To determine the function modeled by the given points \((1, 3)\), \((2, 6)\), \((3, 9)\), and \((4, 12)\), we first notice that these points suggest a linear relationship.
We can represent a linear function in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
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Finding the Slope (m): The slope can be calculated from any two points using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Using the points \((1, 3)\) and \((2, 6)\): \[ m = \frac{6 - 3}{2 - 1} = \frac{3}{1} = 3 \]
This calculation holds for any pair of points. For example, using points \((2, 6)\) and \((3, 9)\): \[ m = \frac{9 - 6}{3 - 2} = \frac{3}{1} = 3 \]
So, the slope is \(3\).
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Finding the Y-Intercept (b): To find the y-intercept, we can choose one of the points and use the slope to solve for \(b\). Using the point \((1, 3)\): \[ y = mx + b \implies 3 = 3(1) + b \] \[ 3 = 3 + b \implies b = 3 - 3 = 0 \]
Thus, the y-intercept is \(0\).
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Writing the Function: The equation of the line based on the slope and y-intercept found is: \[ y = 3x + 0 \quad \text{or simply} \quad y = 3x \]
Final Answer: The function modeled by the pattern is \(y = 3x\), since the slope is \(3\) and the y-intercept is \(0\).
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