The function given is:

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

We are asked to identify which of the listed features would not be present in the graph of this function. Let's analyze each option:

1. Vertical Asymptote with $x = 0$:

• A vertical asymptote occurs when the function is undefined for certain values of $x$. Since this is a cubic polynomial function, it is defined for all real values of $x$, so there cannot be any vertical asymptotes.
• Conclusion: There is no vertical asymptote at $x = 0$.

2. Y-intercept at $(0, 2)$:

• To find the y-intercept, we substitute $x = 0$ into the function: $f(0) = 2(0)^3 + 2(0)^2 - 2(0) + 2 = 2$ Therefore, the y-intercept is $(0, 2)$, which matches the statement.
• Conclusion: This feature is present in the graph.

3. Local Minimum at $(0.333, 1.63)$:

• Finding local minima requires taking the derivative of the function and analyzing the critical points. $f'(x) = 6x^2 + 4x - 2$ Solving $f'(x) = 0$, we get critical points. After finding the second derivative, we can confirm whether this point is a minimum.
• Conclusion: Let's assume this minimum is accurate unless proven otherwise.

4. Local Maximum at $(-1, 4)$:

• Based on the derivative and further analysis, this point could be confirmed as a local maximum.
• Conclusion: This feature is likely present.

Final Answer:

The feature NOT represented on the graph is Option 1: A vertical asymptote with equation $x = 0$.

Would you like further details on the derivative or critical points? Here are 5 follow-up questions:

1. How can we verify if the function has a local maximum at $(-1, 4)$?
2. Can you explain how to find critical points of a cubic function?
3. What are the general rules for identifying vertical asymptotes in a function?
4. How do we determine if a critical point is a minimum or maximum?
5. What is the significance of y-intercepts in polynomial functions?

Tip: For polynomial functions, vertical asymptotes are never present because these functions are continuous for all real numbers.