To compare the linear function and the arithmetic sequence, let's analyze each one step by step:
Linear Function Analysis:
Given points:
- (-1, 10)
- (0, 14)
- (1, 18)
- (2, 22)
1. Rate of Change: The rate of change (slope) of a linear function can be calculated using two points. We can find the difference in y-values (vertical change) over the difference in x-values (horizontal change).
Using the points (0, 14) and (1, 18):
- Change in y = 18 - 14 = 4
- Change in x = 1 - 0 = 1
- Rate of change = Change in y / Change in x = 4 / 1 = 4
So, the rate of change of the linear function is 4.
2. Y-intercept: The y-intercept is the value of y when x = 0. From the table, we see that when x = 0, y = 14.
So, the y-intercept of the linear function is 14.
Arithmetic Sequence Analysis:
Given explicit formula:
- \( a_n = 10 + 4(n-1) \)
1. Rate of Change (Common Difference): The common difference in an arithmetic sequence is the additional amount added to each term to get to the next term. From the formula, we can infer that the common difference is represented by the coefficient of \( (n-1) \).
- Therefore, the common difference (rate of change) of the arithmetic sequence is 4.
2. First Term: The first term of the arithmetic sequence \( a_1 \) can be found by substituting \( n = 1 \) into the explicit formula.
- \( a_1 = 10 + 4(1 - 1) = 10 + 4(0) = 10 \)
So, the first term of the arithmetic sequence is 10.
Final Comparison:
- The rate of change of this linear function is 4, which is the same as the rate of change (common difference) of the arithmetic sequence, which is also 4.
Summary:
- Rate of Change of Linear Function: 4
- Y-intercept of Linear Function: 14
- Rate of Change (Common Difference) of Arithmetic Sequence: 4
- First Term of Arithmetic Sequence: 10
Arithmetic sequences can be represented as a linear function, as demonstrated through this analysis, where both exhibit a constant rate of change.
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