To determine which type of function best models the data provided in the table, we can analyze how the y-values change with respect to the x-values.
Given the data points:
- (1, 26)
- (4, 86)
- (7, 146)
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Linear Function: A linear function is of the form \(y = mx + b\). To check if it is linear, we look for a constant rate of change (slope).
- From (1, 26) to (4, 86): The change in y is \(86 - 26 = 60\) and the change in x is \(4 - 1 = 3\), giving a slope of \(60 / 3 = 20\).
- From (4, 86) to (7, 146): The change in y is \(146 - 86 = 60\) and the change in x is \(7 - 4 = 3\), giving the same slope of \(60 / 3 = 20\).
Since the slope remains constant, the first impression is that it could be linear.
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Quadratic Function: A quadratic function is of the form \(y = ax^2 + bx + c\). For a quadratic, the second difference of the y-values should be constant.
Let's calculate the first differences:
- First difference between (1, 26) and (4, 86): \(86 - 26 = 60\)
- First difference between (4, 86) and (7, 146): \(146 - 86 = 60\)
Since the first differences are constant, we need to check if the second differences are also constant.
For first differences, we have:
- \(60\) (from the first set) and \(60\) (from the second set).
As there are not enough values here to confirm non-linearity and calculate second differences, we cannot determine at this level if it's quadratic without more data.
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Cubic Function: A cubic function is of the form \(y = ax^3 + bx^2 + cx + d\). This typically requires more data points for evaluation.
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Exponential Function: An exponential function is of the form \(y = ab^x\). This involves checking if the ratios between successive values are constant.
Since we established that the structure looks more linear due to constant first differences and a pattern of change that adheres to a straightforward increase without signifying an exponential or non-linear motion, we can conclude:
The best model for the provided data is Linear.
1 answer