Question

Write a function rule for the table?

x y
-1 -4
0 -3
1 -2
2 -1

Answer 1

Answer: `y=x-3`

Explanation:

First, we can see that the function for this table is linear since each time `x` increases by `1`, `y` also increases by `1`. (Note: In general, we can see that a function is linear when the slope `m=(y_2-y_1)/(x_2-x_1)` between each data set is constant.)

Since we have established that the function given is indeed linear, we can use either point-slope form or slope-intercept form to find the function rule. In this case, since we are given a y-intercept `(0,3)`, we will use slope-intercept form: `y=mx+b`, where `m` is the slope and `b` is the y-intercept

Our first step in this process will be finding the slope:
`m=(y_2-y_1)/(x_2-x_1)`

Since the function is linear, we can choose any two data points, but choosing a data point in which either `x` or `y` is `0` will simplify the calculations. So, we will use `(0,-3)` and `(1,-2)`. Plugging into the slope formula:
`m=(-3-(-2))/(0-1)=-1/-1=1`

Since we are given the y-intercept `(0,-3)` we can simply plug `b` into the slope-intercept form formula and we find the function rule:
`y=mx+b`

`y=1x-3`

`y=x-3`, which is our final answer