The equation given is:

$y = x^4 - 12x^3 + 47x^2 - 60x + 6$

We are asked to determine which graph corresponds to this equation. To approach this:

1. Behavior at extremes (as $x \to \infty$ or $x \to -\infty$): Since the leading term is $x^4$, the graph will tend to rise to $\infty$ as $x \to \infty$ and $x \to -\infty$. This means the ends of the graph will both go upwards.

2. Critical points and local behavior: We can find turning points and the general shape of the graph by analyzing the derivative, but visual inspection can also provide clues.

Key observations:

• Even-degree polynomial: The graph should have the ends both going upwards.
• Number of turning points: The derivative of this quartic will give us insight into the number of local minima and maxima. From the structure of the polynomial, we expect around 3 turning points (maximums and minimums).

Now, looking at the provided graphs:

• The top-left graph has ends that both go up, and it features three turning points, which is consistent with the expected behavior of the quartic equation. This is the graph of the equation.

Other graphs either:

• Do not have both ends rising (e.g., bottom-left graph).
• Have incorrect local behavior, like fewer turning points or incorrect curvature near the axis.

Thus, the top-left graph is the correct one.

Would you like a more detailed explanation of how this conclusion was reached or assistance with anything else?

Here are 5 related questions to expand on this concept:

1. How do you determine the number of turning points for a polynomial function?
2. What role do the coefficients of each term in a polynomial play in shaping its graph?
3. How can the first and second derivatives help you sketch the graph of a polynomial?
4. What is the significance of inflection points in the context of polynomial functions?
5. How can synthetic division help in identifying roots of a polynomial?

Tip: When analyzing the graph of a polynomial function, always check the end behavior first by looking at the leading term’s degree and coefficient.