Athena is now helping out with her final project at the firm, designing a concert hall. She loves music so she's very excited. But first things first, she needs to learn a bit about how sound travels, so she can help think about the acoustics of the space. As a starting point, if you're listening to an orchestra or choir in this concert hall, would the music sound louder, sound softer, or sound the same, if we walked further away from the stage where the musicians are.
It would sound softer. You can test this with pretty much any sound.
We just established that the volume of a sound, or its loudness, gets lower or decreases as the distance from the sound source increases. I've drawn four graphs for you here and I'd like you to tell me which ones show loudness decreasing as distance increases. Please note loudness is on the vertical axis in all these graphs and distances on the horizontal axis.
Both of these middle graphs show loudness decreasing as distance increases. This one shows loudness as a constant. It doesn't change at all as distance changes. And this one shows loudness increasing as distance increases. So neither of those is right.
You narrowed down the relationship between loudness and distance to two different graphs in the last quiz. So I'm just going to go ahead and tell you that this one shows the actual relationship between these two quantities. Loudness varies as 1 over the distance from the source of the sound. Now, if you wrote these with more standard mathematical variables, x and y, you'd have the equation y equals 1 over x. We actually worked with this in the last lesson, and this is an example of a rational function.
Just to refresh our memories then, what is a rational function? Is it a polynomial function whose coefficients are all real numbers? A polynomial function whose coefficients are all rational numbers? A function that's one polynomial multiplied by another polynomial? Or is it a function that is one polynomial divided by another polynomial? Please pick what you think is the best answer to describe all rational functions.
A rational function is a function that's one polynomial divided by another polynomial. Rational functions are like rational numbers, except, instead of an integer divided by an integer, such as 3/4's, it's a polynomial divided by another polynomial. We already saw a simple example earlier, in y equals 1 over x. But they can get much more complicated, and therefore, much more interesting. You might even have something like this. Y equals negative 3x cubed plus 2x squared minus 7, all divided by X to the 4th minus X. Note that actually all of the other choices that I gave you here, also technically are rational functions. They just all have denominators of 1. Remember that constants are polynomials, too. Most of the rational functions we'll see though, look more like these 2.
The family of rational functions has a ton of variety in it, since there are so many different polynomials out there. But there's one restriction that all of them are subject to. There's a certain value that the denominator is not allowed to equal. What is it.
The denominator of a rational function cannot be equal to 0. Because then we would be, well, dividing by 0, and we know that's a major problem.
For each of these 2 rational functions then, what is x not allowed to be equal? Remember, we want to avoid having a denominator equals 0.
For each equation we can just set the denominator equal to 0 and then solve for x. When we do that, we find that for a, x cannot be 0, and for b, x cannot be
What's happening in these two functions is something totally new for us. For each one, we get a value of x that's not allowed right in the middle of the domain. What does this actually mean? Well, here's the graph of one of the two functions we were just working with f of x equals 1 over 3x. What do you notice about the behavior of the graph just before x equals 0, so to the left of the y axis and just after x equals 0, to the right of the y axis?
Looking to the left of the graph, just before we reach 0, it looks like we fall way below negative infinity, and just after x equals 0, the graph reappears way up high, and then comes back down for positive infinity. So, as we approach x equals 0 from the left the graph falls low, but as we approach from the right, it rises way high.
In the previous lesson we mentioned the term asymptote. And in this past quiz you saw that the graph of f of x equals 1 over 3x never actually reaches x equals 0. The y axis. Now if we look at this line on the graph since its already drawn for us. We can see that it splits the graph. So that the curve falls down toward negative infinity on one side. And comes back down from positive infinity on the other side. As I mentioned before, this line, x equals function we were looking at before. F of x equals 4 over X minus 1. Based on this graph, what do you think the vertical asymptote for this function is? Make sure that what you write is the equation for a vertical line. Since that's what a vertical asymptote is.
The vertical asymptote is the line x equals 1. This is the value of x that makes the denominator of a rational function eguals 0. X can equal one for f of x equals four over x minus 1 because well, there's a vertical asymptote right there. Its all starting to make sense and come together. If I draw in the line, x equals 1, we can see that yes in fact, it does not look But the curve of the rational function is going to touch this line at all. It's just going down to negative infinity here, and coming back down from positive infinity over here. Never touching this line.
You've looked at the behavior of f of x equals 4 over x minus 1, around x equals 1, right in the middle of the graph. But what's the behavior doing far, far to the left, and far, far to the right? It doesn't look like any other curve we've seen before in the course. Which of these statements right here probably describes the behavior of this function, as we move far, far away from the y axis? Does the curve approach the line y equals o, the x axis, from above but never touch it, as x goes either to positive infinity, or negative infinity? Does the curve approach y equals 0 from above but not touch it, as x goes to infinity. And does it approach y equals 0 from below as x goes to negative infinity? Does the curve approach Y equals 0 from below, as x goes to positive infinity, and from above as x goes to negative infinity? Or does the curve approach Y equals 0 from below, as x goes to positive infinity and to negative infinity?
The second choice is the correct one. As x gets bigger and bigger, going toward positive infinity, the graph is getting closer and closer to the line y equals zero from above the X axis. And as x gets closer and closer to negative infinity, over here on the left side of the graph. The curve of a rational function is also getting closer to y equals zero but from the bottom this time. In either case however, the curve never touches the X axis.
This line y equals, that our curve heads towards, but never quite reaches as x goes to positive infinity and to negative infinity is called a horizontal asymptote. Let's look at another rational function and see if we can find its horizontal asymptote. In fact, why don't we return to that other function we were working with before. f of x equals 1 over 3x. What is its horizontal asymptote? Make sure that the answer you give is the equation of a horizontal line.
The horizontal asymptote for this graph is also y equals 0. We can see that the curve is never going to touch the x axis on either side of the graph.
For both examples we've looked at with Horizontal Asymptotes so far, the Horizontal Asymptote was at y equals 0. Do you think this is the case for all rational functions?
The answer is no. If you think back to other curves you've looked at, like parabolas for example. We saw that we could shift the curves up and down, and we can do the same thing with rational functions, even with the graphs you've already worked with. In fact, here's what the graph of 4 over x minus 1 would look like If we shifted it up 1, we can see that now the horizontal asymptote is at y equals 1, instead of at y equals 0.
The equation of the curve that I just showed you in the last solution is this, f of x equals 4 over x minus 1 plus 1. How can we write this equation in the normal form for the equation of a rational function as one polynomial divided by another polynomial. So, you want just a single fraction here, where both the numerator and the denominator are polynomials. Please fill this in and make sure that the common factor between the numerator and the denominator is 1 in this case.
When we transformed f of x equals 4 over x minus 1 plus 1 into a single fraction, by converting this 1 to have the same denominator as the first term, we end up with, x plus 3 over x minus 1. So, you can see that yes, in fact, this is a rational function, since both of these are polynomials.
This last example actually gives us a clue as to how to find the horizontal asymptote of a rational function. We started off with f of x equals 4 over x minus 1 plus 1. And as x gets bigger, this fraction is going to get smaller and smaller. Eventually we're going to end up with f of x equals 1 plus something that's almost zero. This means, as we saw from the graph, that our horizontal asymptote will be at y equals 1. So how can we reverse the process that we went through of combining two fractions? In other words, if we started out with x plus 3 over x minus 1, how can we get back to 1 plus 4 over X minus 1? Should we divide the numerator by the denominator? Should we divide the denominator by the numerator? Should we subtract the denominator from numerator? Or should we subtract the numerator from the denominator?
We should divide the numerator by the denominator. If you divide x plus 3 by x minus 1, we end up with 1 with a remainder of 4, which is exactly what we see in the answer here. Looks like we've got a method.
Let's try this method of dividing the numerator by the denominator to find the horizontal asymptote with this new function. F of x equals 3x minus 8 over x minus 5. What do you get if you divide 3x minus 8 by x minus 5? Put your answer in this box.
I like to use long division to do problems like these. And if we do, we end up with an answer of 3 plus 7 over x minus 5, since we had a remainder of 7.
Considering the way that we just rewrote 3x minus 8 over x minus 5 as 3 plus 7 over x minus 5. What is the horizontal asymptote of that rational function we are working with?
The horizontal asymptote is at y equals 3. We know this because as x gets very, very big, this fraction is going to go to 0. And that means that the function is going to approach the value 3.
Let's check that there really is a horizontal asymptote at y equals 3, by using this x f of x table. Use a calculator to find the value of f of x for each of these values of x. Rand your answer correctly to five decimal places.
And our values of f of x, are 4.4, 3.07368, 3.00704, 3.0007 and 3.00007. We can see that as x gets bigger and bigger, f of x is in fact getting closer and closer to three.
We're about to take a look at the graph of f of x equals 3x minus 8 over x minus 5. But just before we do, let's find it's vertical asymptote. Where do you think it is? Remember, we just need to check what value of x is going to make the denominator equal to 0. Write in the equation of the asymptote right here.
The vertical asymptotes is the line x equals 5, since if we put 5 in for x, we get 5 minus 5, or 0 in the denominator.
The vertical and horizontal asymptotes of this rational function divide the coordinate plane into four different sections. Which of these four sections does the graph lie in? Check off the proper boxes to show approximately the points that the graph is going to go through.
The graph belongs in this bottom left section and in this top right section. It goes through all 3 points that are marked in each.
Let's try another rational function. We'll look for both the horizontal and the vertical asymptotes. But let's just start out with the vertical. The function I'd like us to look at is f of x equals 8x squared minus 1 over 4x squared minus 9. Please write the vertical asymptotes in this box, separating the equations by commas.
The vertical asymptotes of this rational function are x equals 3/2, and x equals negative 3/2. Here's how we find them. As before, we're looking for the values of x that will make the denominator equal to 0. So we set 4x squared minus 9 equal to 0, and then solve for x. When we take the square root of both sides, you'd get just X on the left-hand side, and we'd get plus or minus the square root of 9/4's on the right-hand side, which is just equal to plus or minus 3 over 2. So this time we have 2 vertical asymptotes.
The next thing I want us to find, is the horizontal asymptote of this function. Remember that the way we found this before, was we divided the numerator by the denominator. So, why don't we take a step back and just do that first? So let's start by just doing that division. What is 8 x squared minus 1 divided by 4 x squared minus 9?
The answer to our division problem is 2 plus 17 over 4x squared minus 9.
Now let's find that horizontal asymptote. Remember what we found as the answer to our division problem in the last quiz? What that gives us is another way to write the rational function we have here. Think about what happens as x gets really, really big to the value of the function. That should tell you the horizontal asymptote. Remember that this should be the equation of some horizontal line.
Our horizontal asymptote is y equals 2. We can see, from the other way to write this rational function, 2 plus 17 over 4x squared minus 9. As x gets really big, the denominator, here, will get very, very large. Which means, this fraction as a whole, will get to be very tiny. That means, that the value of the function, is going to approach 2, as x gets really big.
Now I'd like you to find the locations of the x and y-intercept of the same rational function. Actually, if you could just give me the x-coordinate of each x-intercept and the y-coordinate of each y-intercept, I would be very happy. Just separate your answers by commas if there's more then one in a given box. And if there aren't any, write none.
To find the x-coordinates of the x-intercepts of this rational function, we need to set the numerator equal to 0. Since if that equals 0, then the whole fraction equals 0, which means that f of x equals 0. And that's exactly what's happening at an x-intercept. When we solve for x, we get x equals plus or minus plus or minus 0.354 point. Now for our y-intercepts, all we need to do is plug in zero for x. This gives us a y value of 1 9th. So, there we go. We have two x-intercepts and one y-intercept.
What other information can we learn about this function? Well how about this? Is it even, is it odd, or is it neither one of those? Remember that for an even function, f of x will be equal to f of negative x. And for an odd function, f of x is equal to negative f of negative x. So, you can try plugging in different positive and negative values and see if it fits into either of these two categories.
This function is even. Let's check this out using the numbers 1 and negative 1 for values of x. When we plug in either 1 or a negative 1 into f of x, we get negative 7 over 5 in both cases. Now, ideally, we check this with more than one set of numbers. And if you don't trust me, you can totally go back and do that. But I declare this function, even.
Once again, I've drawn the vertical, and the horizontal asymptotes, for this rational function, on our coordinate plane. And we can see, that this time, the graph is divided into six different sections, by these asymptotes. In which sections of the plane does the graph reside? Think about where its x and y-intercepts are, and also, try plugging in points, from these different areas, and see which ones work for this equation.
These are the boxes that should have been checked off. We can find curves of the graph in three different areas, over here, down here, and up here again. Let me show you the full version of the graph so you can get a real picture of what it looks like. These black curves represent the graph of this function. It's pretty incredible to think about how different this looks from any other rational function we've look at so far. This is just testament to the incredible wide variety of graphs and functions we have in the ratinoal function family.
In this lesson we looked at Rational Functions that have horizontal and vertical asymptotes and we also talked about how to find them. Get ready for the next lesson, where we'll learn how to find the domain and the range for rational functions. Well also talk about one more kind of asymptote. So get excited.