To determine which function best models the data represented by the scatterplot, we need to analyze the options given based on the observed positive relationship between the x and y values.
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Linear Model (Option A: y = 25x + 20): This is a linear function, which means it has a constant rate of change. Based on the data points, we need to check if this option aligns with them. For instance, plugging in x = 7 gives y = 25(7) + 20 = 175, which is significantly lower than the y value of 402 at that x value. So, this doesn't fit well.
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Quadratic Model (Option B: y = 8x^2 + 10): This is a quadratic function. Let's see if it makes sense for our points. x = 7 gives y = 8(7^2) + 10 = 8(49) + 10 = 392 + 10 = 402. This matches the point (7, 402). For x = 11, y = 8(11^2) + 10 = 8(121) + 10 = 968 + 10 = 978, which also matches. For x = 13, y = 8(13^2) + 10 = 8(169) + 10 = 1352 + 10 = 1362, which matches too. This function fits all three points perfectly and follows the trend.
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Exponential Model (Option C: y = 2^x + 12): This is an exponential function. For x = 7, y = 2^7 + 12 = 128 + 12 = 140, which is too low. For x = 11, y = 2^11 + 12 = 2048 + 12 = 2060, which is too high. This function does not fit well.
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Linear Model (Option D: y = 160x - 100): Testing this function, for x = 7: y = 160(7) - 100 = 1120 - 100 = 1020, which is too high. For x = 11: y = 160(11) - 100 = 1760 - 100 = 1660, which is again too high. This does not fit the data.
After evaluating the options, the quadratic function y = 8x^2 + 10 (Option B) best models the data since it fits all given points perfectly and follows the positive trend of the scatterplot.