Plotting lines

Today we looked at plotting lines. To be able to plot a line, we need to have at least two points. If we know two points on the line, simply draw a line through these two points and continue it through.

There are a few different techniques to find two points. We need to look at the information we are given in our question, and then use this to find our two points.

Firstly, we may be told both our y and x intercepts (such as the questions in part 5 of our worksheet). If we know these, we already have our two points. Our x-intercept is the point where the line cuts through the x-axis, so we can mark this as a point. The y-intercept is the point where our line cuts through our y-axis, so we can also mark this point. Now that we have two points marked, we can draw a straight line through this and we have our solution.
For example: Sketch the line that has an x-intercept of 4 and a y intercept of -2

Firstly, we need to draw our Cartesian plane:

Next, we can go ahead and mark our x-intercept at the point 4 on the x-axis. 

Next, we can also mark our y-intercept at the point -2 on the y-axis. 

Now that we have our two points, simply use your ruler and draw a straight line through these two points, and we are done. 

Now sometimes we may have to do some more work to find our two points. For example, all we may be given is the equation of our line (like those in part 10 of the worksheet). If we have these, we do already know one of our points. Remembering that our equation for a line is y= mx + c where "m" is the gradient and "c" is the y-intercept. So if we have an equation given to us, we can easily just read off the value for "c" and use this as a point for our y-intercept, like we did above. 

The next step is a little more complicated. We need to find another point. We do this by giving our "x" in our equation a value, and then finding out what "y" is equal to at this x coordinate. I will usually use the point x=1, as this is often very easy to solve. 

I will go through an example, and hopefully this method will make a little more sense. 

Example: Sketch the line for the equation y = 4x - 1

Again, we first need to draw our Cartesian plane.

Next, we can simply read off our y-intercept from our equation. In this equation, our y-intercept ("c" value) is -1. We can now mark this as a point on our Cartesian plane. 

Now, we want to find another point. To do this, we will substitute in a value for x, and find out what y is equal to at this x-coordinate. For this example I will let x equal 1. 

So, substituting this value of 1 in wherever there is an x in our equation gives us:

y = 4 x 1 - 1 

y = 3

So this tells us that on this line, when x=1, y=3. So we now have another point (1, 3). We can now go ahead and mark this on our Cartesian plane. 

Now we have our two points, we can now go and draw our line through these. 

Done!

Now there is one more special case. If you see on your worksheet in question 6, we are asked to sketch graphs, but the equations for these graphs are in a slightly different style to what we are used to. They are written like x = # or y = # or something like this. 

If we look at, say, y = 4 for an example. What this is basically saying is that we want a line that at every single point y is equal to 4. This means we will have a straight line like this: 

Looking at this line we can see that at every single point, y is equal to 4. It doesn't matter what x is equal to, y is always equal to 4, which is what our equation asked. 

Similarly, if we get an equation of, say, x = -3, we want to draw a straight line so that at every single point, x is equal to -3. So it would look like this:

 Again, we can see that at every single point, x is equal to -3, which satisfies our equation. 

You should now have the skills to work through questions 5, 6 and 10 on the worksheet (which can be found in the "Worksheets" tab). 

As always, if you have any questions feel free to comment on this post, or email or chat to me in person.