Let's go through the comparison step by step:
Linear Function Analysis:
Given values for the linear function:
| x | y | |----|----| | -1 | 10 | | 0 | 14 | | 1 | 18 | | 2 | 22 |
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Rate of Change: The rate of change (slope) of a linear function can be determined using the formula: \[ \text{Slope} = \frac{y_2 - y_1}{x_2 - x_1} \] Using any two points from the table, let’s take (0, 14) and (1, 18): \[ \text{Rate of Change} = \frac{18 - 14}{1 - 0} = \frac{4}{1} = 4 \]
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Y-intercept: The y-intercept is the value of \( y \) when \( x = 0 \). From the table, when \( x = 0 \), \( y = 14 \). \[ \text{Y-intercept} = 14 \]
Arithmetic Sequence Analysis:
The explicit formula is given as: \[ a_n = 10 + 4(n - 1) \]
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Rate of Change (Common Difference): In an arithmetic sequence, the rate of change (common difference) is the coefficient of \( n \) when you write the explicit formula in standard form. Here, the common difference is 4 (as seen from the formula). \[ \text{Common Difference} = 4 \]
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First Value of the Arithmetic Sequence: The first term of the sequence, when \( n = 1 \): \[ a_1 = 10 + 4(1 - 1) = 10 + 0 = 10 \]
Comparison Summary:
- Rate of change of the linear function: 4
- Y-intercept of the linear function: 14
- Rate of change of the arithmetic sequence: 4
- First value of the arithmetic sequence: 10
Conclusion:
The rate of change of this linear function is equal to the rate of change of this arithmetic sequence. Both share the same rate of change of 4, and the arithmetic sequences can indeed be represented as linear functions of the form \( y = mx + b \), where \( m \) is the rate of change (common difference) and \( b \) is the y-intercept.
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