To determine which equation describes the function in the table, we can analyze the values of x and y provided in the table:
- For \(x = 0\), \(y = 3\)
- For \(x = 1\), \(y = 3.5\) (which can be written as \(\frac{7}{2}\))
- For \(x = 2\), \(y = 4\)
- For \(x = 3\), \(y = 4.5\) (which can be written as \(\frac{9}{2}\))
Now, we can try to find the pattern or linear relationship in the values.
Observing the changes in y:
- From \(y = 3\) to \(y = 3.5\) (an increase of 0.5) when \(x\) increases from 0 to 1.
- From \(y = 3.5\) to \(y = 4\) (an increase of 0.5) when \(x\) increases from 1 to 2.
- From \(y = 4\) to \(y = 4.5\) (an increase of 0.5) when \(x\) increases from 2 to 3.
Thus, for every increase of 1 in \(x\), \(y\) increases by \(0.5\).
This suggests a linear relationship that can be expressed in slope-intercept form \(y = mx + b\), where \(m\) is the slope.
The slope \(m\) is \(0.5\) (since \(0.5\) is equivalent to \(\frac{1}{2}\)). To find \(b\), we look at the value when \(x = 0\). When \(x = 0\), \(y = 3\), which means \(b = 3\).
Thus, we can express this in function notation: \[ f(x) = \frac{1}{2}x + 3 \]
Therefore, the correct response is:
f(x) = 1/2x + 3