Certainly! Let's describe the end behavior of the polynomial function \( g(x) = 2x^2 - 3 \).

### End Behavior of \( g(x) \)

The polynomial \( g(x) = 2x^2 - 3 \) is a quadratic function, and the term that dictates its end behavior is the leading term \( 2x^2 \).

#### Key Points:

1. **Degree and Leading Coefficient:**
   - The degree of the polynomial is 2 (since the highest power of \( x \) is 2).
   - The leading coefficient is 2.

2. **End Behavior Analysis:**
   - As \( x \to \infty \):
     \[
     2x^2 \to \infty
     \]
     Since the leading coefficient is positive and the degree is even, the function will approach positive infinity.

   - As \( x \to -\infty \):
     \[
     2(-x)^2 = 2x^2 \to \infty
     \]
     Again, due to the positive leading coefficient and even degree, the function will approach positive infinity.

### Conclusion

The end behavior of \( g(x) = 2x^2 - 3 \) is:
- As \( x \to \infty \), \( g(x) \to \infty \)
- As \( x \to -\infty \), \( g(x) \to \infty \)

Thus, the function increases without bound as \( x \) moves in both the positive and negative directions.

### Summary for Both Functions:

- For \( f(x) = -6x^3 + 8x^2 + 10x - 17 \):
  - As \( x \to \infty \), \( f(x) \to -\infty \)
  - As \( x \to -\infty \), \( f(x) \to \infty \)

- For \( g(x) = 2x^2 - 3 \):
  - As \( x \to \infty \), \( g(x) \to \infty \)
  - As \( x \to -\infty \), \( g(x) \to \infty \)

Would you like details or have any questions? Here are five more questions you might want to ask:

1. How do you find the vertex of a quadratic function?
2. What are the steps to factor a quadratic polynomial?
3. How can you determine the axis of symmetry for a quadratic function?
4. How do you graph a quadratic function using its vertex and intercepts?
5. What are the methods to solve quadratic equations?

**Tip:** For quadratic functions, the leading coefficient and the degree determine whether the parabola opens upwards or downwards, influencing the end behavior.

Explore the end behavior of the quadratic function 2x^2 - 3. Learn how the leading coefficient and degree affect the function's behavior as x approaches positive and negative infinity. Suitable for high school students studying quadratic functions and their properties.

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