Let's break down the function step by step: Given: \( f(x) = 3x^6 + x^5 + 2x^4 - 3x^3 + 2x^2 - 7 \) --- ## **Step 1: End Behaviour** **Explanation:** The end behaviour of a polynomial is determined by its leading term (the term with the highest degree). - Here, the leading term is \( 3x^6 \). - Since the degree (6) is **even** and the leading coefficient (3) is **positive**: - As \( x \to +\infty \), \( f(x) \to +\infty \) - As \( x \to -\infty \), \( f(x) \to +\infty \) **End Behaviour:** Both ends go up (like a "U" shape). --- ## **Step 2: Number of Turning Points Possible (Maximum)** **Explanation:** A polynomial of degree \( n \) can have **at most \( n-1 \) turning points**. - Here, degree = 6, so possible turning points = \( 6-1 = 5 \) **Maximum Turning Points:** 5 --- ## **Step 3: Number of Turning Points for This Particular Function** **Explanation:** You would need to analyze the derivative or the graph to know exactly how many turning points this particular function has, but the question may only ask for the maximum unless you are expected to graph or differentiate. Without graphing or further calculation, we state the maximum possible. **Turning Points for this function:** Cannot be determined exactly without more information; at most 5. --- ## **Step 4: y-intercept** **Explanation:** The y-intercept is found by substituting \( x = 0 \): \[ f(0) = 3(0)^6 + (0)^5 + 2(0)^4 - 3(0)^3 + 2(0)^2 - 7 = -7 \] **y-intercept:** \(-7\) --- ## **Final Summary Answer** 1. **End Behaviour:** Both ends go up (\( f(x) \to +\infty \) as \( x \to \pm\infty \)). 2. **Maximum Turning Points:** 5 3. **Turning Points for this function:** At most 5 (exact number requires further calculation) 4. **y-intercept:** \(-7\)

Certainly! Here's the solution broken down into two steps with explanations, followed by the final answers: --- ### **Step 1: Describe the end behaviour and determine the maximum number of turning points** **Explanation:** The end behaviour of a polynomial depends on its leading term. The given polynomial is: \[ f(x) = 3x^6 + x^5 + 2x^4 - 3x^3 + 2x^2 - 7 \] - The leading term is \( 3x^6 \), which has degree 6 (even) and a positive coefficient. - For an even degree polynomial with a positive leading coefficient: - As \( x \to +\infty \), \( f(x) \to +\infty \) - As \( x \to -\infty \), \( f(x) \to +\infty \) **Therefore, the end behaviour is:** **Both ends go up (the graph rises to infinity in both directions).** --- ### **Step 2: Find the number of possible and actual turning points, and the y-intercept** **Explanation:** - The maximum number of turning points for a degree 6 polynomial is \( 6 - 1 = 5 \). - The actual number of turning points can be less than or equal to this maximum; without graphing or derivative analysis, we state the maximum possible. - To find the y-intercept, substitute \( x=0 \): \[ f(0) = 3(0)^6 + (0)^5 + 2(0)^4 - 3(0)^3 + 2(0)^2 - 7 = -7 \] **Final answers:** - **Maximum number of turning points possible:** 5 - **Number of turning points for this specific function:** At most 5 (exact number requires further analysis) - **Y-intercept:** \(-7\) --- ### **Summary:** 1. **End Behaviour:** Both ends go up (\( +\infty \) as \( x \to \pm\infty \)). 2. **Maximum Possible Turning Points:** 5 3. **Actual Turning Points (for this function):** Up to 5 (exact count depends on further analysis). 4. **Y-intercept:** \(-7\)