Vertical and Horizontal Shifts of Quadratic Graphs - Concept
Multiplying rational expressions is basically two simplifying problems put together. When multiplying rationals, factor both numerators and denominators and identify equivalents of one to cancel. Dividing rational expressions is the same as multiplying with one additional step: we take the reciprocal of the second fraction and change the division to multiplication.
Any time you're asked to graph anything in Math class you can always make an x and y table of values and that's pretty effective but you guys it takes all day I understand like I know it can be a real drag. That's why a lot of teachers make available these graphing calculators to their students. You might have one available to you but even if you don't, I want to show you guys shortcuts for graphing parabolas that you can graph them without having to make the table of values. I'm going to demonstrate on this calculator so that you believe me but you guys will be able to use these patterns these shifting rules not only in your Algebra class but in all of your upcoming Math classes, so let's go ahead and look at the calculator.
I'm not going to spend too much time showing you guys how to use the calculator hopefully you'll be learning that from your Math teacher, but I do want to show you some really cool things about parabolas. First of all, let's look at our parent function y equals x squared, I'll just go ahead and type that in and I'm going to first show you what the table of values look like. The table of values if you look here these show me my x input values, these show me my y output values and as I look through I can see that as if I were to substitute in -3 and square it, I will get +9 substitute in -2 squared I get +4 that's the kind of idea that I'm working with, with this function y equals x squared.
I'm going to scroll down a little bit and show you guys how you can identify the vertex of a parabola by just looking at the table. Here's what I mean, to find the vertex think about the place where your y values changed direction. Look at our y values they're going down going down going down then here they start going up going up going up, what that means is that this is the vertex of my parabola the coordinates x for 0 y for 0 that's the vertex of my parabola. I can also tell using symmetry because this 1 shows up on both sides of the 0 the 4 shows up on both sides 9 that's symmetry.
Let's look at the graph. If I look at the graph of y equals x squared you can see it's a smooth curvy parabola shape here's my vertex at 0,0 it goes to the point 1, 1 2, 4 blah blah blah, oops 2, 4 is about there blah blah blah okay so you guys get the general idea of what the graph of y equals x squared looks like.
Any time you're asked to graph an equation that involves an x squared it's always going to have that same parabola shape. What I'm going to do is go in to here and show you how to graph a few different equations at the same time and we'll look at how they change when I stick in different numbers. Okay, so first I have my parent function y equals x squared, I'm going to add another function in there. I'm going to add y equals x squared plus 2, we're going to see what that looks like on the graph. I'm also going to add y equals x squared take away 3. Our computer is going to graph all 3 of those at the same time and we'll talk about them as they show up. Okay so before we look at the graph I want to point out to you that x squared is going to show as a bold line then we're going to see x squared plus 2 and then we're going to see x squared minus 3 there it was that was pretty fast. Here is my x squared bold, this is x squared plus 2, this is x squared take away 3. Do you notice what the plus 2 and minus 3 did? It just moved the the vertex, it moved it along the y axis the plus 2 took the exact same parabola shape moved it up 2 the minus 3 same parabola shape moved down 3. It looks like these guys are actually wider and skinnier but they're not it's hard to tell on this little screen but all of these parabolas are equally wider skinny that's going to be important matter.
Okay so now you guys can see that when I have a plussing number out here or a subtracted number an added or subtracted number it just moves the vertex along the y axis. Let me show you something different what would happen if? I were to put those added or subtracted numbers inside parentheses like this? I'm going to do x takeaway 2 inside the parenthesis and then squared thinking about the order of operations. I'll also try another one. I'll have x plus 3, x plus 3 inside the parentheses and then squared. Think about what that might do to our vertex. let's look at the graph remember that x squared is going to show up as bold then we'll see x takeaway 2 quantity squared and then we'll see the quantity x plus 3 squared. Okay there's x squared this is x takeaway 2 and that's x plus 3. It's weird you guys it's counter intuitive usually when we're adding numbers that means on the graph it moves to the right but in this case we subtracted 2 it's like the opposite usually we think subtracting 2 would move the vertex in this direction but really subtracting 2 move the vertex here, be really careful with that. Same thing when I had plus 3, intuitively I will think that will mean take my parabola move the vertex 3 in this direction but what happened was it moved 3 in this direction. That's something that's really tricky that's hard to get the hold of. What I want you guys to remember from this video is that if it's inside the parentheses, it represents a horizontal shift. If my number was outside the parentheses, it represents a vertical shift.
The last thing I want to show you before you move on is what would happen if I combined them. I'm going to try to do x takeaway 2 inside my parentheses and then square that and then I want to a plus 3 at the end. Before I show you the graph try to predict what's that graph is going to mean. This minus 2 is going to represent something, the plus 3 is going to represent something else. Well let me just tell you before we look at the graph how you could predict this, this minus 2 means my vertex is going to mean be moved 2 to the right, this plus 3 means my vertex is going to moved up 3. Let's look at the graphs you guys believe me. There's my parent function, I moved my vertex over 2 and up 3 that's really cool you guys it might not sound cool right now but when you get a homework about graphing parabolas you're going to think that's really cool. You don't have to make a table you can just use these shifting rules the -2 tells you a horizontal shift but it's counter intuitive the +3 tells you the vertical shift up and down, so you guys the last thing I want to leave you with is the table of values. Remember how we're looking at the vertex's let's look at the let's try to find the vertex for my y two functions which is the one that has the side to side and vertical movement.
Remember when you're looking for the vertex in a table, you're looking for where your y values change from increasing and decreasing. Like look at my y's getting smaller getting smaller getting smaller oh! Oh! They start getting bigger getting bigger again. That tells me this is my vertex the coordinate 2, 3 is the vertex of that parabola and I could tell from the table because it's where my y values changed direction. They were decreasing and then they started increasing, so guys good luck with these, these shifting roles are important to start practicing now because they're going to show up in your future Math classes and more importantly they make graphing a heck of a lot easier.