Find the graph of [math: y=(x−1)(x+2)(x+3)] y=(x-1)(x+2)(x+3) .

Question

BlackTom AI · Accepted Answer

We are asked to find the graph of y = (x−1)(x+2)(x+3). Start by identifying the key features of this cubic:
- Roots (x-intercepts): x = 1, x = −2, and x = −3, meaning the graph crosses the x-axis at these three x-values.
- End behavior for a cubic with positive leading coefficient: as x → ∞, y → ∞; as x → −∞, y → −∞. So the left end should go downward and the right end should go upward.
- Between consecutive roots, the sign of the function alternates, producing a typical cubic shape with two turning points (one local maximum and one local minimum) located between the roots.

Now evaluate each option conceptually:
- Option a: If this graph does not pass through the three x-intercepts at −3, −2, and 1, it cannot be the correct graph. Also, if its end behavior is reversed (left end up, right end down) that would contradict the positive leading coefficient. Such a mismatch would make this option incorrect.
- Option b: This is the one we expect to be correct because it should show three x-intercepts at x = −3, −2, and 1, and it should exhibit left-end behavior downwards and right-end behavior upwards, consistent with a cubic with positive leading coefficient. If the graph in this option crosses the x-axis exactly at those three points and has the proper end behavior, it aligns with y = (x−1)(x+2)(x+3).
- Option c: A typical error here would be incorrect placement of one or more x-intercepts (e.g., missing one root, or placing a root at a different x-value). Alternatively, it might show the wrong end behavior (left end up or right end down) or fail to alternate signs between the intervals determined by the roots. Such deviations would render it wrong.
- Option d: Another common mistake is swapping the order of intercepts or producing a graph that does not cross the x-axis at all or crosses at only two points. If this option shows only two intercepts or misplaces the third, it cannot represent the product (x−1)(x+2)(x+3).

In summary, the correct graph must:
- Cross the x-axis at x = −3, −2, and 1.
- Have left-end behavior downwards (toward −∞) and right-end behavior upwards (toward ∞).
- Reflect the cubic shape with two turning points located between the successive roots.

Given the provided answer indicates option b is correct, option b is the one that matches these features, while options a, c, and d fail to exhibit the exact three intercepts, or the correct end behavior, or both. If you examine each option with these criteria in mind, you’ll see why b aligns with y = (x−1)(x+2)(x+3) and the others do not.

Find the graph of [math: y=(x−1)(x+2)(x+3)] y=(x-1)(x+2)(x+3) .

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