https://file.aiquickdraw.com/mathbot/file/1725245975451lrze79cm.jpg

c) If the graph is that of a​ function, determine what kinds of symmetry it has. Select all that apply.

A.

It is symmetric with respect to the origin.

B.

It is symmetric with respect to the​ x-axis.

C.

It is symmetric with respect to the​ y-axis.

D.

The graph is not symmetric with respect to the​ x-axis, y-axis, or the origin.

E.

The graph is not a function.

The graph in the image is a parabola, which opens upwards and has its vertex at the point \((-1, -4)\).

To determine the symmetry of the graph:

1. **Symmetry with respect to the y-axis**: A function is symmetric with respect to the y-axis if for every point \((x, y)\) on the graph, the point \((-x, y)\) is also on the graph. This is the case for parabolas that open vertically and are centered on the y-axis. In this case, the vertex is not on the y-axis (it is at \((-1, -4)\)), so the graph is **not symmetric with respect to the y-axis**.

2. **Symmetry with respect to the x-axis**: A graph is symmetric with respect to the x-axis if for every point \((x, y)\), the point \((x, -y)\) is also on the graph. This is not the case here, as the parabola does not mirror itself across the x-axis.

3. **Symmetry with respect to the origin**: A graph is symmetric with respect to the origin if for every point \((x, y)\), the point \((-x, -y)\) is also on the graph. This is not true for this parabola, so the graph is **not symmetric with respect to the origin**.

4. **Is the graph a function?**: Yes, the graph represents a function. It passes the vertical line test, meaning that for every x-value, there is exactly one y-value.

Given these observations, the correct options are:

- **D. The graph is not symmetric with respect to the x-axis, y-axis, or the origin.**
- **It is still a function**.

Would you like more details or have any questions?

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Here are five related questions:

1. What is the equation of this parabola in standard form?
2. How can we find the axis of symmetry of a parabola?
3. What are the necessary conditions for a graph to have symmetry with respect to the x-axis or y-axis?
4. What does it mean geometrically for a graph to have symmetry with respect to the origin?
5. How does the vertex form of a quadratic equation help in identifying symmetry?

**Tip**: Always verify if a graph represents a function by using the vertical line test.

Explore the symmetry properties of parabolas in mathematics. Learn how to determine symmetry with respect to the x-axis, y-axis, and the origin. This problem involves analyzing a parabolic graph and identifying its symmetry properties. Suitable for high school students and those preparing for advanced math examinations.

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