Let's analyze the linear function and the arithmetic sequence step by step:
Linear Function:
Given the values:
| x | y | |----|----| | -1 | 10 | | 0 | 14 | | 1 | 18 | | 2 | 22 |
Rate of Change of the Linear Function: To find the rate of change, we look at how y changes with respect to x.
- From (-1, 10) to (0, 14): Increase by \(14 - 10 = 4\) for an increase in x of \(0 - (-1) = 1\).
- From (0, 14) to (1, 18): Increase by \(18 - 14 = 4\) for an increase in x of \(1 - 0 = 1\).
- From (1, 18) to (2, 22): Increase by \(22 - 18 = 4\) for an increase in x of \(2 - 1 = 1\).
Thus, the rate of change (slope) is 4.
Y-Intercept of the Linear Function: The y-intercept occurs when \(x = 0\). From the table, when \(x = 0\), \(y = 14\). Therefore, the y-intercept is 14.
Arithmetic Sequence:
The explicit formula for the arithmetic sequence is: \[ a_n = 10 + 4(n - 1) \]
Rate of Change (Common Difference) of the Arithmetic Sequence: The common difference in the arithmetic sequence can be seen from the formula. The value \(4\) indicates that each term increases by \(4\) for each increment in \(n\). So, the rate of change (common difference) is 4.
First Value of the Arithmetic Sequence: To find the first term (\(a_1\)): \[ a_1 = 10 + 4(1 - 1) = 10 + 0 = 10 \] Thus, the first value of the arithmetic sequence is 10.
Comparing Rates of Change:
The rate of change of the linear function is 4, and the rate of change of the arithmetic sequence is also 4. Therefore, the statement becomes: "The rate of change of this linear function is equal to the rate of change of this arithmetic sequence."
Conclusion on Relationship:
Arithmetic sequences can be represented as a linear function. Therefore, we conclude that: "Arithmetic sequences always can be represented as a linear function."
Summary:
- 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 always can be represented as a linear function.
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