To determine the end behavior of the graph of the function $f(x) = 2x^6 - 2x^2 - 5$, we need to focus on the term with the highest degree, which is $2x^6$.

Key Points to Consider:

1. Degree and Leading Coefficient:

   • The degree of the polynomial is 6 (which is even).
   • The leading coefficient is 2 (which is positive).

2. End Behavior:

   • Since the degree is even and the leading coefficient is positive, the end behavior of the function will be:
     • As $x \to +\infty$, $f(x) \to +\infty$
     • As $x \to -\infty$, $f(x) \to +\infty$

Graph Characteristics:

• Both ends of the graph will rise towards positive infinity.

Therefore, any graph that shows both ends of the curve rising upwards (as $x$ becomes very large or very small) represents the correct end behavior.

Related Questions:

1. How does changing the leading coefficient from positive to negative affect the end behavior?
2. What is the significance of the degree of a polynomial in determining its end behavior?
3. How does the presence of other terms in the polynomial (like $-2x^2$ and $-5$) influence the overall graph, aside from the end behavior?
4. What happens to the end behavior if the degree of the leading term is odd instead of even?
5. How would a graph look if the polynomial was of the form $f(x) = -2x^6 - 2x^2 - 5$?

Tip:

Remember that the leading term of a polynomial largely determines the end behavior, so always identify the highest-degree term and its coefficient first when analyzing graphs.