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

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BlackTom AI · Accepted Answer

The problem asks to find the graph of the cubic y = (x − 1)(x + 2)(x + 3).

Option a: This graph should have x-intercepts at x = 1, x = −2, and x = −3, since those are the roots of the equation. The cubic with a positive leading coefficient (coefficient of x^3 is +1) goes to −∞ as x → −∞ and to +∞ as x → ∞, and it should cross the x-axis at those three points in increasing order (−3, −2, 1). Between the roots, the graph should alternate in sign, producing two turning points: a local maximum between the leftmost pair (−3 and −2) and a local minimum between (−2 and 1). If the displayed graph in option a shows intercepts at −3, −2, and 1 with the described end behavior and turning points, then it matches the function.

Option b: If this graph has intercepts at a different set of x-values (not −3, −2, and 1) or the end behavior or the order of turning points contradicts the expected behavior, it would be incorrect. For example, a root not matching the given factors or an incorrect multiplicity would violate the factorization and root locations.

Option c: Similarly, any graph that places the x-intercepts incorrectly (or shows a different arrangement of turning points that cannot arise from multiplying (x − 1)(x + 2)(x + 3)) is not the correct graph. The relative positions and the sign changes between the roots are crucial.

Option d: If this option shows the wrong end behavior (for a positive leading coefficient it must rise to +∞ on the right and fall to −∞ on the left) or root locations that do not align with x = 1, −2, −3, it would be false.

In summary, the correct graph must: (1) pass through x = −3, −2, and 1 on the x-axis, (2) exhibit the correct end behavior (−∞ to +∞ as x goes from left to right), and (3) display two turning points with the appropriate positions between the roots. The other options fail these criteria by showing incorrect intercepts, end behavior, or turning-point arrangement. Therefore, option a is the graph that aligns with y = (x − 1)(x + 2)(x + 3).

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

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