To identify the function that describes the given table of values, we need to analyze the relationship between \( x \) and \( y \).
From the provided values:
- \( (10, 8) \)
- \( (5, 4) \)
- \( (0, 0) \)
- \( (-5, -4) \)
Let's check the slope \( m \) using two points from the table, say \( (10, 8) \) and \( (0, 0) \):
Using the formula for the slope: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{8 - 0}{10 - 0} = \frac{8}{10} = \frac{4}{5} \]
Now that we have the slope, we can express \( y \) in terms of \( x \): \[ y = \frac{4}{5}x \]
Now we can check which option corresponds to this function.
- Option a: \( y = \frac{5}{4}x \)
- Option b: \( y = \frac{4}{5}x \) (this matches!)
- Option c: \( y = -\frac{5}{4}x \)
- Option d: \( y = x - 2 \)
Since option b is \( y = \frac{4}{5}x \), it accurately represents the relationship between \( x \) and \( y \) in the table.
Thus, the correct answer is:
b. \( y = \frac{4}{5}x \)