Let's go through each part of the problem based on the provided function $f(x) = 2x^3 + 3x^2 + 3$.

Part (a)

To find the maximum number of turning points of a polynomial function, we use the fact that the maximum number of turning points for a polynomial of degree $n$ is $n - 1$. Here, the function $f(x) = 2x^3 + 3x^2 + 3$ has a degree of 3.

• Maximum number of turning points: $3 - 1 = 2$.

This matches the answer filled in on the worksheet.

Part (b)

For part (b), you are asked to identify the correct graph of the function $f(x) = 2x^3 + 3x^2 + 3$ using the specified window settings. The correct graph should have the characteristics of a cubic function with up to 2 turning points, as discussed.

• From the given options, Option C is marked as correct. This graph is consistent with the shape of a cubic polynomial and the expected number of turning points.

Part (c)

The end behavior of a polynomial function is determined by its leading term. In this case, the leading term of $f(x) = 2x^3 + 3x^2 + 3$ is $2x^3$.

• As $x \to \infty$, $f(x) \to \infty$.
• As $x \to -\infty$, $f(x) \to -\infty$.

Thus, the end behavior resembles the function $y = 2x^3$.

Would you like further details on any part of this explanation?

Related Questions:

1. How can we determine the turning points of a cubic function algebraically?
2. Why does the degree of the polynomial dictate the maximum number of turning points?
3. How does the leading coefficient affect the end behavior of a polynomial function?
4. What would the end behavior be if the leading term was negative?
5. How can graphing utilities help verify turning points and end behavior of functions?

Tip:

For polynomials, always look at the highest-degree term to determine the end behavior. This term alone dictates how the graph behaves as $x$ approaches positive or negative infinity.