To create a quadratic model for the dataset, let's first identify the x-values and their corresponding y-values:
- (6, -1)
- (5, 0)
- (1, 6)
- (9, 2)
- (14, 3)
A quadratic model takes the form of \( f(x) = ax^2 + bx + c \).
To find the quadratic equation that best fits the given data, we can use methods like polynomial regression, but given that we don’t have tools for calculations right now, we will proceed with analysis of the options provided.
The given options are:
- \( f(x) = 2x + 5 \) (Linear, not quadratic)
- \( f(x) = x^2 + 7 \) (Quadratic, but needs verification)
- \( f(x) = r^2 + 5 \) (This seems incorrect as it introduces 'r' without context)
- \( f(x) = x + 5 \) (Linear, not quadratic)
To test which one might fit the data points, we will analyze the second option, \( f(x) = x^2 + 7 \).
For each x-value, calculate \( f(x) \):
-
For \( x = 6 \): \( f(6) = 6^2 + 7 = 36 + 7 = 43 \) (not equal to -1)
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For \( x = 5 \): \( f(5) = 5^2 + 7 = 25 + 7 = 32 \) (not equal to 0)
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For \( x = 1 \): \( f(1) = 1^2 + 7 = 1 + 7 = 8 \) (not equal to 6)
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For \( x = 9 \): \( f(9) = 9^2 + 7 = 81 + 7 = 88 \) (not equal to 2)
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For \( x = 14 \): \( f(14) = 14^2 + 7 = 196 + 7 = 203 \) (not equal to 3)
Thus, \( f(x) = x^2 + 7 \) does not fit the data points.
Since the dataset does not seem to perfectly fit any of the given functions (especially since 4 of the options are clearly not quadratic), the most reasonable course of action is to point out that none of the provided options seem to correctly model the data points given.
Therefore, based on the provided options, I recommend recalculating based on actual derived values or using a regression method for precise coefficients \( a, b, \) and \( c \) for a true quadratic fit—until that point, none of the provided answers seem adequate for the data provided.
3 answers