# Write a function rule for the table?

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

May 25, 2018

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 $m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$ 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 $\left(0 , 3\right)$, we will use slope-intercept form: $y = m x + b$, where $m$ is the slope and $b$ is the y-intercept

Our first step in this process will be finding the slope:
$m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$

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 $\left(0 , - 3\right)$ and $\left(1 , - 2\right)$. Plugging into the slope formula:
$m = \frac{- 3 - \left(- 2\right)}{0 - 1} = - \frac{1}{-} 1 = 1$

Since we are given the y-intercept $\left(0 , - 3\right)$ we can simply plug $b$ into the slope-intercept form formula and we find the function rule:
$y = m x + b$

$y = 1 x - 3$

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