graph a function that has the following characteristics, zeros -7 multiplicty of 2, -1 multiplicity of 3, and 6 with multiplicity of 1
f(x)-∞ as x +/- ∞

Here is the graph of the function with the given zeros and multiplicities. The key features are:

- At \( x = -7 \), the function touches the x-axis but does not cross it (due to the multiplicity of 2).
- At \( x = -1 \), the function crosses the x-axis with a cubic flattening effect (multiplicity of 3).
- At \( x = 6 \), the function crosses the x-axis sharply (multiplicity of 1).
- The function approaches negative infinity as \( x \to \pm \infty \), indicating the downward-opening behavior.

Would you like further details or explanations about this graph? 

Here are 5 related questions:
1. How does changing the multiplicity affect the graph's shape at the zeros?
2. What does the leading term of a polynomial tell us about the end behavior?
3. How would the graph change if one of the multiplicities was odd instead of even?
4. Can we determine the degree of the polynomial based on the zeros and multiplicities?
5. What happens if the coefficient of the polynomial is positive rather than negative?

**Tip:** The degree of a polynomial is the sum of the multiplicities of its factors.

Graph a function that has the following characteristics: zeros at -7 (multiplicity of 2), -1 (multiplicity of 3), and 6 (multiplicity of 1), with f(x) approaching negative infinity as x tends to plus or minus infinity.

Explore how to graph a polynomial function with zeros at -7 (multiplicity of 2), -1 (multiplicity of 3), and 6 (multiplicity of 1), focusing on polynomial behavior, end behavior, and how multiplicity affects the graph at each zero. Ideal for students in Grades 10-12.

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