Determine whether the function is​ even, odd, or neither. Then determine whether the​ function's graph is symmetric with respect to the​ y-axis, the​ origin, or neither.
​g(x)equals=x squared minus 4 xx2−4x
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Part 1
Determine whether the function is​ even, odd, or neither. Choose the correct answer below.

even

odd

neither
Part 2
Determine whether the graph of the function is symmetric with respect to the​ y-axis, the​ origin, or neither. Select all that apply.

origin

neither

​y-axis

To determine whether the function \( g(x) = x^2 - 4x \) is even, odd, or neither, and to determine the symmetry of its graph, we need to follow these steps:

### Part 1: Even, Odd, or Neither

1. **Definition of Even Function:**
   A function \( f(x) \) is even if \( f(-x) = f(x) \) for all \( x \) in the domain of \( f \).

2. **Definition of Odd Function:**
   A function \( f(x) \) is odd if \( f(-x) = -f(x) \) for all \( x \) in the domain of \( f \).

3. **Calculate \( g(-x) \):**
   \[
   g(-x) = (-x)^2 - 4(-x) = x^2 + 4x
   \]

4. **Compare \( g(-x) \) with \( g(x) \) and \(-g(x)\):**
   - \( g(x) = x^2 - 4x \)
   - \( g(-x) = x^2 + 4x \)
   - \( -g(x) = -(x^2 - 4x) = -x^2 + 4x \)

5. **Determine the nature of \( g(x) \):**
   - \( g(-x) \neq g(x) \)
   - \( g(-x) \neq -g(x) \)

Since \( g(-x) \) is neither equal to \( g(x) \) nor \(-g(x)\), the function \( g(x) \) is neither even nor odd.

### Part 2: Symmetry of the Graph

1. **Symmetry with respect to the \( y \)-axis:**
   - A function \( f(x) \) is symmetric with respect to the \( y \)-axis if \( f(-x) = f(x) \).
   - Since \( g(-x) \neq g(x) \), the graph is not symmetric with respect to the \( y \)-axis.

2. **Symmetry with respect to the origin:**
   - A function \( f(x) \) is symmetric with respect to the origin if \( f(-x) = -f(x) \).
   - Since \( g(-x) \neq -g(x) \), the graph is not symmetric with respect to the origin.

3. **Conclusion:**
   - Since \( g(x) \) does not satisfy either condition for symmetry with respect to the \( y \)-axis or the origin, the graph is symmetric with respect to neither.

### Summary

**Part 1: The function is neither even nor odd.**

**Part 2: The function's graph is symmetric with respect to neither the \( y \)-axis nor the origin.**

Would you like more details or have any questions?

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### Related Questions:

1. How do you determine if a function is even or odd from its equation?
2. What is the significance of symmetry in the graph of a function?
3. How does the graph of \( g(x) = x^2 - 4x \) look?
4. Can a function be both even and odd? Explain why or why not.
5. What types of symmetry can a function's graph have?
6. How does the degree of the polynomial affect the symmetry of its graph?
7. How can transformations affect the symmetry of a graph?
8. What are some examples of even and odd functions?

### Tip:

When analyzing the symmetry of a function, always test \( f(-x) \) and compare it to \( f(x) \) and \(-f(x)\) to determine if the function is even, odd, or neither.

Explore whether the function g(x) = x^2 - 4x is even, odd, or neither, and determine its symmetry with respect to the y-axis, origin, or neither. This problem involves analyzing the definitions of even and odd functions and testing symmetry conditions. Suitable for advanced high school students and beyond.

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