Let's analyze the given linear function and the arithmetic sequence.
For the Linear Function:
We have the data points:
| x | y | |----|----| | -1 | 10 | | 0 | 14 | | 1 | 18 | | 2 | 22 |
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Rate of Change of the Linear Function:
- The rate of change (slope) can be calculated using any two points.
- Using points (0, 14) and (1, 18): \[ \text{Rate of Change} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{18 - 14}{1 - 0} = 4 \]
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Y-Intercept of the Linear Function:
- The y-intercept is the value of y when x = 0.
- From the table, when \( x = 0 \), \( y = 14 \).
- Thus, the y-intercept is 14.
For the Arithmetic Sequence:
The explicit formula is: \[ a_n = 10 + 4(n - 1) \]
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Rate of Change (Common Difference) of the Arithmetic Sequence:
- The common difference is the coefficient of \( n \) in the explicit formula.
- Hence, the rate of change (common difference) is 4.
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First Value of the Arithmetic Sequence:
- The first term (when \( n = 1 \)) can be calculated as follows: \[ a_1 = 10 + 4(1 - 1) = 10 + 0 = 10 \]
- Thus, the first value is 10.
Comparison of Rates of Change:
- The rate of change of the linear function is 4.
- The rate of change of the arithmetic sequence is 4.
Conclusion:
- The statement "The rate of change of this linear function is _the rate of change of this arithmetic sequence."** can be filled with "equal to".
- Additionally, "Arithmetic sequences _can be represented as a linear function" can be filled with "can", as arithmetic sequences indeed can be represented as linear functions.
Final 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 can be represented as a linear function.