# 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**

**Gradient** | How steep the line is. |

**y**-INTERCEPT
| The point on the $y$-axis where the line crosses through. |

**Horizontal**
| A straight line going between left and right. Think of the "horizon" to help you remember. |

**Vertical**
| A straight line going between up and down. |

## 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, $m>0$:

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.*