solve this question please <---**Question Number:** 11 **Graph Description:** * Type: Graph of a function on a Cartesian coordinate system. * Coordinate Axes: X-axis and Y-axis intersect at the origin O (0,0). The X-axis has ticks labeled at -3, -2, -1, 1, 2, 3. The Y-axis has ticks labeled at -10 and 10. * Curve: A smooth curve that crosses the X-axis at approximately x = -3 and x = 2, and touches the X-axis at x = 0. The curve goes through the origin (0,0). The curve has a local maximum in the second quadrant (between x = -2 and x = -1) and a local minimum in the fourth quadrant (between x = 1 and x = 2). As x approaches negative infinity, the curve goes upwards (towards positive infinity). As x approaches positive infinity, the curve goes upwards (towards positive infinity). **Question Stem:** Which of the following could be the equation of the graph above? **Options:** A) y = x(x - 2)(x + 3) B) y = x^2(x - 2)(x + 3) C) y = x(x + 2)(x - 3) D) y = x^2(x + 2)(x - 3)

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Let's identify which polynomial function matches the given graph. Looking at the graph, we can see it has x-intercepts at x equals negative 3, x equals 0, and x equals 2. Notice that at x equals 0, the curve touches the x-axis but doesn't cross it, indicating a root with multiplicity 2. The curve crosses the x-axis at x equals negative 3 and x equals 2, suggesting roots with multiplicity 1. Let's analyze the options to find the matching function. Let's compare options A and B, which both have roots at x equals negative 3, 0, and 2. Option A is y equals x times x minus 2 times x plus 3, which is a degree 3 polynomial with odd degree. This means it has opposite end behaviors as x approaches positive and negative infinity. Option B is y equals x squared times x minus 2 times x plus 3, which is a degree 4 polynomial with even degree. This gives the same end behavior in both directions. Looking at x equals 0, option A crosses the x-axis, while option B touches but doesn't cross it, indicating a root with multiplicity 2. This matches our graph's behavior at the origin. Now let's examine options C and D and compare them with option B. Option C is y equals x times x plus 2 times x minus 3, which has x-intercepts at x equals negative 2, 0, and 3. Option D is y equals x squared times x plus 2 times x minus 3, which also has x-intercepts at x equals negative 2, 0, and 3, but with x equals 0 having multiplicity 2. Looking at our original graph, the x-intercepts are at negative 3, 0, and 2. This means both options C and D have incorrect roots and cannot match our graph. Only option B has the correct x-intercepts at negative 3, 0, and 2, with the proper behavior at each intercept. Let's confirm that option B is the correct answer by verifying all key features of the graph. Option B is y equals x squared times x minus 2 times x plus 3. First, it has the correct x-intercepts at x equals negative 3, 0, and 2. At x equals 0, the graph touches the x-axis without crossing it, indicating a root with multiplicity 2. At x equals negative 3 and x equals 2, the graph crosses the x-axis. As a degree 4 polynomial with a positive leading coefficient, the graph approaches positive infinity as x approaches either positive or negative infinity. The graph also has a local maximum in the second quadrant and a local minimum in the fourth quadrant. All these features perfectly match our original graph, confirming that option B is the correct answer. Let's summarize what we've learned about identifying polynomial functions from their graphs. When analyzing a graph, we need to identify the x-intercepts or roots and their multiplicities, which tell us whether the graph crosses or touches the x-axis at those points. We also need to examine the end behavior, which is determined by the degree of the polynomial and its leading coefficient. Even-degree polynomials with positive leading coefficients approach positive infinity in both directions. Additionally, we should note any local maxima and minima. For this problem, we determined that option B, y equals x squared times x minus 2 times x plus 3, is the correct answer because it has the correct roots at x equals negative 3, 0, and 2, with the root at x equals 0 having multiplicity 2. As a degree 4 polynomial with a positive leading coefficient, it has the correct end behavior, and it matches all the critical points and curve features of the original graph.