which graph shows the same end behavior as the graph of f(x) = 2x^6 - 2x^2 - 5?

Answer:

To determine the end - behavior of the polynomial function (f(x)=2x^{6}-2x^{2}-5), we consider the leading term. The leading term of the polynomial is (2x^{6}).

For a polynomial function (y = a_nx^n+a_{n - 1}x^{n - 1}+\cdots+a_1x + a_0), the end - behavior is determined by the leading term (a_nx^n).

If (n) is even and (a_n>0), as (x\to+\infty), (y\to+\infty) and as (x\to-\infty), (y\to+\infty).

In the function (f(x)=2x^{6}-2x^{2}-5), the degree (n = 6) (even) and the leading coefficient (a_n=2>0). So, as (x\to+\infty), (f(x)\to+\infty) and as (x\to-\infty), (f(x)\to+\infty). The graph will open upwards on both the left and the right.

Explanation:

Step1: Identify the leading term

The leading term of (f(x)=2x^{6}-2x^{2}-5) is (2x^{6}).

Step2: Analyze the degree and coefficient

The degree (n = 6) (even) and (a = 2>0).

Step3: Determine end - behavior

As (x\to\pm\infty), (y\to+\infty).