Ellipse Worksheet: Graphing and Identifying Key Features of Ellipses

Problem Overview:

The task involves identifying and writing the equations of ellipses from their graphs (Questions 1-3) and graphing given ellipse equations while identifying key features such as the center, vertices, co-vertices, and foci (Questions 4-7).

Solution:

#### Part 1: Writing the Equation of the Conic Section (Ellipses) from Graphs (Questions 1-3)

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,
- \(b\) is the semi-minor axis length.

We will analyze each graph to determine the center, \(a\), and \(b\), and then write the equation.

---

##### Question 1:
- Graph Analysis:
- The center of the ellipse is \((0, 0)\).
- The ellipse is horizontally oriented (major axis is horizontal).
- The vertices are at \((-4, 0)\) and \((4, 0)\), so the semi-major axis \(a = 4\).
- The co-vertices are at \((0, -2)\) and \((0, 2)\), so the semi-minor axis \(b = 2\).

- Equation:
\[
\frac{x^2}{4^2} + \frac{y^2}{2^2} = 1 \quad \Rightarrow \quad \frac{x^2}{16} + \frac{y^2}{4} = 1
\]

##### Question 2:
- Graph Analysis:
- The center of the ellipse is \((0, 0)\).
- The ellipse is vertically oriented (major axis is vertical).
- The vertices are at \((0, -6)\) and \((0, 6)\), so the semi-major axis \(a = 6\).
- The co-vertices are at \((-3, 0)\) and \((3, 0)\), so the semi-minor axis \(b = 3\).

- Equation:
\[
\frac{x^2}{3^2} + \frac{y^2}{6^2} = 1 \quad \Rightarrow \quad \frac{x^2}{9} + \frac{y^2}{36} = 1
\]

##### Question 3:
- Graph Analysis:
- The center of the ellipse is \((2, 0)\).
- The ellipse is horizontally oriented (major axis is horizontal).
- The vertices are at \((-2, 0)\) and \((6, 0)\). The distance from the center \((2, 0)\) to each vertex is \(4\), so the semi-major axis \(a = 4\).
- The co-vertices are at \((2, -3)\) and \((2, 3)\), so the semi-minor axis \(b = 3\).

- Equation:
\[
\frac{(x-2)^2}{4^2} + \frac{y^2}{3^2} = 1 \quad \Rightarrow \quad \frac{(x-2)^2}{16} + \frac{y^2}{9} = 1
\]

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#### Part 2: Graphing Ellipses and Identifying Key Features (Questions 4-7)

For each ellipse equation, we will:
1. Identify the center \((h, k)\).
2. Determine the lengths of the semi-major axis \(a\) and semi-minor axis \(b\).
3. Find the vertices and co-vertices.
4. Calculate the foci using the formula \(c = \sqrt{a^2 - b^2}\).
5. Graph the ellipse.

---

##### Question 4:
\[
\frac{(x+1)^2}{16} + \frac{(y-2)^2}{25} = 1
\]

- Center: \((h, k) = (-1, 2)\)
- Semi-major axis \(a\): Since \(25 > 16\), the major axis is vertical, and \(a = \sqrt{25} = 5\).
- Semi-minor axis \(b\): \(b = \sqrt{16} = 4\).
- Vertices: Along the major axis (vertical), the vertices are \((-1, 2 \pm 5)\), i.e., \((-1, 7)\) and \((-1, -3)\).
- Co-vertices: Along the minor axis (horizontal), the co-vertices are \((-1 \pm 4, 2)\), i.e., \((3, 2)\) and \((-5, 2)\).
- Foci: \(c = \sqrt{a^2 - b^2} = \sqrt{25 - 16} = \sqrt{9} = 3\). The foci are \((-1, 2 \pm 3)\), i.e., \((-1, 5)\) and \((-1, -1)\).

- Summary:
- Center: \((-1, 2)\)
- Vertices: \((-1, 7)\) and \((-1, -3)\)
- Co-vertices: \((3, 2)\) and \((-5, 2)\)
- Foci: \((-1, 5)\) and \((-1, -1)\)

---

##### Question 5:
\[
\frac{(x-1)^2}{36} + \frac{y^2}{9} = 1
\]

- Center: \((h, k) = (1, 0)\)
- Semi-major axis \(a\): Since \(36 > 9\), the major axis is horizontal, and \(a = \sqrt{36} = 6\).
- Semi-minor axis \(b\): \(b = \sqrt{9} = 3\).
- Vertices: Along the major axis (horizontal), the vertices are \((1 \pm 6, 0)\), i.e., \((7, 0)\) and \((-5, 0)\).
- Co-vertices: Along the minor axis (vertical), the co-vertices are \((1, 0 \pm 3)\), i.e., \((1, 3)\) and \((1, -3)\).
- Foci: \(c = \sqrt{a^2 - b^2} = \sqrt{36 - 9} = \sqrt{27} = 3\sqrt{3}\). The foci are \((1 \pm 3\sqrt{3}, 0)\).

- Summary:
- Center: \((1, 0)\)
- Vertices: \((7, 0)\) and \((-5, 0)\)
- Co-vertices: \((1, 3)\) and \((1, -3)\)
- Foci: \((1 + 3\sqrt{3}, 0)\) and \((1 - 3\sqrt{3}, 0)\)

---

##### Question 6:
\[
\frac{x^2}{36} + \frac{(y+1)^2}{25} = 1
\]

- Center: \((h, k) = (0, -1)\)
- Semi-major axis \(a\): Since \(36 > 25\), the major axis is horizontal, and \(a = \sqrt{36} = 6\).
- Semi-minor axis \(b\): \(b = \sqrt{25} = 5\).
- Vertices: Along the major axis (horizontal), the vertices are \((0 \pm 6, -1)\), i.e., \((6, -1)\) and \((-6, -1)\).
- Co-vertices: Along the minor axis (vertical), the co-vertices are \((0, -1 \pm 5)\), i.e., \((0, 4)\) and \((0, -6)\).
- Foci: \(c = \sqrt{a^2 - b^2} = \sqrt{36 - 25} = \sqrt{11}\). The foci are \((0 \pm \sqrt{11}, -1)\), i.e., \((\sqrt{11}, -1)\) and \((- \sqrt{11}, -1)\).

- Summary:
- Center: \((0, -1)\)
- Vertices: \((6, -1)\) and \((-6, -1)\)
- Co-vertices: \((0, 4)\) and \((0, -6)\)
- Foci: \((\sqrt{11}, -1)\) and \((- \sqrt{11}, -1)\)

---

##### Question 7:
\[
\frac{(x+1)^2}{16} + \frac{(y-2)^2}{39} = 1
\]

- Center: \((h, k) = (-1, 2)\)
- Semi-major axis \(a\): Since \(39 > 16\), the major axis is vertical, and \(a = \sqrt{39}\).
- Semi-minor axis \(b\): \(b = \sqrt{16} = 4\).
- Vertices: Along the major axis (vertical), the vertices are \((-1, 2 \pm \sqrt{39})\).
- Co-vertices: Along the minor axis (horizontal), the co-vertices are \((-1 \pm 4, 2)\), i.e., \((3, 2)\) and \((-5, 2)\).
- Foci: \(c = \sqrt{a^2 - b^2} = \sqrt{39 - 16} = \sqrt{23}\). The foci are \((-1, 2 \pm \sqrt{23})\).

- Summary:
- Center: \((-1, 2)\)
- Vertices: \((-1, 2 + \sqrt{39})\) and \((-1, 2 - \sqrt{39})\)
- Co-vertices: \((3, 2)\) and \((-5, 2)\)
- Foci: \((-1, 2 + \sqrt{23})\) and \((-1, 2 - \sqrt{23})\)

---

Final Answers:

\[
\boxed{
\begin{aligned}
1. & \quad \frac{x^2}{16} + \frac{y^2}{4} = 1 \\
2. & \quad \frac{x^2}{9} + \frac{y^2}{36} = 1 \\
3. & \quad \frac{(x-2)^2}{16} + \frac{y^2}{9} = 1 \\
4. & \quad \text{Center: } (-1, 2), \text{ Vertices: } (-1, 7) \text{ and } (-1, -3), \text{ Co-vertices: } (3, 2) \text{ and } (-5, 2), \text{ Foci: } (-1, 5) \text{ and } (-1, -1) \\
5. & \quad \text{Center: } (1, 0), \text{ Vertices: } (7, 0) \text{ and } (-5, 0), \text{ Co-vertices: } (1, 3) \text{ and } (1, -3), \text{ Foci: } (1 + 3\sqrt{3}, 0) \text{ and } (1 - 3\sqrt{3}, 0) \\
6. & \quad \text{Center: } (0, -1), \text{ Vertices: } (6, -1) \text{ and } (-6, -1), \text{ Co-vertices: } (0, 4) \text{ and } (0, -6), \text{ Foci: } (\sqrt{11}, -1) \text{ and } (-\sqrt{11}, -1) \\
7. & \quad \text{Center: } (-1, 2), \text{ Vertices: } (-1, 2 + \sqrt{39}) \text{ and } (-1, 2 - \sqrt{39}), \text{ Co-vertices: } (3, 2) \text{ and } (-5, 2), \text{ Foci: } (-1, 2 + \sqrt{23}) \text{ and } (-1, 2 - \sqrt{23})
\end{aligned}
}
\]