Straight line graphs

​​In a nutshell

Straight lines join two points on a coordinate grid together and can continue past those points infinitely. There are three types of straight line graphs: horizontal, vertical and diagonal. Straight lines can be described using two quantities: the gradient and the $$y$$-intercept.

Definitions

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​​Different gradients

You need to be able to recognise if a line has a positive, negative or zero gradient. You can denote the gradient with $$m$$.​

Positive gradient

If a line has a positive gradient, then it is a diagonal line, moving up as it moves to the right. The line below has a positive gradient, or in other words, $$m>0$$​:

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The bigger the value of $$m$$, the steeper the line. So a line with gradient $$5$$ is steeper than a line with gradient $$2$$.​

​​Negative gradient

A line with a negative gradient is also a diagonal line, but it moves down as it moves to the right. The line below has a negative gradient. In other words, $$m<0$$​:

The more negative the value of $$m$$, the steeper the line. So a line with gradient $$-7$$ is steeper than a line with gradient $$-3$$.

Zero gradient

A line can have a gradient of $$0$$. So $$m=0$$. A line with a zero gradient is horizontal, for example, the line below:

Vertical lines

How steep is a vertical line? A vertical line is 'infinitely steep', but infinity is not a number so you can't say that $$m$$ is equal to infinity. Gradient is not referred to when discussing vertical lines. 

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Location on the grid

The gradient tells you how steep a line is, and the $$y$$-intercept tells you where the line sits on the coordinate grid. This is denoted by $$c$$. 

The $$y$$-intercept of a line is where the line crosses the  $$y$$​-axis. It is straightforward to read or mark the $$y$$-intercept of a graph.  You can use the equation of a line to do this.

A vertical line will not have a $$y$$​-intercept. In the special case of the vertical line being on the $$y$$-axis, the line touching the $$y$$-axis is everywhere. 

Example 1

What can you conclude about the gradient and the $$y$$-intercept of the graph below?

The line is moving upwards as it moves to the right, so its gradient is positive. It crosses the $$y$$-axis at $$-1$$, so the $$y$$-intercept is $$\underline{-1}$$​. 

Example 2

​Consider a graph of a horizontal line that passes through the point $$(7,5)$$. What can you conclude about the line's gradient and its $$y$$-intercept?

If a line is horizontal, it must have a zero gradient. 

Gradient = $$\underline{m=0}$$​​

A horizontal line has the same $$y$$-coordinate all the way along it. This line passes $$(7,5)$$, therefore must have $$y$$-coordinate $$5$$ at every point. 

​$$y$$​-intercept = $$\underline{c = 5}$$​

The gradient and the $$y$$-intercept are:

​$$\underline{m=0, c=5}$$​​

Note: Even though this example was about a graph, a diagram was not actually necessary! Sometimes you can answer graph questions without a picture.
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