# How do you graph the function y=x-3?

Jan 31, 2017

See explanation

#### Explanation:

The graph of $y = x - 3$ is almost the same as $y = x$. The difference is that every point on $y = x$ has been lowered by 3.

Thus, instead of the y-intercept being at $y = 0$ it is at $y = - 3$ instead.

Consider the generic equation of $y = m x + c \textcolor{w h i t e}{.}$ where $m$ is the gradient (slope).

If you compare this to both $y = x$ and $y = x - 3$ you will observe that there is no value shown for $m$. That is because it is there but it's value is 1 and it is good mathematical practice to NOT write $1 x$

The gradient of 1 $= \left(\text{count of up or down")/("count of along - left to right}\right)$

This must mean that the (count of up or down) = (the count of along")

The y-axis passes through the x-axis at $x = 0$ so if we substitute the value of 0 for $x$ we get:

$y = x - 3 \text{ "->" } y = 0 - 3$

So at $x = 0$ we have $y = - 3$

So the x-intercept is at the point $P \to \left(x , y\right) = \left(0 , - 3\right)$

Similarly the graph crosses the x-axis at $y = 0$ so by substitution we have:

$y = x - 3 \text{ "->" } 0 = x - 3$

$0 + 3 = x - 3 + 3$
$3 = x + 0$
$x = 3$
So the y-intercept is at the point $P \to \left(x , y\right) = \left(3 , 0\right)$