To compare the linear function and the arithmetic sequence, we can break down the information you've provided.
Linear Function
From the linear function you provided:
- Points: (0, 14), (1, 18), (2, 22), and (1, 10) seems contradictory to the other pairs.
We can interpret this data. Let’s calculate the slope (rate of change) from the two consistent points:
Using the points (0, 14) and (1, 18):
- Slope (m) = (y2 - y1) / (x2 - x1) = (18 - 14) / (1 - 0) = 4.
Using the points (1, 18) and (2, 22):
- Slope (m) = (22 - 18) / (2 - 1) = 4.
Thus, the rate of change of the linear function is 4.
Y-Intercept
The y-intercept is the y-value when x = 0. From the point (0, 14), we find that the y-intercept of the linear function is 14.
Arithmetic Sequence
The explicit formula for the arithmetic sequence is given as: \[ a_n = 10 + 4(n - 1) \]
Rate of Change
- The common difference (rate of change) is 4, as indicated by the "+ 4" term in the formula.
First Value
To find the first value of the arithmetic sequence, we can simply set \( n = 1 \): \[ a_1 = 10 + 4(1 - 1) = 10 + 4 \cdot 0 = 10 \]
So, the first value of the arithmetic sequence is 10.
Summary
- Rate of Change of Linear Function: 4
- Y-Intercept of Linear Function: 14
- Rate of Change of Arithmetic Sequence: 4
- First Value of Arithmetic Sequence: 10
Conclusion
Both the linear function and the arithmetic sequence have the same rate of change of 4. The arithmetic sequence can be represented as a linear function, specifically in the format of \( y = mx + b \) where \( m \) is the rate of change and \( b \) is the y-intercept.
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