solve this---**Chart Description:** The image contains a graph of a function plotted on a Cartesian coordinate system. - The horizontal axis is the x-axis, labeled 'x'. - The vertical axis is the y-axis, labeled 'y'. - The origin is marked by the letter 'O'. - The x-axis has tick marks labeled -3, -2, -1, 1, 2, 3. - The y-axis has tick marks labeled -10, 10. - The graph is a smooth curve. - The curve intersects the x-axis at approximately x = -3. - The curve touches the x-axis at the origin, x = 0, forming a local maximum there (y=0). - The curve intersects the x-axis at approximately x = 2. - The curve goes down to a local minimum between x=1 and x=2, with a y-value below -10. - The curve appears to be a polynomial function. **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 the equation of this polynomial function. The graph shows a curve with 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, forming a local maximum. This indicates that x equals 0 is a root with multiplicity 2, giving us a factor of x squared. At x equals negative 3 and x equals 2, the curve crosses the x-axis, indicating roots with multiplicity 1, giving us factors of x plus 3 and x minus 2. Looking at the end behavior, as x approaches positive or negative infinity, y approaches positive infinity, which is consistent with an even-degree polynomial with a positive leading coefficient. Now let's analyze the behavior at each x-intercept to determine the factors of our polynomial. At x equals negative 3, the graph crosses the x-axis, indicating a factor of x plus 3 with multiplicity 1. At x equals 0, the graph touches the x-axis without crossing, forming a local maximum. This behavior indicates a factor of x squared with multiplicity 2. At x equals 2, the graph crosses the x-axis again, giving us a factor of x minus 2 with multiplicity 1. Combining these factors, we get y equals x squared times x minus 2 times x plus 3, which corresponds to option B. Let's compare all four options by graphing them. Option A has a degree of 3, which is odd, so its end behavior would have y approaching positive infinity on one side and negative infinity on the other. This doesn't match our graph. Option B has a degree of 4, which is even, with a positive leading coefficient, so y approaches positive infinity on both sides. It also has the correct roots: x equals 0 with multiplicity 2, and x equals 2 and negative 3 with multiplicity 1. Options C and D have roots at x equals 0, negative 2, and 3, which don't match our graph's x-intercepts. Therefore, option B is the only equation that correctly represents the given graph. Let's verify that option B is indeed the correct answer. The equation y equals x squared times x minus 2 times x plus 3 gives us a polynomial of degree 4, which is even. This means the end behavior has y approaching positive infinity as x approaches either positive or negative infinity, which matches our graph. The x-intercepts occur at x equals negative 3, x equals 0, and x equals 2, exactly as shown in the graph. At x equals 0, the factor of x squared creates a local maximum where the curve touches but doesn't cross the x-axis. When expanded, this polynomial becomes y equals x to the fourth plus x cubed minus 2x squared minus 6x. All the key features of this function perfectly match the given graph, confirming that option B is the correct answer. To summarize what we've learned: When identifying a polynomial function from its graph, we need to analyze several key features. First, identify all x-intercepts, which correspond to the roots of the equation. Second, determine the behavior at each root - whether the graph crosses the x-axis or just touches it. A root with odd multiplicity causes the graph to cross the x-axis, while a root with even multiplicity causes the graph to touch but not cross. Third, examine the end behavior, which is determined by the degree of the polynomial and the sign of its leading coefficient. In our problem, option B with equation y equals x squared times x minus 2 times x plus 3 is the correct answer because it has roots at x equals negative 3, 0, and 2, with the appropriate multiplicities, and its even degree with positive leading coefficient matches the end behavior of the graph.