To determine which function has a greater rate of change than the function represented in the table, we first need to calculate the rate of change for the data given in the table.
The table shows the following pairs of \( (x, y) \):
- For \( x = 2 \), \( y = 7 \)
- For \( x = 3 \), \( y = 10.5 \)
- For \( x = 4 \), \( y = 14 \)
- For \( x = 5 \), \( y = 17.5 \)
- For \( x = 6 \), \( y = 21 \)
To find the rate of change, we can calculate the differences in \( y \) and \( x \) values between consecutive points:
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From \( (2, 7) \) to \( (3, 10.5) \): \[ \text{Change in } y = 10.5 - 7 = 3.5 \] \[ \text{Change in } x = 3 - 2 = 1 \] \[ \text{Rate of change} = \frac{3.5}{1} = 3.5 \]
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From \( (3, 10.5) \) to \( (4, 14) \): \[ \text{Change in } y = 14 - 10.5 = 3.5 \] \[ \text{Change in } x = 4 - 3 = 1 \] \[ \text{Rate of change} = \frac{3.5}{1} = 3.5 \]
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From \( (4, 14) \) to \( (5, 17.5) \): \[ \text{Change in } y = 17.5 - 14 = 3.5 \] \[ \text{Change in } x = 5 - 4 = 1 \] \[ \text{Rate of change} = \frac{3.5}{1} = 3.5 \]
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From \( (5, 17.5) \) to \( (6, 21) \): \[ \text{Change in } y = 21 - 17.5 = 3.5 \] \[ \text{Change in } x = 6 - 5 = 1 \] \[ \text{Rate of change} = \frac{3.5}{1} = 3.5 \]
Since the rate of change for the function represented in the table is consistently \( 3.5 \), we will compare this with the rate of change for the functions listed in the options.
A) \( y = 4x \) gives a rate of change of \( 4 \).
B) \( y = x + 2 \) gives a rate of change of \( 1 \).
C) \( y = 2x + 3 \) gives a rate of change of \( 2 \).
D) \( y = 3x + 4 \) gives a rate of change of \( 3 \).
From this analysis, we see that only option A \( y = 4x \) has a greater rate of change (4) than the table function (3.5).
Thus, the correct answer is A) y = 4x.