Grant has recruited Athena, with all of her experience designing things, to create a new logo for the Gleaming Glasses company. It's going to have a pair of glasses on it, of course, and Athena wants to make sure that the glasses are exactly symmetric. Now, she really likes the look of the function f of x equals x to the 4th, what I have graphed right here. So, she's trying to figure out how to use it to make the glasses. We need to figure out how to help Athena transform this function. Flip it over, move it from side to side, like this, and also move it up and move it down. All these different kinds of transformations are definitely going to be required for the design of this logo. Let's help her out.
Let's start thinking back to the different ways that we transformed parabolas before. And then maybe we'll be able to generalize this to work with other functions. As we've seen so many times before, here's the graph of f of x equals x squared. What would happen to our graph though, if we changed the equation? For example, let's say I want to change it to f of x equals x squared minus 2. What direction would the graph shift in? Up, down, left, or right, and by how many units?
The answer is that the graph would shift down by 2 units. We can see that that's happened in the graph over here. This was the original function, f of x equals x squared. And this is the transformed one, f of x equals x squared minus 2. So, subtracting 2 shifts us down 2. Interesting. You probably remember this from the section on the section on [UNKNOWN] earlier on in the course but it never hurts to do some review.
In the last quiz, we transformed the function f of x equals x squared, into the function f of x equals x squared minus 2. And we saw that the way this affects our graph is, it moves it down by 2 units. We called this a vertical shift downward. But remember, we're trying to help Athena with her function. And I'm not sure yet if we're going to need to move things vertically. We do know, however, that she's going to want to move her function horizontally. So why don't we try that for a second. So let's say that, once again, we start out with the function, f of x equals x squared. This time however, I know that I want to get a graph that's exactly the same as this one, but shifted over to the right by 3 units. What equation do I need for this new graph? Please write it right here.
To shift our graph to the right by 3, we need to replace x in the original equation for the original graph, with x minus 3. So we end up with f of x equals the quantity x minus 3, squared. Now you also could of course write y equals the quantity x minus 3 squared.
Now that we've seen a couple of different transformations, let's look at several at once. Each of these equations represents some sort of transformation of our original graph, f of x equals x squared 2. What I would like you to do is match the letter next to each of these equations over here with the way that it transforms this function. Then, if there's any missing information like the number of units the graph has shifted by, please fill that in as well. One thing to note is this last choice down here, reflected in the x axis. What this means is that if we have our standard x,y axis here, we have some curve drawn like this, maybe a sort of parabola. If it's reflected in the x axis, that means it's been flipped across the X axis or flipped over the x axis, like this. So, this graph below would be a reflection in the x axis of the one above. Now it's up to figure out what equation makes that happen. Good luck.
Here are our answers. Equation e, would shift our graph up by 6 units, that's why we have a plus 6 outside the original function. And if you want to shift down instead, you need to subtract. In this case, we're subtracting by 1 with equation d. Equation c, which of the graph left by five units since we're adding to x. And graph a would do essentially the opposite, it would shift the graph right, this time, by four units. Finally, graph b here, f of x equals negative x squared, represents that reflection in the x-axis of the original graph. This negative sign out in front of the original function takes all of the function values of the original graph and flips them across the x-axis, makes them negative.
Now, we know how easy it is to transform a function in a lot of different ways. We can make a graph move to the left, we can make it move to the right. We can make it move up, and we can make it move down, too. We can even flip it upside down, amazing. Now, we know that this kind of flipping is called a reflection in the x-axis since the x-axis is the line that we're flipping the graph across. However, we have this other axis, too, the y-axis. What do you think a reflection in the y-axis would look like? And how do we make that happen in our equation for a function? The function that we were just looking at and that I was just transforming is this black one right here, f of x equals the square root of x. Now, graphs A and B here are graphs of different functions, functions that are different transformations of this original function. Which one represents this function reflected in the x-axis? And which one, for instance this function, reflected in the y-axis? Please fill in A or B in each of these boxes.
Since graph A is exactly the same as this function except flipped across the x-axis, it represents a reflection of f of x in the x-axis. Graph B is identical to f of x, except flipped across the y-axis. So, it represents a reflection in the y-axis. Cool. We have all sorts of mirror images all over the place.
But how in general do reflections like the ones we were just looking at show up in equations? Well, as a simple system to work with, let's just look at these three points for now. You have the point 1,1, the point 4,2, and the point 9,3. If we take these three points and reflect them in the x-axis, what new points will we end up with? Please write the coordinates of each point in their reflection that corresponds to each original point.
Remembering that these three points that lie above the X axis, were the points we started out with. These three below the X axis are the ones we and up with, after reflection in the x-axis. Their coordinates are (1,-1), (4,-2) and (9,-3).
What pattern do you notice between the original points and those that are reflected in the x-axis? How do each of the coordinates change? First start out with a general point. Let's call it x comma y. What point will I end up with after reflection in the x-axis? Fill in the new coordinates right here.
As we can see from the three points we looked at first, the x-coordinate of our point is not going to change. It's still going to just be x. The y-coordinate however, needs to switch signs, since we have positive y here, we need negative y here.
We just saw that in order to reflect points in the x axis, we need to switch any y coordinates to negative y. And in fact, this is exactly what we do with functions as well. So for example, if we start out with the function we were looking at before, y equals the root of x. Then in order to reflect this in the x axis, we need to switch y to be negative y, which will gives us the equation negative y equals the square root of x. What I'd like you to do is to rewrite this equation. Negative y equals the square root of x, so that y is all by itself on the left hand side of the equation. So you'll end up with y equals something. What goes on the right hand side?
To transform negative y equals the root of x, so that y is by itself on the left hand side, we need to multiply both sides of the equation by negative one. Then we end up with this, y equals negative square root x. So, just like we saw before with polynomials, negating the entire function, or in other words, the entire expression involving x flips the graph upside down. Or in other words, reflects it in the x axis.
When we reflected this function in the x axis, we ended up with a new equation, this one. So we can see that all that we did was replace the square root of x by negative square root x. Now in terms of functions, if we said that our original equation was y equals f of x, then what we see with our reflection in the x axis is y equals negative f of x. Reflection in the X axis just involves negating the entire function that we're working with. But what about reflection in the y-axis, we've seen one example of a graph showing that before, but what happens with the equation? Well, going back to our three simple points that we saw before, where did they end up if we reflected them in the y-axis? Please fill in the coordinates of the new reflected points here
The points we will end up with after a reflection in the y axis, are negative
This means that for some general point, x,y, a reflection in the y axis produces what new point? Please write in, its coordinates here.
This time, it's our x coordinate that changes. The y coordinate stays the same, but the x coordinate becomes negative x instead of x. We can see this happening in our graph. All the y values of these corresponding points are the same, but their x values have been made negative, since we flipped across the y axis.
We just saw that in order for a function to undergo a reflection in the y-axis, we need to replace x in the original function with negative x. So going back to our equation, y equals the root of x, if we want to reflect the graph of this function in the y-axis, what do we need to replace with what in the equation? And what final equation will that give us?
To reflect the graph of y equals the square root of x in the y-axis, we need to replace x with negative x, which gives us a final equation of y equals the square root of negative x
Let's put our new understanding of transforming functions to use, and try to graph some interesting equations. If we take the function f of x equals negative x minus one cubed. We can compare it to the function f of x equals x cubed, to help us figure out how this function might look when it's graphed, since we already know what this one looks like. So, first things first. What is the minus 1 right here doing? Please fill in this sentence properly to tell me about this transformation. Which direction does this replacement shift the graph and by how many units.
Replacing x with x minus 1 in any function will move the graph to the right by
So far, we've figured out how to do one transformation to f of x equals x cubed. We've shifted it to the right by one unit. However, we know that this wasn't the function we were actually looking to end up with. We wanted f of x equals negative x -1 cubed. So, what does this negative sign out front do? This negative sign is going to cause our function to be reflected in some way. But which axis will it be reflected in? Please write either x or y in this box.
Putting a negative sign out in front of the entire function means we're going to reflect the function in the x-axis. Let's take a look at our graphs, finally. So here we have our progression starting with our first function, f(x) equals x cubed. Then we translate it to the right by 1. To give us f of x equals the quantity x minus 1 cubed. And then finally, we reflect this in the x axis, giving us f of x equals negative x minus 1 cubed.
Recently, we've been predicting changes in graphs based on changes in equations. So let's work the other way around for a second. Here's the graph of a function. I'm going to call this graph A. And I'd like us to consider this our original graph. Now, graph B over here is a transformed version of graph A. Now, I didn't size these graphs perfectly, so B's a little bit shrunk. But the only thing that's happening. In order for graph A, to become graph B. Is first, it's shifted in some direction. Either right, left, up, or down. By some number of units. And then it's reflected. Either in the x axis. Or in the y axis. I would like you to feel in the sentence properly. To describe this transformation, from A to B.
First graph A was shifted to the right by 1 unit. We can see that based on where the x intercepts are in graph A. They're at negative 2, negative 1, 1 and over by 1 to the right. The final transformation that's happening here is the graph is reflected in the x axis. We can see that here, the ends are both pointing upward, and here they're both pointing downward.
We've seen how to make horizontal and vertical shifts happen for all sorts of functions. And we've also seen how to reflect in the x and y axis. These types of transformations are typically refererred to as rigid transformations, because the basic shape of the graph doesn't change. All we're doing is translating it around the coordinate plane. Or flipping it over. Only its position changes. We can also, however, have nonrigid transformations. These involve the stretching or shrinking of a function. Nonrigid transformations distort the shape of the original function. We're going to deal with this stretching and shrinking with something familiar right now. Parabolas. Here are a bunch of parabolas that have been stretched in different ways. I'd like you to match each one with the equation it corresponds to. Please write the letter of the parabola in the proper box.
Graph A is represented by the equation y equals 1/3 x squared, B goes with y equals 1/2 x squared, C goes with y equals 3x squared, D is y equals 2x squared, and E is y equals x squared.
Now, before when we talked about parabolas, we said that changing the coefficient out in front of the x squared changed the width of the parabola. Putting a fraction out in front, like 1 3rd or one half, made the graph really wide. And putting a bigger number like 2 or 3, a number greater than 1 as the coefficient, made the graph skinnier, like we see in C and D. However, let's think about this from a vertical point of view instead now. Instead of thinking about these as making graphs wider or skinnier, we can think about these coefficients actually making the graph taller or shorter. Changing this coefficient, changes what the y value is going to be for any given x value. Graph E down here, is our parent problem, y equals x squared. While we see that point 2 comma 4, here. On graph B, we can see that when the x coordinate is 2, the y coordinate is not all the way up at 4. It's only halfway as tall. You can imagine that if I drew a vertical line going from the x-axis all the way up to where the line meets the graph, we can compare how tall graphs are to one another by picking the same x value and then seeing how tall that line is. How many units in the y direction it takes to hit our graph. But, what determines if a graph is stretching, like in graph D, or shrinking like in graph B? Remember, this is in the y direction, the vertical direction. Something clearly has to happen in their equations to make this happen in their graphs. Lets use good old y equals x squared, as our starting point, the function that we're going to transform. When x squared, in this function, is multiplied by a value that is greater than zero but less than one, will the graph be vertically stretched or vertically shrunk? Now, on the other hand, if x squared is multiplied by a value that is greater than 1, what will happen to the graph?
Multiplying x squared in this function, by a number greater than 1, makes it grow taller. So this is a vertical stretch. We can see this happening here in the equation y equals 2 x squared. On the graph of y equals x squared we get the point 2, 4, but on the graph of y equals 2 x squared, we got the point 2, by a factor of 2. This graph is taller than this one. Now on the other hand, if we multiply x squared instead by a number that is still positive but is less than one, then the graph will be vertically shrunk. Like this graph, y equals 1 half x squared. Here instead of the 0.24 that we see on this graph, we have the
We've been working with parabolas quite a bit. So let's play with some cubic functions for a minute. There are two right here. We have the graphs of f of x equals x cubed, this green curve, and g of x equals 3x cubed, this red curve. Now what I'd like you to do is write the equation of a function whose graph will lie between these two graphs. And we'll still intersect the origin. We're going to call this graph h of x. Let me show you an example of what h of x might look like since there are a number of different equations that it could be. So h of x might look something like this purple line. Think about how we need to stretch or shrink f of x to make it lookmore like h of x.
There are actually a ton of different right answers for what h of x could be. The actual equation I graphed over here is h of x equals three halves x cubed. Instead of this three halves though, you could have any coefficient. Maybe lets call that coefficient A for now. That's between the values of 1 and 3. We want h of x to be stretched taller than f of x but not quite as tall as g of x.
Let's finally get back to helping Athena. She knows that she wants to build her glasses frame logo using the shape of y equals x to the 4th. However, she wants the center of her glasses frame to be at the origin. So, she's figured out she needs to move this function to the right and down one, to create the bottom half of the right side of the frame. That should give her this graph. So this area right here will be the bottom of the right side of the frame, however we need the equation of this function. Please write the right-hand side of the equation here, I've already given you the left-hand side, it's just y.
To move the function to the right by one unit. We need to replace the x. In the original function with x minus 1. So at that point, with just the translation to the right. We have y equals the quantity x minus 1 to the 4th. Now to move down by one unit. We need to subtract 1. So this is our final equation. For this graph over here. Y equals x minus 1 to the 4th minus 1.
Athena is one quarter of the way done with designing her logo. She has the bottom half of the right-hand side of the frame. But how can she complete it? What equation do we need for the top half? So what then is the equation of this function, which is the reflection in the X axis of this equation. Please write the right-hand side of the new function right here.
Reflecting in the x axis requires replacing y with negative y. Which means negating the entire function. So I've made the whole right side of the function negative. Now we could also of course distribute this negative to the two terms that we have. That would give us negative, the quantity x minus1 to the 4th, plus 1. Either way works.
We're now half way there. We have the equations of two of the lines that we are going to need to help Athena design her glasses logo. This is the right-hand side of the frame. And the other half of the frame should just be exactly like this one, simply reflected in the y-axis. So, here are the two functions we've already found. And over here, I'd like you to just write the right-hand side of each of their reflections in the y-axis. So, these are the two functions that are represented on the left-hand side of the graph over here.
To make a reflection in the Y axis happen, we simply need to replace x in our original function with negative x in the transformed function. So, here are the four functions that Athena needs to make her final glasses frame logo. This is pretty cool. With a couple of domain restrictions and a few other functions added in, she could complete her picture like this. Or maybe she'd rather let the functions themselves complete the glasses. Of course, adding in a couple of other lines and points for wipers and nozzles. Either way, I think we've seen just how powerful transformation of different kinds of functions can be. You've learned about shifts, horizontally and vertically, reflections in the X-axis and the Y-axis, and stretches and shrinks in the vertical direction. These are all super powerful tools that can be used with any kind of function. Awesome job. I'm sure Grant will be very pleased with this logo.