Transformations of the 1/x Function

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  • 0:02 The 1/x function
  • 1:32 Vertical & Horizontal Shifts
  • 4:19 Slope Transformations
  • 6:18 Lesson Summary
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Lesson Transcript
Instructor: Elizabeth Foster

Elizabeth has been involved with tutoring since high school and has a B.A. in Classics.

In this lesson, you'll learn about the function f(x) = 1/x. You'll also learn about the different transformations that can be applied to the equation to change the graph of this function.

The 1/x Function

f(x) = 1/x looks like it ought to be a simple function, but its graph is a little bit complicated. It's really not as bad as it looks, though! Let's examine it more closely.

1 over x function
1overxfunction

If you follow the function's behavior from left to right, you can see that it's a decreasing function, a function where f(x) decreases as x increases. That makes sense because x is the number on the bottom of a fraction. The bigger the denominator, the smaller the fraction. With negative numbers, it works the same way.

Another important point about this function is that it has two asymptotes. An asymptote is a line that the function gets closer and closer to but never crosses. Can you spot the asymptotes on this graph? In this graph, the asymptotes are the x-axis and the y-axis. The curve of the function gets closer and closer and closer to the axes, but it never quite crosses them. Mathematically, you can see this if you look at the function:

  • x can never equal zero because then you'd be dividing by zero, which doesn't work.
  • y can never equal zero because you can't divide one by anything to get zero. The only way you can divide something by something else and get zero is if zero is on top.

You'll need to understand all of that to understand the topic of this lesson, which is all the different transformations of the function f(x) = 1/x. We can transform this function in all kinds of ways by adding or subtracting numbers to the equation in various places.

1 over x function shifts
1overxfunction

Vertical and Horizontal Shifts

We'll start with some fairly easy transformations that keep the graph looking exactly the same, but shift it up and down a little. If you do all the division and then add some number to the end, you'll move the graph up. If you subtract some number, you'll move the graph down.

Conceptually, this is because you're doing all the division work and then adding d to the y-value at the very end. So the division determines the shape of your graph, and d gives you a bigger y-value for any given x. For example, here everything is just shifted up by 5 units because for every value of x you get the same value you would have gotten for 1/x, plus 5 more.

1 over x function vertical shift of 5 units
1overxfunction

If you add some number to x on the bottom of the fraction, you'll move the function horizontally without changing its shape. Here's where it gets different: if you add c units, the function will move to the left by c units. If you subtract c units, the function will move to the right by c units.

How does this work conceptually? The bigger the bottom of a fraction is, the smaller the total value of the fraction. So, if you take some value x on the bottom of the fraction and add some value c to it, the resulting fraction will have a smaller total value than just plain 1/x.

On the other hand, if you subtract some value from x, the resulting fraction will be bigger. So, for any given value of x in our transformed fraction, adding something to it will give us a smaller value of y, and subtracting something to it will give us a bigger value of y.

Another way to look at it is to start with the y-values of 1/x. If you wanted to get those same y-values from 1/(x + 5), you would have to subtract 5 from every value of x. So, for any given y-value, the x-value that gets you there is moved 5 units to the negative side of the graph, which is left.

These two simple transformations up and down shift the asymptotes of the function. f(x) = 1/x + 5 has an asymptote at x = 5, not at x = 0. That's because we can now get a value of 0 out of this function. If we plug in -1/5 for x, we get f(x) = -5 + 5, which is equal to 0. But we can't get 5 because to get 5 1/x would have to equal 0, which is impossible.

Slope Transformations

Now, let's look at transformations that change the shape of the function, instead of just its location on the x and y-axes. We'll start with what happens when you multiply the top of the fraction by some number. This will flatten out the function.

Let's think about this conceptually. If a is greater than 1, then for any given value of x (1 * a)/x will be greater than 1/x. So, every value of x in the new function generates a bigger value of y than that same value of x did in the original function 1/x.

So far, that sounds like the same thing we did with adding a number to the function. But that's not all! You can see, from the comparison chart, that multiplying the function by a constant, causes it to behave differently than just adding a constant.

1 over x function comparison chart
1overxfunction

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