To determine which equation correctly describes the function from the table, we first need to observe the relationship between \(x\) and \(y\).
The given points are:
- When \(x = 0\), \(y = -5\)
- When \(x = 1\), \(y = -9\)
- When \(x = 2\), \(y = -13\)
- When \(x = 3\), \(y = -17\)
To identify the function, let's look at the differences in \(y\) as \(x\) increases:
- From \(x = 0\) to \(x = 1\): \(-9 - (-5) = -4\)
- From \(x = 1\) to \(x = 2\): \(-13 - (-9) = -4\)
- From \(x = 2\) to \(x = 3\): \(-17 - (-13) = -4\)
The \(y\) values decrease consistently by 4 as \(x\) increases by 1, suggesting a linear relationship. The slope \(m\) is \(-4\).
Next, we can use the slope-intercept form of the line, \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
Using the point \( (0, -5) \):
- The slope \(m = -4\)
- The y-intercept \(b = -5\)
Thus, the equation that describes the relationship is: \[ y = -4x - 5 \]
Now, we can express this in function notation: \[ f(x) = -4x - 5 \]
The correct answer from the provided responses is: f(x) = -4x - 5.