Solve this question---**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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In this problem, we need to identify the equation that matches the given graph. The graph shows a polynomial function with specific features. Let's analyze the key characteristics: The curve crosses the x-axis at x equals negative 3 and x equals 2, and touches the x-axis at x equals 0. The curve has a local maximum in the second quadrant and a local minimum in the fourth quadrant. As x approaches either positive or negative infinity, the curve goes upward toward positive infinity. Let's analyze the x-intercepts of the graph. The curve has three x-intercepts: at x equals negative 3, x equals 0, and x equals 2. At x equals negative 3 and x equals 2, the curve crosses the x-axis, indicating roots with multiplicity 1. At x equals 0, the curve touches the x-axis but doesn't cross it. This behavior indicates a root with multiplicity 2. These x-intercepts correspond to the factors in our equation. A root at x equals negative 3 gives us the factor (x plus 3). A root at x equals 2 gives us the factor (x minus 2). And a double root at x equals 0 gives us the factor x squared. Now let's analyze the end behavior of the graph. As x approaches negative infinity, y approaches positive infinity. Similarly, as x approaches positive infinity, y also approaches positive infinity. This behavior indicates a polynomial with an even degree and a positive leading coefficient. When we expand the expression x squared times (x minus 2) times (x plus 3), we get x to the fourth plus x cubed minus 6x squared. This confirms that our polynomial has degree 4, which is even. The leading coefficient is 1, which is positive. This matches the observed end behavior of the graph. Now let's evaluate each of the given options to determine which one matches our graph. Option A is y equals x times (x minus 2) times (x plus 3). This has the correct roots, but it's a degree 3 polynomial, which is odd. An odd-degree polynomial would have opposite end behaviors, but our graph approaches positive infinity in both directions. So option A is incorrect. Option B is y equals x squared times (x minus 2) times (x plus 3). This has the correct roots with the right multiplicities: x equals 0 with multiplicity 2, x equals 2, and x equals negative 3. It's a degree 4 polynomial, which is even, and has a positive leading coefficient. This matches our graph's end behavior. Option C is y equals x times (x plus 2) times (x minus 3). This has the wrong roots: 0, negative 2, and 3. Option D is y equals x squared times (x plus 2) times (x minus 3). This also has the wrong roots: 0 with multiplicity 2, negative 2, and 3. Therefore, option B is the correct answer. Let's summarize what we've learned from this problem. First, the x-intercepts of a graph determine the factors in its equation. In our case, the graph has roots at x equals negative 3, x equals 0 with multiplicity 2, and x equals 2. Second, the end behavior of the graph—both ends pointing upward—indicates an even-degree polynomial with a positive leading coefficient. Third, when analyzing polynomial graphs, we need to consider both the locations of the roots and their multiplicities, which affect how the graph behaves at those points. After evaluating all options, we found that option B, y equals x squared times (x minus 2) times (x plus 3), matches all the key features of our graph. This expands to x to the fourth plus x cubed minus 6x squared. Therefore, B is the correct answer.