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
  1. Mark two points on the line, as far apart as possible
  2. Connect the two points with one horizontal line and one vertical line to form a triangle
  3. Measure the step - how far the line has gone to the right on the horizontal line
  4. Measure the rise - how far the line has gone up (or down) on the vertical line
  5. 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.
Work out the answer to each of these questions then click on the button marked Click on this button below to see the correct answer to see whether you are correct.

What is the gradient of line (a)?

What is the gradient of line (b)?

 

Plotting a graph from its equation

The equation of a line (e.g. y = x + 3) usually tells us how to find a y-coordinate using an x-coordinate.
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–2–10123
y = 2x + 1–3–11357

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 Click this button to see if you are correct 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 Click on this button to see the correct answer to see the answer.

Question 1
For each of the lines below, decide if the gradient is positive or negative:
A

B

C

D

E

F

Question 2
Calculate the gradient of each of the lines below, giving your answer as a decimal where necessary:
(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Question 3
Determine the gradient of each of the following lines:
(a)

(b)

(c)

(d)

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
(a) Complete the following table
x –2 –1    0   . 1 2 3
y = x – 2   –2

(b) Plot the six points on the graph and draw a line through them.
(b) When you have finished, click

(c) What is the gradient of the line?

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Question 5
A line has equation y = 2x – 2
(a) Complete the following table
x –1 0 1    2   . 3 4
y = 2x – 2    2

(b) Plot the six points on the graph and draw a line through them.
(b) When you have finished, click

(c) What is the gradient of the line?

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Question 6
A line has equation y = ½x + 2
(a) Complete the following table
x –2 0 2    4   . 6 8
y = ½x + 2    4

(b) Plot the six points on the graph and draw a line through them.
(b) When you have finished, click

(c) What is the gradient of the line?

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Question 7
Look back through the last three questions and see if you can see a link between the equation of each line and its gradient.
Look at the four equations of lines below and try and predict what the gradient would be (without actually plotting the graph).

(a) y = 2x + 4

(b) y = 3x – 9

(c) y = 10x + 1

(d) y = 5x + 3


You have now completed Unit 14 Section 3
Your overall score for this section is
Correct Answers
You answered questions correctly out of the questions in this section.
Incorrect Answers
There were questions where you used the Tell Me button.
There were questions with wrong answers.
There were questions you didn't attempt.
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Produced by A.J.Reynolds May 2008