Find the graph of $ y=(x-1)(x+2)(x+3) $.

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The problem asks to find the graph of y = (x - 1)(x + 2)(x + 3). First, identify the key features from the factors: - The roots (zeros) occur at x = 1, x = -2, and x = -3, since each factor is zero there. - The leading term is x^3, which means as x → ∞, y → ∞ and as x → −∞, y → −∞ (odd-degree polynomial with positive leading coefficient). Next, consider the sign changes between the roots to determine the shape: - For x < -3, all three factors are negative (x−1 < 0, x+2 < 0, x+3 < 0); the product is negative. - Between -3 and -2, (x−1 < 0, x+2 < 0 or > depending on exact x, x+3 > 0); evaluating one sample point (e.g., x = -2.5) gives a positive product; thus the graph crosses from negative to positive at x = -3. - Between -2 and 1, choose a point like x = 0: (−1)(2)(3) = −6, so the product is negative; the graph crosses from positive to negative at x = -2. - For x > 1, all factors except possibly (x−1) are positive, so the product is positive; the graph crosses from negative to positive at x = 1. Therefore, the cubic should have: - x-intercepts at x = −3, −2, and 1, - end behavior down on the left and up on the right, and - three real turning points corresponding to the three roots (a w-shaped cubic). Now, regarding the options presented in the problem: the provided data shows an image with several graphs and the answer indicated as 'b.', but the answer options are missing from the input (the field answer_options is empty). This means I cannot explicitly compare each listed graph option to the correct features above, since the exact candidate graphs are not shown in the text. What you can do to determine the correct graph yourself (given a set of candidate graphs): - Check the x-intercepts: verify they occur at x = −3, −2, and 1. - Verify the end behavior: as x → −∞, y → −∞; as x → ∞, y → ∞. - Confirm the order of the intercepts along the x-axis matches the ascending roots (−3, then −2, then 1). - Look for the typical cubic shape with two local extrema (a local maximum between −3 and −2, and a local minimum between −2 and 1) that connect through the three intercepts. Since the options are not provided here, I cannot state which specific option (a, b, c, or d) corresponds to the correct graph. If you can share the labeled options or the images for each graph, I can pinpoint which one aligns with the above characteristics and explain why it is correct and why the others are not.

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

Find the graph of \( y=(x-1)(x+2)(x+3) \).

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