in this video we'll look at reciprocal graphs a reciprocal graph is a graph in the form y equals k over x where k is a constant such as five seven minus two uh we're going to draw the graph y equals 1 over x to begin with so i've created a little xy table and i've chosen some values for x so let's work that out so 1 divided by x so 1 divided by 10 is not 0.1 1 divided by 5 is not 0.2 1 divided by 2 is a half 1 divided by 1 is 1 1 divided by 0.5 is 2 and 1 divided by 0.1 is 10 and 1 divided by zero you can't do one divided by zero divided by zero is undefined so you get then um on your calculator math error um here we don't have a value so uh let's unplot those points so 10 across 0.1 up 5 across 0.2 up and so on you should end up with something that looks like this takes that ship so what you'll notice whenever you plot it is that it approaches the x-axis now remember we'll never actually reach the x-axis because one divided by 101 divided by thousand one divided by a million will always give you a no point sum of an answer okay so we'll never actually reach zero it'll get very close we'll always approach it um and whenever you divide by decimal numbers like not 0.1 0.01 and so on it'll get very very large and so again it'll approach the y-axis so it looks something like that let's now draw do the other side so 1 divided by 0.1 would be minus 10. 1 divided by 0.5 would be equal to -2 1 divided by minus 1 is minus 1 1 divided by minus 2 is minus 0.5 1 divided by minus 5 is minus 0.2 and 1 divided by minus 10 is minus 0.1 again whenever you plot those you're going to get something that looks like this okay so it looks like this you'll have these two lines the y-axis and the x-axis it approaches those but it never reaches them that's called an asymptote so these are called asymptotes the y-axis remember the equation for the y-axis is x equals zero and also the x-axis is an asymptote and that's equation is y equals zero so they're called asymptotes those lines that the graph approaches but never reaches okay um so that's what the graph 1 over x would look like if you were to do 2 divided by x well if you do 2 divided by 0.1 you get 20 2 divided by 0.2 you get 4 2 divided by 1 is 2. so what happens is your answers would be double so the graph will just move further out so here's a typical question that might look something like that where you've been asked to sketch 2 over x and 4 over x on the same diagram so what you would do is you just sketch your two over x these are very bad diagrams so that's a two over x well don't do that and you'd also draw your four over x on the same diagram and you would just try to show that four over x is further out than the actually i've got something gonna do that but again uh you just sort of show that your four over x is further out than your two over x okay so we've drawn a graph in the form of y equals one over x or y equals k over x where k is a positive number so i've done 1 divided by x 1 1 divided by 10 1 divided by 5 1 divided by 2. well what would happen if it was y equals minus 1 over x well let's have a look at an xy table for that so x y and let's just start off with the positive side so zero one or zero not point one not point five two ten one two ten okay well minus one divided by zero again is undefined so that y axis will still be an asymptote minus one divided by not point one would be minus ten so that would be down here somewhere minus 1 divided by 0.5 would be minus 2 so it would be there somewhere minus 1 divided by 1 is minus 1 minus 1 divided by 2 is minus a half and minus 1 divided by 10 would be minus 0.1 notice it is it looks the same but it's in the other quadrant so it looks something like this okay likewise doing the negatives if you had your table and you had minus numbers remember negative divided by negative is a positive so what would happen is it would look something like this so if you've got y equals k over x and you've got a positive number so just say for instance y equals 1 over x so y equals 10 over x the graph will look like this where it's in this quadrant and it comes down and approaches the x-axis and likewise it's in this quadrant whenever x is negative and it'll approach the x-axis whenever it's negative something like y equals minus 2 over x or y equals minus 1 over x then it would look like this would be in the overall quadrants because remember a negative divided by positive is a negative and the negative divided by a negative is a positive okay so to recap a reciprocal graph is something in the form k over x the graph whenever it's positive will look like this the x-axis and the y-axis are asymptotes because it will never actually have a value of those particular points and if the number is negative on the top line then it'll be in the other quadrants so it'll look like that and finally if the number is larger on the top line it'll be further out because whatever you do