In today's video, we're looking at reciprocal graphs, which are ones where y is equal to something over x. And here, we're representing that something with the letter a. So the a could be any value that you can think of.
The most common one you'll see though is y equals 1 over x. So we're going to take a closer look at this one. To draw it, you need to draw a curve like this in the top right, and then a similar curve in the bottom left. So there are actually two separate parts to it, even though it's just one graph.
As well as being able to recognize the graph, you also need to know that it's symmetrical in this line of y equals x, because this part is symmetrical to this part, and it's also symmetrical in this line of y equals minus x, because this part is symmetrical to this part. If you're ever given a graph but not quite sure if it really is y equals 1 over x, remember that you can just make a table of x and y values and check it for yourself. So in this table, I've picked the x values minus 2, 0, and 2, but you could pick any values that appear on your graph.
and then we can plug each of these x values into the equation to find out what the corresponding y values should be. For example, to find out what y is when x is minus 2, we just plug minus 2 into our y equals 1 over x equation, so it becomes y equals 1 over minus 2, or just y equals minus 1.5. So we know that when x equals minus 2 on our graph, y should equal minus a half, which it does right here. Finding the y value for when x equals 0 is actually a bit trickier, because if we plug in x as 0, we find that y will be equal to 1 over 0, which is basically infinity, so there's no way to plot it on a graph. This is why if we look at where x equals 0 on our graph, which is basically the y-axis, there's no corresponding y-value for our line.
Or in other words, our curve doesn't cross the y-axis, because theoretically it would only cross the axis at infinity, which would never be on a graph. And finally, if we did the same thing for when x equals positive 2, we'd find that y equals positive a half. And looking at our graph, we can see that that's right here, and our curve does pass through that point.
So because the curve that we were given fits with what we expect from our table of values, we can be fairly sure that it's the y equals 1 over x curve. Now that we've seen what the simple 1 over x graph looks like, we need to cover what happens when this number on top is something other than 1. And there are really three possibilities you need to know about. Namely, when the value on top is bigger than 1, when the value on top is between 0 and 1, and when the value on top is a negative number. To see how each of these things changes the graph, let's faintly draw the basic y equals 1 over x graph on each of the grids, so that we can compare them to our new lines. When the number on top is bigger than 1, like in y equals 4 over x, the graph basically moves outwards, away from the axes.
And the bigger the value on top, the further it will move. Meanwhile, if the value on top is between 0 and 1, like maybe 0.5, then the graph does the opposite, and moves inwards towards the axes. And finally, if the value on top is negative, like with minus 1 over x, then it will swap quadrants, so that it's in the top left and bottom right instead. Anyways, that's everything you need to know about reciprocal graphs.
So I hope that was helpful, and we'll see you again soon!