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Graphing Radical Equations using Shifts - Concept
MA, Stanford University
Teaching in the San Francisco Bay Area
Alissa is currently a teacher in the San Francisco Bay Area and Brightstorm users love her clear, concise explanations of tough concepts
When graphing radical equations using shifts, adding or subtracting a constant that is not in the radical will shift the graph up (adding) or down (subtracting). Adding or subtracting a constant that is in the radical will shift the graph left (adding) or right (subtracting). Multiplying a negative constant by the equation will reflect the graph over the x-axis. Multiplying by a number larger than one increases the y-values.
When you're graphing radical equations, it's really important you think about what your domain is. Remember with the square root, you can't have a negative radicand or whatever is under the square root sign can't take on negative values because there's no real solutions. What that means is that our graphs are going to be kind of funny shapes, they're going to have dead ends.
What I want to do is show you guys on the graphing calculator how you can do these graphs pretty easily using shifting rules without having to make a table of values. Before I do that though I want to remind you, it's really important on the graphing calculator that you're careful with things like parentheses. For example if I want to type in the square root of x+2 or the square root of x+2 where the whole thing is square rooted, those are 2 really different things. On the calculator, I have to show this guy with parentheses only around the x, this guy would be parentheses around the whole x+2 piece. Let's go ahead and look at the calculator and I'll show you what I mean.
What I have here is first my parent function where y is equal to the square root of x, then I have where y is equal to the square root of x then plus 2, and what I'm going to do next is y is equal to the quantity meaning I have to combine this process first. The quantity of x+2. Think back to what you know to the order of operations and how when you're doing order of operations you do parentheses before you apply any any exponents. The square root is actually an exponent and you'll learn about that later.
Let's look at the graph. First we are going to see our parent function, y equals x squared, there's y equals x squared plus 2, where I took the square root of x and then added 2. This is y equals the square root of the quantity x+2. You can see how the +2 outside the square root sign moved my graph up 2 but the square root with the parentheses and the +2 under the square root sign moved my graph to the left 2. That's the same shifting rules you're going to be seeing with parabolas, with cubics, with absolute values and with all kinds of graphs that you're doing in your math class.
Let's look at a couple of different other ones. I have y equals the square root of x. What happens if I do y equals the negative square root of x? Let me pause really quickly because I bet a lot of you are saying, "wait a second lady. There's no such thing as the square root of a negative number." Well that's mostly right. I can square root a number and then negativise it and that's what this process means. That means take the square root of x and then stick a negative sign in front of it. First let me show you the table, so you can see what I'm talking about. This is y equals square root of x right here. This is y equals square root of x with an equal sign in front of it. Look at here at the four. That's probably the thing that makes most sense. I have four, the square root of four is 2 and the square root of four with a negative sign in front would be -2.
Now let's look at the graphs. This is going to be a little bit tricky so I'm going to show y equals square root of x in bold. That will show up first and then I'm going to show y equals square root of excuse me, negative square root of x. Bold, regular. [IB] you guys you're probably looking at that and thinking, "hey lady that's just a parabola." Yeah you're right. There's a parabola except this represents 2 different functions. Don't get confused. I have y equals square root of x, that's this bold thing. This is y equals negative square root of x. What happens is it got reflected across the x axis. The negative sign in front of the square root reflected it across the x axis.
Let's look at a couple more. Instead of having a negative square root of x, I'm going to try 2 times the square root of x and we'll see what that looks like. Try to predict in your head what you think it might look like. What I'm doing is the square root of x multiplying my result by 2. Let's graph those and see what happens. Again my square root of x is going to show up as bold and my 2 square roots of x is going to show up regular. Bold, regular. [IB] it affected how steep the curve happened. This is y equals the square root of x, that's y [IB] square root of x times 2. And it makes sense because my y values are being multiplied by 2 so it goes up more steeply. My y value or my change in y is greater.
I'm going to add one more to that just so you guys can really feel the pattern make sense. I'm going to add one half or 0.5 times the square root of x. Well, if the coefficient of 2 made the graph wider, the square root of a half, what do you think that's going to do? [IB] you're right, it makes it skinnier or it makes that change less steep.
So we have always the same shape, but if the value of a is negative, it's going to be reflected across the x axis and if the value of a is an integer, absolute value excuse me is an integer, it's going to make that shape wider and if the value of x, absolute valued is a fraction or a decimal, it's going to make it skinnier.
Okay, let's go back and do one that puts everything all together. Let's just say you're coming to your homework and your teacher asks you to graph some nasty square root function that has all kinds of things going on to it, and your options are to make a table of values which would take all day [IB] all day but really like 10 minutes and that's kind of long. You could take all day or you guys could use these shifting rules that I'm showing you. What I'm going to do is the square root of the quantity x take away 2 close parentheses, so that's going to move side to side then I'm going to add 3 it's going o move it up and down. Don't forget I also have this one half out front, which is going to change how wide or skinny it is.
Here comes the graph. My parent function and the thing moved side to side. It moved over 2 and up three because of the twos and threes I had in my equation. Isn't this weird how it has a dead end right there? That's really weird looking to me and the reason why it's there is because of the domain. Let's go back to the function and talk about what I mean with the domain.
Domain is what a possible input for your x numbers and remember that you can't have a negative value under the square root. Which tells me that the quantity x take away 2 has to be larger than or equal to zero. If I saw that x-2 is larger than or equal to zero, I'm going to get x numbers are larger than or equal to zero. That's my domain and it's going to be confirmed when I look in my table here. My x numbers, I get error unless my x numbers are larger than or equal to zero. Or excuse me larger than or equal to 2.
Error means something important. It doesn't mean you typed something wrong into your calculator, it just means these x values are outside the domain. And again I just want to leave you with the graph. How the end of that parabola got moved out of the origin over 2, up 3because my domain is x values that are greater than or equal to 2.
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