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

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.

Work out the answer to each of these questions then click on the button marked 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

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

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

A Positive Negative B Positive Negative C Positive Negative D Positive Negative E Positive Negative F Positive Negative

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?

You need to upgrade your Flash Player Go to http://www.adobe.com/go/getflashplayer.