Question

use a graphing utility to graph f and g in the same viewing rectangle. then use the zoom out feature to show that f and g have identical end - behavior.
f(x)=x^{3}-6x - 2
g(x)=x^{3}
now use the zoom out feature for the graph found in the previous step. choose the correct graph below.

Explanation:

Step1: Analyze end - behavior of polynomials

The end - behavior of a polynomial function \(y = a_nx^n+\cdots+a_0\) is determined by the leading term \(a_nx^n\). For \(f(x)=x^{3}-6x - 2\) and \(g(x)=x^{3}\), the leading term for both is \(x^{3}\). When \(x\to+\infty\), \(y = x^{3}\to+\infty\) for both functions, and when \(x\to-\infty\), \(y = x^{3}\to-\infty\) for both functions.

Step2: Consider the general shape of cubic functions

Cubic functions with a positive leading coefficient (\(a = 1\) in both \(f(x)\) and \(g(x)\)) start from the bottom - left and go to the top - right as \(x\) increases.

Step3: Evaluate the graphs

When we zoom out, the non - leading terms (\(-6x - 2\) in \(f(x)\)) become less significant, and the graphs of \(f(x)\) and \(g(x)\) will look very similar in terms of end - behavior. We need to look for graphs where both functions have the same left - hand and right - hand behavior (both go down as \(x\to-\infty\) and up as \(x\to+\infty\)).

Answer:

Without seeing the actual options clearly (as the image has some issues with readability), in general, the correct graph will show two cubic - shaped curves, one for \(y = x^{3}-6x - 2\) and one for \(y = x^{3}\), that have the same end - behavior: as \(x\to-\infty\), the curves go down and as \(x\to+\infty\), the curves go up. If we assume the options are labeled A, B, C, D in order, you should choose the option where both graphs are cubic with the correct end - behavior as described above.