Figuring out the type of function from a table of values requires either a graph or to look at y increases when x increases. (Or decreases.)
The function is linear because as x increases, y increases at the same amount no matter what two x values you pick. If you increase by 1 on the x, you always increase by 1/2 on x. And if you double the increases on x, the same for y.
Since it's a linear function, it has a slope. Slope, m, is the difference of the y coordinates divided by the difference of the x coordinates. Choose (4, 2) and (0,0) as (x₁, y₁) and (x₂, y₂) respectively for this, but any pair works.
m = [tex] \frac{y_{2}-y_{1}}{x_{2}-x_{1}} = \frac{2-0}{4-0} = \frac{2}{4} = \frac{1}{2} [/tex]
Our slope is 1/2. To write this in functional form, we can write this equation in slope-intercept form, y = mx + b where m is our slope and b is our y-intercept. We know from above that (0,0) is on the graph so 0 is both the x-intercept and the y-intercept. (If you substitute x = 0, y = 0, and m = 1/2 you would that b = 0 as well.)
So m = 1/2 and b = 0, and we report the linear equation in slope-intercept form, y = 1/2x + 0, which we give as y = 1/2x. Our equation represents y in terms of x, so it's a function too.
Thus, the function is y = [tex] y = \frac{1}{2}x [/tex].
