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

First, examine the given function y = (x − 1)(x + 2)(x + 3). This is a cubic with three distinct real zeros at x = 1, x = −2, and x = −3, since each factor equals zero at those x-values.
Option A: Consider the x-intercepts. If this graph shows intercepts at x = 1, −2, and −3 in any order, that would be consistent with the roots. If the graph instead shows intercepts at different x-values, it is incorrect. Also note that all roots are simple (multiplicity 1), so the graph should cross the x-axis at each root rather than touch and bounce.
Option B: Evaluate end behavior. Since the leading term is x^3 with positive coefficient (the product of x terms yields x^3 with coefficient +1), as x → ∞, y → ∞, and as x → −∞, y → −∞. The graph must rise to the right and fall to the left, with the end behavior reflecting an odd-degree, upward-opening cubic overall.
Option C: Analyze the turning points. A cubic with three distinct real zeros will have two turning points: one local maximum and one local minimum in between the zeros, with the approximate ordering of x-values for the extrema depending on the coefficients. The graph should show a crest between the leftmost and middle roots and a trough between the middle and rightmost roots, with the overall shape transitioning from negative to positive across the left region to the rightmost region.
Option D: Check for symmetry or other features. The given cubic is not even or odd symmetric, so any claim of symmetry about the y-axis or origin would be incorrect. Also, since all zeros are simple, the graph should cross the axis at each root, not merely touch it.

In evaluating each option, it helps to sketch the qualitative features: three x-intercepts at x = −3, −2, and 1; two turning points with the crest lying between the leftmost two zeros and the trough between the middle and rightmost zeros; end behavior that falls to the left and rises to the right. Compare these features to each provided graph by inspecting where the graph crosses the x-axis and the general up-down pattern between zeros. Any option that misplaces an intercept, alters the number or location of turning points, or gives inconsistent end behavior would be incorrect. Likewise, any indication of symmetry or a root multiplicity different from 1 would signal inaccuracy.

By systematically matching the zeros, the number and placement of turning points, and the tail behavior to the given cubic, you can identify which option aligns with y = (x − 1)(x + 2)(x + 3). The option that best fits all these criteria (three simple real zeros at −3, −2, and 1; two turning points with the expected arrangement; and the correct end behavior) is the one corresponding to the correct graph according to the evaluation above.

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

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