To determine which equation describes the function in the table, we need to analyze the relationship between x and y in the provided data points. The given points are:
- (0, 3)
- (1, 3.5)
- (2, 4)
- (3, 4.5)
From these points, we can observe how y changes as x changes. Let's list the changes in y:
- From x = 0 to x = 1, y changes from 3 to 3.5 (an increase of 0.5).
- From x = 1 to x = 2, y changes from 3.5 to 4 (an increase of 0.5).
- From x = 2 to x = 3, y changes from 4 to 4.5 (an increase of 0.5).
This suggests a linear relationship, as y is increasing by a constant amount (0.5) for each unit increase in x.
To find the equation, let's consider the slope. The slope (m) can be calculated as:
m = (change in y) / (change in x) = 0.5 / 1 = 0.5.
Now, we can use the point-slope form of a linear equation. We know one point is (0, 3). Using the slope-intercept form \(y = mx + b\), we substitute m and b (where b is the y-intercept) to get:
\(y = 0.5x + 3.\)
In function notation, this would be represented as:
\(f(x) = \frac{1}{2}x + 3.\)
Now let's check the answer choices:
- f(y) = 12x + 3 (not fitting).
- f(y) = \frac{1}{2}x + 3 (this fits).
- f(x) = 12x + 3 (not fitting).
- f(x) = 3x + \frac{1}{2} (not fitting).
The equation that describes the function in the table using function notation is:
f(y) = \frac{1}{2}x + 3.