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Syllabus Edition
First teaching 2021
Last exams 2024
Straight Line Graphs (y = mx + c) (CIE IGCSE Maths: Core)
Revision Note
Finding Equations of Straight Lines
Why do we want to know about straight lines and their equations?
- Straight Line Graphs (Linear Graphs) have lots of uses in mathematics – one use is in navigation
- We may want to know the equation of a straight line so we can program it into a computer that will plot the line on a screen, along with several others, to make shapes and graphics
How do we find the equation of a straight line?
- The general equation of a straight line is where;
- is the gradient,
- is the -axis intercept (or simply, the -intercept)
- To find the equation of a straight line you need TWO things:
- the gradient, , which you can put straight into
- get this from the question directly, or from a diagram
- any point on the line- substitute this point into (as you already know ) and solve to find
- if given two points which you used to find the gradient, just choose either one of them for the point to find
- the gradient, , which you can put straight into
- You may be asked to give the equation in the form
(especially if is a fraction)
If in doubt, SKETCH IT!
What if the line is not in the form y=mx+c?
- A line could be given in the form
- It is harder to identify the gradient and intercept in this form
- We can rearrange the equation into , so it is easier to identify the gradient and intercept
- This is easiest done when the values of and are known
- e.g. The equation can be rearranged to
- So the gradient is and the line intercepts the y-axis at the point
- e.g. The equation can be rearranged to
Worked example
Find the equation of the straight line with gradient 3 that passes through (5, 4).
We know that the gradient is 3 so the line takes the form
To find the value of c, substitute (5, 4) into the equation
Replace c with −11 to complete the equation of the line
y = 3x − 11
First find m, the gradient
Identify two points the line passes through and work out the rise and run
Line passes through (2, 4) and (10, 0)
∴ rise = 4 (4 - 0)
run = 8 (10 - 2)
!! Remember we also need to consider whether the line is "uphill" or "downhill" to find the gradient !!
The slope is downward in the left to right sense, so it is a negative gradient
∴ gradient,
To find the value of c, substitute one of the points identified earlier into this equation
Here we've used (10, 0) as 0's often make the maths easier
Replace c with 5 to complete the equation of the line
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Drawing Linear Graphs
How do we draw the graph of a straight line from an equation?
- Before you start trying to draw a straight line, make sure you understand how to find the equation of a straight line – that will help you understand this
- How we draw a straight line depends on what form the equation is given in
- There are two main forms you might see:
y = mx + c and ax + by + c = 0
- Different ways of drawing the graph of a straight line:
- From the form y = mx + c
(you might be able to rearrange to this form easily)
go 1 across, m up (and repeat until you can draw the line)
plot c on the y-axis - From ax + by + c = 0
put x = 0 to find y-axis intercept
put y = 0 to find x-axis intercept
(You may prefer to rearrange to y = mx + c and use above method)
- Work out and plot two points on the line
Pick a value of x, substitute this into the equation of the line (whichever form it is in) to find the corresponding y coordinate
Repeat this for a different value of x
Plot the two points and join them up
It is often a good idea to pick and plot a third point, as a check
All three points will be on the line if they're correct - if not, at least one of the points is incorrect!
How do we draw the graph of a straight line from a table of values?
- Some questions will not give you a diagram, nor tell you a graph is a straight line (linear) graph
- Instead they will give you a table of x and y values to complete using the equation of the straight line
- The line is then drawn using the values in the table as coordinates
- If you recognise that the equation (or pattern in the values) will give a straight line graph then you will know if any points are incorrect when you plot them
- For example, complete the table of value below for the equation
-3 -2 -1 0 1 2 3 2 4
Substitute each x value in turn to find the corresponding y valueand so on for the other missing y values
Then you would plot each coordinate and join them up to draw the straight line graph.
If any coordinates do not lie on the line you should go back and check your calculations for that one.
Examiner Tip
- If plotting a straight line from a table of values then there is no need to plot them all
- You'll need to plot at least 2 points
- We recommend you plot a third too as a way of checking the first two
- To complete a table of values, you can use the TABLE feature/mode of your calculator, if it has one
- Most modern calculators have this feature but it may be hidden under on screen menus rather than having a dedicated key
Worked example
On the axes below, draw the graphs of and .
For , first plot c, which is (0, −1)
Then, as m = 3, . So plot a point 3 up and 1 right. Repeat at least once more and then join the points with a straight line. Extend the line to the edges of the grid.
The steps for are the same, but first we need to rearrange into the form y = mx + c
Now we can plot c, which is (0, 3)
For the gradient, . So plot a point 3 down and 5 right. There isn't space on the grid to repeat this so join the points with a straight line. Extend the line to the edges of the grid.
Parallel Lines
What are parallel lines?
- Parallel lines are lines that have the same gradient, but are not the same line
- Parallel lines do not intersect with each other
- You can easily spot that two lines are parallel when they are written in the form , as they will have the same value of (gradient)
- and are parallel
- and are not parallel
- and are not parallel; they are the exact same line
How do I find the equation of a line parallel to another line?
- As parallel lines have the same gradient, a line of the form will be parallel to a line in the form , where is the same for both lines
- If then they would be the same line and therefore not parallel
- If you are asked to find the equation of a line parallel to , you will also be given some information about a point that the parallel line, passes through;
- You can then substitute this point into and solve to find
Worked example
Find the equation of the line that is parallel to and passes through (2,1)
As the gradient is the same, the line that is parallel will be in the form:
Substitute in the coordinate that the line passes through:
Simplify:
Subtract 6 from both sides:
Final answer:
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