Let's analyze the linear function and the arithmetic sequence provided.
Linear Function:
The table given shows the following pairs of \(x\) and \(y\):
- For \(x = -1\), \(y = 10\)
- For \(x = 0\), \(y = 14\)
- For \(x = 1\), \(y = 18\)
- For \(x = 2\), \(y = 22\)
To find the rate of change (slope) of the linear function, we can calculate the difference in \(y\) values for consecutive \(x\) values:
- From 0 to 1: \(18 - 14 = 4\)
- From 1 to 2: \(22 - 18 = 4\)
- From -1 to 0: \(14 - 10 = 4\)
The rate of change of the linear function is 4.
Next, to find the y-intercept, we observe that the y-intercept occurs when \(x = 0\). From the table, when \(x = 0\), \(y = 14\).
The y-intercept of the linear function is 14.
Arithmetic Sequence:
The explicit formula for the arithmetic sequence is given by: \[ a_n = 10 + 4(n - 1) \]
This formula also has a common difference (rate of change) mentioned, which can be recognized from the formula itself:
- The rate of change (common difference) of the arithmetic sequence is 4.
The first value of the arithmetic sequence can be found by substituting \(n = 1\) in the formula: \[ a_1 = 10 + 4(1 - 1) = 10 + 0 = 10 \]
So, the first value of the arithmetic sequence is 10.
Both the linear function and the arithmetic sequence have a rate of change of 4, confirming that the rate of change of this linear function is equal to the rate of change of this arithmetic sequence.
Lastly, we can say that arithmetic sequences can be represented as a linear function because both share the property of having a constant rate of change.
Putting it all together:
- The rate of change of the linear function is 4.
- The y-intercept of the linear function is 14.
- The rate of change (common difference) of the arithmetic sequence is 4.
- The first value of the arithmetic sequence is 10.
- The rate of change of this linear function is equal to the rate of change of this arithmetic sequence.
- Arithmetic sequences can be represented as a linear function.
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