Maths year 10/11: gradients, intercepts and equations of straight lines

A treasure map tells you how to get to the treasure.

From a start point it tells you how many steps you need to go in each direction.

On this map from the start you need to go 5 steps along to the right and then 3 steps up.

Graph paper is just a grid of squares.

You draw a horizontal and a vertical line on the paper which cross at a point called the **origin**.

You need an origin because you can go along left and right and go up and down like on the map.

Sometimes they say **go along the corridor and up the stairs**.

The vertical scale is called the Y axis. Remember the vertical part of the Y.

The horizontal scale is called ther X axis.

** Q:** If you started at point A on the map what are the steps to reach the treasure

Every point on the grid can be found with two numbers.

One along the vertical scale labelled y and one on the horizontal scale labelled x

These are called co-ordinates and written like (x , y) where x and y are numbers.

The origin is the point (0, 0)

On the grid shown, (1,2) is one along the x scale right from zero and two along the y scale up from zero.

**Q:** What are the co-ordinates of the other points

A straight line is one that does not curve.

It is the shortest distance between two points.

A straight line that goes through a set of co-ordinates has an equation

Look at the vertical line that goes through (2, -1), (2, 0) and (2, 3). All the x co-ordinates are 2. This line has the equation x = 2

Q1: What is the equation of the horizontal line?

ANSLook at these lines on the grid

What do you notice about the co-ordinates on the red line?

The co-ordinates are (–1, –1), (1, 1), (2, 2) and (3, 3)

So the x-co-ordinates are the same as the y co-ordinates.

We write this as y = x and say the **equation** of this line is **y = x**.

The **gradient** of a line is how much it goes up or down in a certain horizontal distance

Gradient is how steep a line is.

Use a triangle to work out a gradient as shown above. Divide the height by the width (3 ÷ 3 = 1)

So the gradient of the red line is 1.

**Q1**: What is the equation of the blue line.

**Q2**: What is the gradient of the blue line.

**Q3**: What is this gradient:

**Q4**: What is this gradient:

**Q5**: What does the line look like with a gradient of 0.

**Q6**: What does the line look like with a gradient of infinity.

The equation of a straight line is:

where m is the gradient and c is the intercept. (in some countries it's b not c

This is the y-intercept and is the y value at that point

Looking at this line, the gradient, m = 1

and the intercept c (the y value) = 2

So the equation of this line is **y = x + 2**

**Q1**: What is the equation of line A and line B

**Q2**: Do you notice anything about these lines