## 2.1 Defining a straight line

Before we disucss a 'line of best fit', we will first discuss how to define a straight line. Consider the below graph of a straight (linear) line:

When we plot a straight line as we have done above, it can be defined by two things:

1. The "$$y$$-intercept", i.e. the value of $$y$$ at which the line crosses the $$y$$ axis. This occurs when $$x = 0$$
2. The "slope" of the line. The slope tells us how 'steep' or 'flat' the line is. It also tells us how much $$y$$ increases (or decreases) for each unit increase in $$x$$.

We can write down the equation of a line in a way you may be familiar with:

$y = mx + c,$

where:

• $$m$$ is the slope of the line
• $$c$$ is the $$y$$-intercept.

By studying the above graph, see if you can answer the following questions:

1. What is the $$y$$-intercept?

The equation of the line is provided in the above graph - you can use this equation to identify the value of $$c$$ (i.e. the $$y$$-intercept).

10

1. What is the slope?

The equation of the line is provided in the above graph - you can use this equation to identify the value of $$m$$ (i.e. the slope).

5

1. What would be the value of $$y$$ when $$x = 2$$?

$$y = 5x + 10 = 5\times 2 + 10 = \ldots$$?

20

To further explain the slope, let's zoom in on the above graph:

Looking at this zoomed in version of the graph, we can see that as we move from the yellow point to the red point, the following happens:

• $$x$$ increases by one unit (from 2 to 3)
• $$y$$ increases by 5 (from 20 to 25).

No matter where we are on the line, increasing $$x$$ by one unit will always result in an increase of $$y$$ by 5, which is the slope.