Unit 14 Section 3 : Plotting Graphs Given Their Equations
In this section we introduce the idea of the gradient of a line, which is a measure of how steep it is.
We will also see how to plot the graph of a straight line using the equation of the line. There is a link
between the equation of a line and its gradient, which you should see by the end of the section.
Gradient of a line
The gradient of a line is found as shown in the diagram below:

Steps to find the gradient
 Mark two points on the line, as far apart as possible
 Connect the two points with one horizontal line and one vertical line to form a triangle
 Measure the step  how far the line has gone to the right on the horizontal line
 Measure the rise  how far the line has gone up (or down) on the vertical line
 Divide the rise by the step to get the gradient


Example Question
Look at the two lines on the diagram below.

Work out the gradient of each line.
In (a) the step is 4 and the rise is 2, so we divide 2 by 4 to find the gradient.
The gradient is 0.5
In (b) the step is 5 and the rise is –5 (because the line goes downwards), so the gradient is found by dividing –5 by 5.
The gradient is –1

NOTE: Looking from left to right, lines which go upwards have positive gradients and lines which go downwards have negative gradients
A horizontal line has a rise of zero, so the gradient will also be zero.
Practice Question
Look at the two lines on the diagram below.
Plotting a graph from its equation
The equation of a line (e.g. y = x + 3) usually tells us how to find a ycoordinate using an xcoordinate.
We can find the coordinates of several points on a line by picking x values and working out y values.
Example Question
A line has equation y = 2x + 1.
Using x values from –2 to +3, plot the graph of this equation.
The first stage is to draw up a table of x values and work
out the y values using the equation:
x  –2  –1  0  1  2  3 
y = 2x + 1  –3  –1  1  3  5  7 

Next, each pair of x and y values can be plotted on the graph
as coordinates. In this case the coordinates are: ( –2 , –3 ) ,
( –1 , –1 ) , ( 0 , 1 ) , ( 1 , 3 ) , ( 2, 5 ) and ( 3 , 7 ).
Finally the points are joined with a straight line running all the way across the graph:


Practice Question
A line has equation y = x – 3.
(a) Using x values from –1 to +4, work out the missing values in the table below.
(b) Plot the points on the graph below and draw a straight line through the points.
Exercises
Work out the answers to the questions below and fill in the boxes. Click on the
button to find out whether you have answered correctly. If you are right
then will appear and you should move on to the next
question. If appears then your answer is wrong. Click
on to clear your original answer and have another go.
If you can't work out the right answer then click on
to see
the answer.
NOTE: In the next few questions you need to plot points and draw lines on graphs.
To plot a point, just click on the graph with your left mouse button. To draw a line, hold down
the left mouse button at one point on the line and drag the pointer to another point on the line.
When you let go of the button the line will appear and it will automatically cross the whole graph.
If you make a mistake, press the delete key and the graph will be cleared.


Question 4
A line has equation y = x – 2
Question 5
A line has equation y = 2x – 2
Question 6
A line has equation y = ½x + 2
You have now completed Unit 14 Section 3
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Produced by A.J. Reynolds May 2008
