We are given the polynomial function $f(x) = x^3 - 6x - 5$. To match it with the correct graph, let's analyze the behavior of the cubic function:

1. Degree and Shape: Since the degree of the polynomial is 3, it will be an odd-degree polynomial. This means the graph will go off to opposite directions at either end. Specifically, the left side of the graph will go to negative infinity, and the right side will rise to positive infinity, as the leading coefficient of $x^3$ is positive.

2. Turning Points: A cubic function can have up to two turning points, which occur at critical points where the derivative equals zero.

3. Intercepts:

   • Y-intercept: When $x = 0$, $f(0) = -5$, so the y-intercept is at (0, -5).
   • X-intercepts: To find the x-intercepts, we would need to solve $x^3 - 6x - 5 = 0$, but rough approximations or graphing will help us match this better visually.

Given these insights, we can examine the graphs:

• Option A: This graph shows the behavior of a cubic function, with one x-intercept and two turning points.
• Option B: This graph has a similar structure, but the y-intercept looks closer to the origin, whereas we expect it to be at $-5$.
• Option C: This graph also shows cubic behavior, but the intercept and turning point positioning does not seem correct.
• Option D: This shows a quadratic shape, which is not suitable for a cubic polynomial.

Correct Answer: The correct graph is A, which displays the expected cubic behavior with the correct intercept and turning points.

Do you want further details on any specific steps or have questions?

Here are 5 related questions:

1. How do you determine the number of turning points for a polynomial function?
2. What are the characteristics of a cubic function in terms of end behavior?
3. How can you find the exact x-intercepts for a cubic polynomial?
4. What is the role of the leading coefficient in determining the graph's direction?
5. How do derivatives help in finding turning points on graphs?

Tip: Always check the degree of the polynomial to predict the general shape of the graph before looking for finer details like intercepts and turning points.