Let's go step by step to analyze and graph the polynomial function:

$f(x) = -(x+3)^2(x-5)$

Step 1: Determine the End Behavior

The degree of the polynomial function is determined by expanding the factors:

• $(x+3)^2$ contributes degree 2.
• $(x-5)$ contributes degree 1.
• The total degree is $2+1=3$ (odd degree).
• The leading coefficient is negative due to the negative sign in front.

For odd-degree polynomials:

• If the leading coefficient is positive, the left side falls and the right side rises.
• If the leading coefficient is negative, the left side rises and the right side falls.

So, end behavior: $\text{As } x \to -\infty, f(x) \to +\infty$ $\text{As } x \to +\infty, f(x) \to -\infty$

Step 2: Find the x- and y-Intercepts

x-intercepts:

Set $f(x) = 0$:

$-(x+3)^2(x-5) = 0$

This gives us solutions:

• $x+3 = 0 \Rightarrow x = -3$
• $x-5 = 0 \Rightarrow x = 5$

Thus, the x-intercepts are $(-3,0)$ and $(5,0)$.

y-intercept:

Set $x = 0$:

$f(0) = -(0+3)^2(0-5) = -(9)(-5) = 45$

Thus, the y-intercept is $(0, 45)$.

Step 3: Find the Multiplicity of Real Zeros

• $x = -3$ comes from $(x+3)^2$ → multiplicity 2 (even).
• $x = 5$ comes from $(x-5)$ → multiplicity 1 (odd).

Behavior at x-intercepts:

• Multiplicity 2 at $x = -3$ → graph touches the x-axis but does not cross.
• Multiplicity 1 at $x = 5$ → graph crosses the x-axis.

Step 4: Maximum Number of Turning Points

The maximum number of turning points for a polynomial of degree $n$ is:

$n-1 = 3-1 = 2$

Thus, the graph has at most 2 turning points.

Step 5: Plot the Graph

1. Plot the x-intercepts: $(-3,0)$ and $(5,0)$.
2. Plot the y-intercept at $(0,45)$.
3. Since $x = -3$ has multiplicity 2, the graph touches the x-axis at $(-3,0)$.
4. Since $x = 5$ has multiplicity 1, the graph crosses the x-axis at $(5,0)$.
5. End behavior: The graph rises to the left and falls to the right.
6. Additional points can help refine the shape.

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Would you like me to graph this for you? Let me know if you need any clarifications!

Related Questions

1. What happens when a polynomial has an even multiplicity at a zero?
2. How can you determine the number of turning points without graphing?
3. What is the significance of the leading coefficient in determining end behavior?
4. How do you find additional points to refine a polynomial graph?
5. How does the degree of a polynomial affect its shape?

Math Tip

When graphing a polynomial, always check both end behavior and zero multiplicities to determine the overall shape efficiently!