## Inverse Relationships In Graphs

Two things to check here:

·         In the area of the graph to the right of the y-axis, does one variable increase as the other variable decreases?  The area of the graph to the left of the y-axis is a bit tricky – for , as x decreases (goes more negative), y gets larger (becomes less negative).  However, for , as x decreases, y decreases too.

·         Is the graph undefined at x = 0 (where the y-axis usually is)? This should happen regardless of what type of inverse relationship it is, because there’s a divide by ‘x’ or divide by ‘x2’ or something similar. Since x is zero, dividing by ‘0’ gets you an undefined number.  The value of ‘y’ near x = 0 should be either a very large positive number or a very large negative number.

Here’s a heap of graphs that aren’t inverse relationships:

 Not inverse relationships

These graphs are inverse relationships:

This graph doesn’t pass through the origin, which is OK.  To the right of the y-axis, as x increases, y decreases, which is OK.  To the left of the x-axis as x decreases, y increases.  It also appears that the value of y gets very large and positive as you approach x = 0 from the right side of the y-axis.  It also seems to get very large and negative when you approach from the left side of the y-axis.  Because the graph is mirror imaged both across the x-axis and y-axis, it’s probably of the form , or , or  – the power of x in the denominator will be an odd number.

This graph doesn’t pass through the origin which is OK.  To the right of the y-axis, as x increases y decreases, which is OK.  To the left of the y-axis, as x decreases, y decreases too.  This is OK as well – it just means that the inverse relationship is going to be like ,  – the power of ‘x’ in the denominator will be an even number (remember that a negative x raised to an even power will give a positive answer, which makes sense since all the values of y to the left of the y-axis are positive).