To find the equation that describes the function represented by the given points, we can first analyze the data:
Let's examine the points:
- (0, -2)
- (2, 4)
- (4, 10)
- (6, 16)
Next, we can look for patterns in the \(y\) values based on the \(x\) values.
- Calculate the differences in \(y\):
- From \(x = 0\) to \(x = 2\): \(4 - (-2) = 6\)
- From \(x = 2\) to \(x = 4\): \(10 - 4 = 6\)
- From \(x = 4\) to \(x = 6\): \(16 - 10 = 6\)
Since the differences in \(y\) are constant (6), this indicates that the relation is linear.
-
Determine the slope \(m\): The slope (\(m\)) can be calculated as: \[ m = \frac{\text{change in } y}{\text{change in } x} = \frac{6}{2} = 3 \]
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Find the y-intercept \(b\): We can use the slope and one point to find the y-intercept. Using the point (0, -2): \[ y = mx + b \ -2 = 3(0) + b \ b = -2 \]
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Write the function: The equation that describes the function in the form \(y = mx + b\) is: \[ y = 3x - 2 \]
In function notation, we can express this as: \[ f(x) = 3x - 2 \]
Thus, the correct equation that describes the function in the table is: \[ \boxed{f(x) = 3x - 2} \]