When graphing parabolas and other functions, there are certain types of graph dilations that help us graph and explore parabolas more efficiently. When graphing parabolas, dilations occur when the "a" term value is changed. If we change the "a" to a negative, the graph shifts downward instead of upward. As the absolute value of the "a" value increases, the graph becomes "steeper." If we use a fraction as the "a" value, the graph of the parabola becomes wider.
You guys know that any time you are making a graph in Math class, you can make a table of x and y values and substitute the x's in one at a time. But oh my gosh! It takes all day. I understand. I'm a Math teacher, I assign this stuff, I have to do it myself. I know it's a real drag. That's why I personally learn a whole bunch of shortcuts. And when I'm grading homework, I don't make tables of values all the time. I use the shortcuts in my brain to just check students' homework like estimation. That's how you guys can do your graphs. If your teacher says just make a sketch. It doesn't have to be exact and that's what we're going to be looking at here.
We're going to be looking at what happens with a quadratic equation meaning you have x squared as your highest exponent. If it's multiplied by a negative number, like it has a negative a value, or if a is not one. We're going to be checking that out. I'm going to be using one of these calculators that your teacher might have available to you, might not. But these are sometimes useful just to check your work. Okay, so I'm going to go on ahead and turn on the computer and we're going to see what we can do when we explore quadratic equations.
The first thing I'm going to graph is y=x squared. That's what called a parent function. It's like the function that creates the shape that all quadratic equations have. I'm also going to make it so whenever I draw that graph, it's going to show up as bold. At the same time I'm going to be graphing y=-x squared so we can see how they're alike and how they're different.
Let's go ahead and hit graph and see what happens. There's my bold and there's my regular. That means this is the graph of y=x squared and this is the graph of y=-x squared. Look at how it's upside down. A lot of students remember that negative means upside down. It's like a sad face. Negative is sad. So you frown. That's how they remember that this graph is going to be upside down.
Let's go look at the table of values so you guys can see what else is going on. This is my graph or this this column here represents y=x squared. This represents y=x squared but then negativise, you see all these negative signs. That's the points that create the graph that we just saw.
Okay, let's go back and look at a couple of different equations. We have y=x squared. Next let's look move those graphs up and down on the axis. Let's do y=x squared plus two and then we'll do y=-x squared plus two. Again I want you guys to think about what's alike and what's different. Before I hit graph try to predict what this is going to look like. The negative means that it's going to be a frowny face or upside down. And this +2 business means that my vertex is going to be moved up two vertically. Here comes the graph.
My parent function y=x squared plus two and y=-x squared plus two. Sad frowny face. Same vertex though, the vertex got moved from 0 0, up two because of that +2 business.
Let's look at a couple of these equations where the vertex is not on the origin or it's not even on the y axis. What I'm going to do is move the vertex over and up and down by using some parentheses. I'm going to do y=x take away two quantity squared and then plus three. Remember that the stuff inside the parentheses represents a horizontal shift and the stuff outside the parentheses represents a vertical shift in my case it's going to be up three.
I'm also going to be graphing y=-x+2 inside the parentheses squared plus three. So these are, oops, I meant to do minus two. Back up. Back up. Back up. Let's make that a negative sign. Okay, so these are going to be the exact same equation except for the one tiny difference I'm going to have is this negative sign outside. I shouldn't call it a tiny difference, it's actually a big deal. Remember also the order of operations. If you are making a table of values, you would do your x number, subtract two square it, negativise that answer and then plus three.
Okay, let's look at the graphs. There's my parent function, there's my parabola that's opening up because a was a positive value or there's a positive outside the parentheses, this is the negative one. See how the vertex is no longer on the y axis but that pattern of flipping up and down still stayed the same.
Okay, so that's all good. The next thing we are going to be looking at is what happens if a is a number other than one or -1. For example, I'm going to be graphing y=x squared along with y=2 times x squared. Think about what that might mean. I'm also going to graph y=4x squared so we can really start seeing what difference it makes when a is a number bigger than one. Here comes the graph, y=x squared, y=2x squared, y=4x squared. Notice all three of those have the same vertex of 0 0, but y=2x squared is steeper because your y values are being multiplied by two every time. It gets steeper more quickly than y=x squared. Along those same lines, when I multiply by 4x squared it gets even steeper still. What happens when you're a value, absolute value is larger than one, it makes your graph what we call skinny.
That's as opposed if I use a fraction and what I'm going to do next, is half or 0.5 of x squared. We'll also do 0.2 so you guys can really see the couple difference. 0.2 of x squared. Let's graph all three of those, see what happens. There's x squared, half of x squared and 0.2. Ooh, they got wider.
Now I think that's kind of counter intuitive. I would think that fractions would mean like get skinnier, but actually fractions make the parabola get wider. There's x squared, there's half x squared, there's 0.2x squared. So you'll see as my decimal or fraction, makes the a value smaller and smaller, my parabola actually gets wider because my y value is changing in a less rapid way.
Okay, let's go back and do something where we can put it all together. I'm going to leave the y=x squared in bold, but let's do something where we have negative and a fractional value of a. I'm going to use y=-0.2 as my coefficient times x squared, and then we'll see how that looks.
Try to predict before I hit graph what that might look like. The negative means that it's going to be a frowny face, upside because negative's sad like a frown and then 0.2 is going to make my parabola wider, skinny? It's going to make it wide. Let's check. There it is, upside down and wide.
Let's do one that's super super messy putting together everything you guys have ever learned just to show you that once you get the hang of these shifting rules, you can use them to graph any parabola you come across. And by the way these shifting rules don't only apply to parabolas or quadratics. They also apply to absolute values, they're going to apply to cubics, are going to apply to all kinds of different shapes that you're going to see throughout your Math career. So I'm not just showing you this for fun or for shortcuts. These are actually going to be really important once you start moving through your math classes.
Okay, so I'm moving the vertex side to side, moving it up and down and I'm turning it upside down and I'm making it wide. Before I hit graph, let's just review what all of these different numbers mean. Negative means upside down, half means make it wide, this +2 moves my vertex side to side, it's going to move it to, in the left direction even though plus is usually to the right and then +3 is going to move my vertex up three. Let's go ahead and graph it, see what it looks like.
There it is. My vertex got moved over two, up three, upside down and wide. I could have drawn this graph by hand without making a table of values and that would make me feel like a superstar A+ Math student.
Start practicing these shifting rules guys. They're going to make your graphing a lot easier.