Axis Intercepts

An intercept is the place on an axis where a line crosses over it.  If you think about it, any infinitely long straight line is going to have to cross the x-axis and the y-axis somewhere.  The only exceptions to this are vertical or horizontal lines.  But even those must cross one of the axes.  A vertical line must cross the x-axis somewhere, and a horizontal line must cross the y-axis somewhere.

Working out intercepts from the relation

A typical question in an exam might give you a relation, and then ask you to work out whether the gradient is positive or negative, and find the x-axis and y-axis intercepts.  Now, sometimes they’ll ask you to draw a graph of the relation, but if not, then there are quicker ways to work out the answers.

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Say you get given a relation like .  Think about going from left to right on a graph.  Will the value of y increase or decrease?

Well, as we go from left to right, what happens to ‘x’?  It increases in value.  Now we get our value of ‘y’ by multiplying ‘x’ by three, and then adding two.  So if x gets larger moving from left to right, then y as well is going to get larger. 

To be really sure, try a couple of values – let’s do x = 0 and then x = 2.

                                                       For x = 0, y = 2.

                                                       For x = 2, y = 8.

So as x got bigger, which corresponds to moving from left to right on the graph, y also got bigger.  This means that the gradient is positive – we’re going uphill.

What about if it had been ?  Well, as our x values get larger when we move from left to right, what happens to the value of y?  It’s important to notice the ‘–3’ in front of the ‘x’.  This ‘–3’ multiplies x by three and makes it negative.  So as we move left to right and our x values get larger, our value of y is actually going to get smaller.  Once again, we can test by doing a couple of points:

                                                       For x = 0, y = 2

                                                      For x = 2, y = –4

So as our value of x gets larger, our value of y gets smaller.  So for this relation, the gradient is negative.

To work out whether the gradient is positive or negative for a linear relation, you first need to makes sure your relation is in the form, “y = some expression involving x.”  Then, just look at the coefficient of x (the number just in front of the x, multiplying it).  If it’s positive then your gradient is positive.  If it’s negative then your gradient is negative.

Zero and undefined gradients

Only time this won’t work is with horizontal or vertical lines.  If you get a relation like this:


then you’ve got a horizontal line, which has a zero gradient.

If you get something like this:


Then you get a vertical line, which has an undefined gradient.