Find the equation of the following function:

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BlackTom AI · Accepted Answer

To start, let's restate what the problem is asking: find the equation of the given function from its graph and choose among the four options.
Option a presents the cubic: y = -x^3 + 4x^2 - x - 6. Option b is y = x^3 - 4x^2 + x + 6. Option c is y = 2x^3 - 4x^2 + x - 6. Option d is y = x^3 + 4x^2 + x + 6.

First, observe the end behavior of the graph. As x goes far to the left, the graph rises toward positive infinity, while as x goes far to the right, the graph falls toward negative infinity. This suggests a negative leading coefficient for a cubic (the -x^3 term dominates for large |x| and causes the left end to go up and the right end to go down). Among the options, only option a has a negative leading term (-x^3); the others have positive leading terms or different leading coefficients, which would flip the end behavior.

Next, consider the number of turning points. The graph shows two turning points, which is characteristic of a cubic function with a real cubic polynomial. All of the options are cubic polynomials, so this alone does not distinguish them, but it aligns with the possibility for any of them. The key distinguishing feature here is the end behavior, which points to the negative leading coefficient.

Now examine the remaining coefficients briefly to see potential disagreements. Option a’s leading term is -x^3, matching the observed left-up, right-down behavior. The other options introduce a positive leading term or a different sign pattern in the cubic term, which would produce opposite end behavior (left down, right up) or different growth than the graph shows.

Therefore, the combination of the observed end behavior and the negative leading term supports option a as the correct equation. The other choices are inconsistent with the graph because they would produce opposite end behavior or sign patterns that do not match the plotted curve.

Note: If you focus on end behavior alone, you can quickly rule out b, c, and d since their leading terms do not produce the left-end up/right-end down pattern observed in the graph. Then, among the remaining candidates, option a is the one that aligns with the required negative leading coefficient.

MUF0091 Mathematics Unit 1 - Semester 1, 2025 12.3 Study Area 1 revision questions

Find the equation of the following function:

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