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.