To understand why the output values for linear function A will always be different from the corresponding output values for linear function B, we can analyze the two functions based on the provided tables.
Let's summarize the information given:
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Linear Function A:
- Input (x): 1, 3, 5, 7, 9
- Output (y): 3, 7, 11, 15, 19
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Linear Function B:
- Input (x): 1, 3, 5, 7, 9
- Output (y): 4, 8, 12, 16, 20
Now, we can clearly see from the tables:
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Initial Values (y-intercepts):
- For Linear Function A, when \(x = 1\), \(y = 3\).
- For Linear Function B, when \(x = 1\), \(y = 4\).
- The initial values (or outputs for the first input) are different (3 for A and 4 for B).
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Rates of Change (slopes):
- The change in output for Linear Function A (between successive input values) is consistently 4 (e.g., \(7 - 3 = 4\), \(11 - 7 = 4\), etc.),
- The change in output for Linear Function B is also consistently 4 (e.g., \(8 - 4 = 4\), \(12 - 8 = 4\), etc.).
- Therefore, both functions have the same rate of change (slope is the same).
Given these observations:
- The initial values (or outputs) for the two functions are different.
- However, the rates of change are the same.
Thus, the appropriate conclusion is:
The initial values of the two functions are different, and the rates of change of the two functions are also the same.
This is the reason why the output values for linear function A will always be different from the corresponding output values for linear function B.
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