To determine which formula can represent the points (1,2), (2,6), (3,10), and (4,14), we can analyze the points and check for a linear relationship.
We can first calculate the differences in y-values and corresponding x-values:
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From (1, 2) to (2, 6):
- Change in y = 6 - 2 = 4
- Change in x = 2 - 1 = 1
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From (2, 6) to (3, 10):
- Change in y = 10 - 6 = 4
- Change in x = 3 - 2 = 1
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From (3, 10) to (4, 14):
- Change in y = 14 - 10 = 4
- Change in x = 4 - 3 = 1
The change in y is consistently 4 for each change in x of 1, indicating a linear relationship with a slope of 4.
Thus, the slope (m) is 4. We can use the slope-intercept form of a linear equation, \(y = mx + b\), where \(m = 4\).
Now we can substitute one of the points to find the y-intercept (b). We'll use the first point (1, 2):
\[ 2 = 4(1) + b \] \[ 2 = 4 + b \] \[ b = 2 - 4 = -2 \]
So the equation of the line is:
\[ y = 4x - 2 \]
This corresponds to the response:
y = 4x - 2.
Thus, the correct formula that represents the points is \( y = 4x - 2 \).