Linear Graphs
I will do algebra, I will do trigonometry, I will even do statistics, but graphing is where I draw the line.
This lesson/tutorial is all about how to plot a linear line graph. When graphing, the use of the term linear is slightly different to that of which we use with sequences; linear means that the numbers are arranged in or extending along a straight or nearly straight line suggesting that the graph is a straight line graph. It is important that you remember the difference between linear and non-linear graphs.
There are many ways in which we can sketch and plot linear graphs and it is actually a lot easier than what it first looks like. The universal linear equation to solving graphs is usually portrayed as y=mx+c, where y is the y coordinate, m is the gradient, x is the x coordinate and c is the y intercept.
The gradient is the amount in which the values increase or decrease by and can be determined by the equation:
The values of two coordinates along the same line can be substituted into this equation as such: (X1, Y1), (X2,Y2)
The y-intercept is value in which the line crosses the y axis and is represented by c; so if I had and equation such as y=2x+4, the y intercept is 4.
For example take a look at the graph shown to the left. This represent y=x.
In the terms of y=mx+c, the gradient is one (but because we don't write 1x we simplify it so that it is just x) and we can assume that the y intercept is 0 as there is no c in the equation. If you take a look at the graph
So y =1x + 0 is the same as y = x in the form of y=mx+c, again just take another look at the graph on the left to see how this works.
If we were to then take the equation y=x , from above, and change it to become y=x+2, the graph should look like this (graph on the right). Do you notice the differences?
The gradient of the graph remains the same because it's still 1x (or x), but c has now become 2 instead of zero meaning that, if you look at the graph, you can see that the line now crosses the y axis at 2 instead of 0.
Again, taking the equation from above (y = mx+c), we can change it to y=3x+2. It looks different from both of the ones above, right?
This is because now we have kept the y intercept as 2, so it still crosses the y axis at 2, but we have changed the gradient so that as it goes across one value on the x axis it goes up three values on the y axis.
Now we can use this knowledge to sketch linear graphs; if we took the equation from above, by just looking at the equation itself, we can say that when x is 0, y must be 2 because that's the y intercept. We can then substitute x with 1 so:
y=(3x1)+2
y=5
We can then put this into a table:
Notice how the numbers go up in threes; this is due to the gradient being 3. If the gradient was -3 it would be going down in threes and if the gradient was 1/3 the values would go up by a third.
Note:
Linear graphs can come up in many different ways in the exam; it may ask you to fill the values in a table like above. This is easily sorted by substituting the x coordinate into the equation to find y.
They can also come up as rearranging the formula to find what x is (so (y-c)/m).
Thirdly, it may ask to prove that a certain coordinate is on a line with a given equation. To solve this, substitute the x coordinate into the equation.
It could also ask you to find the equation of a line given on a graph. To find this you work out the gradient using the equation from the top of the page and then find c using the graph to find out where it intercepts the y axis.
Questions may ask you to find the value of c and to do this you must rearrange the equation and substitute y and x into it, eventually working out what c is.
EXAMPLE 1
A line has the equation of y=1/2x +5. Fill in the table below and then plot and draw what the graph should look like.
As the equation of the line does not contain a power (indices) it means that the line is linear.
EXAMPLE 2
A line has the equation of y=4x-3. If y is five what is the x coordinate?
x=? y=5 m=4 c=-3
y=mx+c -> x=(y-c)/m
x=(5+3)/4
x=2
EXAMPLE 3
Prove that the coordinates (3, 12) are on the line y=1/3x +11.
y=(1/3x3)+11
y=1+11
y=12
so when x is 3, y is 12.
EXAMPLE 4
What is the equation of the line from this graph?
It crosses the y axis at the value 4, so c = 4.
Then we find the gradient:
(Y2-Y1)/(X2-X1) => (8-4)/(2-0) => 4/2 => 2
So y = 2x + 4
EXAMPLE 5
If the coordinate (2,5) rests on a line with a gradient of one, work out the y intercept of the line.
y=mx+c => 5=1x2+c
c=5-2
c=3
Parallel lines
If two lines are parallel they never meet so, if two linear lines are parallel they have the same gradient. For example, y=2x+7 is parallel to y=2x-5. Despite the y intercept being different the gradient is the same meaning that they are parallel. y=3x-2 and y=-3x-2 are not parallel.
Perpendicular Lines
If two lines are perpendicular it means that they meet at right angles. With linear lines, if they are perpendicular, the two gradients can multiply together to get -1. Or an easier way to calculate the gradient of a perpendicular line is to find the reciprocal of the gradient and then change the sign (so if it is positive it becomes negative). For example y=2x+4 and y=-1/2x+3 are perpendicular.
Perpendicular gradients:
-3 =>1/3
5 => -1/5
1/2 => -2
-1/4 => ?
-4 => ?
