Quizizz worksheet on conic sections focusing on ellipses, including questions about vertices, co-vertices, and center.

Problem Analysis and Solution

The provided image contains questions related to ellipses, specifically focusing on identifying vertices, co-vertices, and the center of an ellipse. Let's solve each question step by step.

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#### Question 1: What are the vertices of the ellipse?

Image Description:
The ellipse is centered at \((3, 1)\) and is oriented horizontally (the major axis is horizontal).

Key Concepts:
- The vertices of an ellipse are the points where the ellipse intersects its major axis.
- For a horizontally oriented ellipse, the vertices are located at \((h \pm a, k)\), where:
- \((h, k)\) is the center of the ellipse,
- \(a\) is the semi-major axis length.

Solution:
1. From the graph, the center of the ellipse is \((3, 1)\).
2. The distance from the center to each vertex along the major axis is the semi-major axis length (\(a\)).
3. Observing the graph, the vertices are located at \((7, 1)\) and \((-1, 1)\).

Thus, the vertices are \((7, 1)\) and \((-1, 1)\).

Correct Answer:
\[
\boxed{A}
\]

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#### Question 2: What are the co-vertices of the ellipse?

Image Description:
The ellipse is centered at \((3, 1)\) and is oriented vertically (the major axis is vertical).

Key Concepts:
- The co-vertices of an ellipse are the points where the ellipse intersects its minor axis.
- For a vertically oriented ellipse, the co-vertices are located at \((h, k \pm b)\), where:
- \((h, k)\) is the center of the ellipse,
- \(b\) is the semi-minor axis length.

Solution:
1. From the graph, the center of the ellipse is \((3, 1)\).
2. The distance from the center to each co-vertex along the minor axis is the semi-minor axis length (\(b\)).
3. Observing the graph, the co-vertices are located at \((3, 7)\) and \((3, -5)\).

Thus, the co-vertices are \((3, 7)\) and \((3, -5)\).

Correct Answer:
\[
\boxed{D}
\]

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#### Question 3: What is the center of the shifted ellipse?

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

Key Concepts:
- The standard form of an ellipse equation is:
\[
\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1
\]
where \((h, k)\) is the center of the ellipse.
- Comparing the given equation with the standard form, we can identify \(h\) and \(k\).

Solution:
1. Rewrite the given equation:
\[
\frac{(x + 3)^2}{4} + \frac{(y - 1)^2}{1} = 1
\]
2. Compare it with the standard form:
\[
\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1
\]
Here, \(h = -3\) and \(k = 1\).
3. Therefore, the center of the ellipse is \((-3, 1)\).

Correct Answer:
\[
\boxed{B}
\]

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#### Question 4: Which graph correctly represents the equation below?

Equation Given:
\[
\frac{(x - 3)^2}{4} + \frac{(y + 2)^2}{10} = 1
\]

Key Concepts:
- The standard form of an ellipse equation 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 (if the major axis is horizontal),
- \(b\) is the semi-major axis length (if the major axis is vertical).
- If \(a^2 < b^2\), the major axis is vertical; if \(a^2 > b^2\), the major axis is horizontal.

Solution:
1. Identify the center \((h, k)\):
\[
h = 3, \quad k = -2 \quad \Rightarrow \text{Center is } (3, -2).
\]
2. Identify \(a^2\) and \(b^2\):
\[
a^2 = 4, \quad b^2 = 10.
\]
Since \(a^2 < b^2\), the major axis is vertical.
3. Determine the lengths of the semi-major and semi-minor axes:
\[
a = \sqrt{4} = 2, \quad b = \sqrt{10}.
\]
4. The ellipse is centered at \((3, -2)\), with a vertical major axis, and the vertices are \((3, -2 \pm \sqrt{10})\).

Graph Analysis:
- The correct graph should have:
- Center at \((3, -2)\),
- Major axis vertical,
- Semi-major axis length \(\sqrt{10}\),
- Semi-minor axis length \(2\).

From the provided graphs, identify the one that matches these criteria.

Correct Answer:
\[
\boxed{\text{[Graph corresponding to the above description]}}
\]

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Final Answers:

1. \(\boxed{A}\)
2. \(\boxed{D}\)
3. \(\boxed{B}\)
4. \(\boxed{\text{[Graph corresponding to the above description]}}\)