### Step 1: Identify the center \((h, k)\)
Here:
- \(h = 0\),
- \(k = -2\).

So, the center of the ellipse is \((0, -2)\).

#### Step 2: Identify \(a\) and \(b\)
Rewrite the equation in standard form:
\[
\frac{x^2}{49} + \frac{(y + 2)^2}{1} = 1.
\]
- \(a^2 = 49 \implies a = \sqrt{49} = 7\),
- \(b^2 = 1 \implies b = \sqrt{1} = 1\).

Since \(a > b\), the major axis is horizontal.

#### Step 3: Determine the vertices and co-vertices
- The vertices are located at \((h \pm a, k)\):
\[
(0 \pm 7, -2) = (7, -2) \text{ and } (-7, -2).
\]
- The co-vertices are located at \((h, k \pm b)\):
\[
(0, -2 \pm 1) = (0, -1) \text{ and } (0, -3).
\]

#### Step 4: Sketch the ellipse
- Plot the center \((0, -2)\).
- Mark the vertices \((7, -2)\) and \((-7, -2)\).
- Mark the co-vertices \((0, -1)\) and \((0, -3)\).
- Draw the ellipse through these points, ensuring it is horizontally elongated.

---

Final Answer:

\[
\boxed{
\begin{array}{ll}
1. & \text{Center: } (2, 2), \text{ Vertices: } (7, 2), (-3, 2), \text{ Co-vertices: } (2, 5), (2, -1) \\
2. & \text{Center: } (5, -2), \text{ Vertices: } (5, 3), (5, -7), \text{ Co-vertices: } (6, -2), (4, -2) \\
3. & \text{Center: } (1, 0), \text{ Vertices: } (6, 0), (-4, 0), \text{ Co-vertices: } (1, 1), (1, -1) \\
4. & \text{Center: } (0, -2), \text{ Vertices: } (7, -2), (-7, -2), \text{ Co-vertices: } (0, -1), (0, -3)
\end{array}
}
\]