**For full marks, full solution is required for all questions.** 1.Given 𝑓(𝑥)=−6𝑥3 +8𝑥2 +10𝑥−17 and 𝑔(𝑥)=2𝑥2 −3, a) Describe the end behaviour of 𝑓(𝑥). (2 marks)

To describe the end behavior of the polynomial function \( f(x) = -6x^3 + 8x^2 + 10x - 17 \), we need to analyze the term with the highest degree, as it will dominate the behavior of the function as \( x \) approaches positive and negative infinity.

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

The polynomial \( f(x) = -6x^3 + 8x^2 + 10x - 17 \) is a cubic function, and the term that dictates its end behavior is the leading term \(-6x^3\).

#### Key Points:

1. **Degree and Leading Coefficient:**
   - The degree of the polynomial is 3 (since the highest power of \( x \) is 3).
   - The leading coefficient is \(-6\).

2. **End Behavior Analysis:**
   - As \( x \to \infty \):
     \[
     -6x^3 \to -\infty
     \]
     Since the leading coefficient is negative and the degree is odd, the function will approach negative infinity.

   - As \( x \to -\infty \):
     \[
     -6(-x)^3 = -6(-x^3) = 6x^3 \to \infty
     \]
     Again, due to the negative leading coefficient and odd degree, the function will approach positive infinity.

### Conclusion

The end behavior of \( f(x) = -6x^3 + 8x^2 + 10x - 17 \) is:
- As \( x \to \infty \), \( f(x) \to -\infty \)
- As \( x \to -\infty \), \( f(x) \to \infty \)

Thus, the function decreases without bound as \( x \) increases and increases without bound as \( x \) decreases.

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

1. How do you find the local maxima and minima of a polynomial function?
2. Can you explain the concept of polynomial division?
3. How do you determine the roots of a cubic equation?
4. What is the significance of the leading coefficient in a polynomial function?
5. How can you graph a polynomial function based on its end behavior and critical points?

**Tip:** When determining the end behavior of a polynomial function, always focus on the term with the highest degree.

Understand how to describe the end behavior of the polynomial function f(x) = -6x^3 + 8x^2 + 10x - 17. Learn about cubic functions, leading coefficients, and how these affect the function's behavior as x approaches infinity and negative infinity. Suitable for high school students studying polynomial functions.

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