Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. Quadratic functions and their graphs Graphing F of X equal AX squared. The graph of F of X equal AX squared is a parabola with X equaling 0. As it acts as a symmetry, its vertex is the origin. If a is greater than 0, the parabola opens upward, and if a is less than 0, the parabola opens downwards. If the absolute value of A is greater than 1, the parabola is narrower than Y equal X ^2. If it's greater than one, it means it's closer to the Y axis. If the absolute value of A is between 01, the parabola is wider than Y equal X ^2 or it's closer to the X axis. Use your graphing calculator to see how changes in A affect the shape of a parabola by graphing the following. If we look at these graphs, we should see that Y equal X ^2 is going to look something like this, and Y equal 2, X squared is going to be closer to the Y axis. Y equal 5, X squared is going to be even closer because it's a bigger number to the Y axis, and Y equal 1/3 X squared is going to be closer to the X axis. That wasn't a good one, but you get the idea. I hope. There we go graphing Y equal a * X -, H ^2. The graph of F of X equal a times the quantity X -, H ^2 has the same shape as the graph of Y equals AX squared. If the H is positive, the graph is shifted H units to the right. If H is negative, the graph is shifted absolute absolute value H units to the left. The vertex is H 0. The axis of symmetry is X equal H The way I personally think about it is if I think about this parenthesis portion equaling 0 and solve for X so X = H the axis of symmetry is the line that if I fold the graph right on top of that line, the left side and right side will be exactly the same. If we use our graphing calculator to see the effect of the left and right positioning of the parabola, we can see that at y = 7 X squared, we're going to go right through the origin. If we have y = 7 X -1 ^2, that's going to shift one to the right. If we have seven X -, 2 ^2, that's going to shift 2 to the right. If we have seven X + 3 ^2, that's going to shift us 3 to the left. Let's look at some more examples. So if we have graphed the following G of X equal X + 4 ^2, if I thought about putting that X + 4 equal to 0, we'd get X equaling -4 for the X value of the vertex. There is no number at the end that's plus or minus, so we haven't moved it up or down any. So the Y is going to be 0. We could think of it as A + 0. The axis of symmetry is always a line. It's going to be X equal -4. It's the X equaling whatever the X value was of the vertex. This is an understood one in front. So our A equal 1 one is positive or greater than 0. So we know that this is going to be going up. If it's going up, that tells us that point has to be a minimum. So a minimum at zero. We put the minimum being the Y value, how high or low it went. Traditionally we make an XY chart. When we're graphing these, we put our vertex in the middle and we give ourselves 2 numbers that are less than and two numbers that are greater than. So if -4 zero is my vertex, I'm going to use -2 negative three -5 negative 6. If I stick in -2 negative 2 + 4 ^2 is going to give me my new Y value. Well -2 + 4 is 22 squared is going to give us 4. So when X is -2, Y is 4. Now due to symmetry and the fact that we're squaring, when I put in -6, I ought to get the same value. So -6 + 4 is -2. But when we square -2, we get 4. So due to the symmetry, the axis of symmetry concept that -2 and -6 should give us the same Y values out if we stick in -3 negative 3 + 4 ^2, 1 ^2, which is just one, should give us the same as if we stick in -5 well -5 + 4 is -1 ^2 is +1. So when we graph this, we'd have -2 positive 4:00, we'd have -3 one. We'd have -4 zero. Oops. Except I wasn't doing that right was I -3 one -4 zero -5 one negative 6/4 when I put in my axis asymmetry, If I fold the graph right along that line, the left side and right side should fall on top of each other. Let's look at another example. This one we have X -, 3 ^2. So we know the vertex is going to occur when that X - 3 goes to 0 or X equal 3. There is no plus or minus at the end. We're going to look at other examples in a moment that do have one. So the vertex is going to be 30. The axis of symmetry is going to be X equal 3 understood coefficient of 1. So it's going to be going up and hence it's a minimum at zero. It remember it's at the height, however high or low it goes. When we put in our T chart, I'm going to have 30 and then I'm going to do a couple numbers that are smaller and a couple numbers that are bigger. If I have my symmetry right, we should get numbers that are the same for the Y value for two of them. So 1 - 3 is -2 negative, 2 ^2 is 4. So 1/4 if I put in five, 5 - 3 ^2 is 2 and 2 ^2 is also four. If I put in two 2 - 3 ^2 -1 ^2 which is one, 4 - 3 ^2 which is 1 quantity squared, we get one. So when we graph these, we get 1/4 21304154, putting in a dotted line for our axis of symmetry, connecting the points. And that's our graph for this next one. Remember, it doesn't matter if it's G of X or F of X or H of X. Those all represent Y. And it doesn't matter if it's G of X equal or the equation equal. So our vertex again is going to be whatever makes that parenthesis portion go to 0. So X is going to be two. There is no number that's added or subtracted with it, so we're still going to have our Y vertex being zero. Our axis asymmetry is going to be X equaling whatever the X value is in the vertex. This time our A is a -1. If the a is negative, it's less than 0, so we know it's going down. So it's got a highest point now or a maximum, and that maximum occurs at 0. Putting in AT chart, we have our X, we have our Y. We're going to put 20 as the dead middle, so we have 013 and four. If I stick in zero -1 * 0 - 2 ^2, well, 0 - 2 is -2 ^2. Remember our order of operations, we have to do that square before we multiply by the -1 in front. So we get -1 * 4 or -4. Now, if I've done it correctly, when I stick in four, I should get the same thing 4 - 2 quantity squared -1 * 2 ^2 -1 * 2 ^2 -1 * 4, which is also -4. We stick in one 1 - 2 ^2 1 - 2 is -1 ^2 so -1 * 1 or -1. You can check that when we put a -3, we also get -1 so zero -4 one -1 2/03 negative 1/4, negative 4, our axis of symmetry, putting in our dotted line for the axis of symmetry, and then graphing our graph should look something like that. OK, so now we're going to get into one that has a fraction, and it's the exact same steps. We're going to look at what makes the parenthesis go to 0. So we get X equal negative 1/2. There isn't a number added or subtracted with this at the moment, so it's just going to be negative a half zero. The axis of symmetry is always X equaling whatever the X value was on the coordinate on the vertex coordinate. So X equal negative 1/2. Our a is negative. If it's negative, it's going down, which makes the highest point or a maximum at whatever that Y value is 0. This time when we do our T chart, we're going to have negative 1/2 of 0. But we want to do it by adding and subtracting whole numbers. So one half, three halves -3 halves -5 halves. And the reason is that 1/2 will go away. So if I have -2 * 3 halves plus 1/2 ^2, well three halves plus 1/2 is 4 halves, which is really just two. So when we have 2 ^2 we get 4 * -2 is going to be -8. Now if we did it correctly -5 halves should also give us out -8. You can check that one. If I put in one half, 1/2 + 1/2 as a whole, 1 ^2 is 1, so -2 * 1 ^2 is -2. Once again, if you put a -3 halves, you'll find that those are the same values. So when we graph this, we're going to have three halves negative 812345678. We're going to 1/2 is going to be -2. OK, let's try this again. Three halves positive X direction. 3 halves is negative 812345678. One half 1/2 positive 1/2 is -2. Negative 1/2 is zero -3. Halves is -2. Negative 5 halves is -8. If we sketch that in, we get a parabola going down. Put in our axis asymmetry as a dotted line through that X = 0. That's the line that if we fold the parabola, the left side and right side fall right on top of that line. Now we're going to actually shift it. We're going to move that Y value of the vertex. So it's going to be exactly the same thing. But now we're going to have a + K at the end. And if it's a + K, it's going to move it up. If it's a -, K, it's going to move it down. So when we look at this next one, same exact steps, we're going to take that inside parenthesis, set it equal to 0 to figure out what our axis is symmetry. And the vertex for the X value is. But now we're going to look at whatever was plus or minus at the end. It's a -, 2. So it tells us we went down 2. The A value is a understood +1 here. If it's a one, it's going up, which makes that point a minimum point. And now the minimum points at -2 because our Y value is -2 if we put in our XY chart -3 negative 2, let's do -2 negative one -4 negative 5 -1 + 3 ^2 - 2. SO 2 ^2 - 2 four -2 is going to give us 2 and -5 + 3 ^2 -2 negative 5 + 3 is -2 ^2, which is 4 - 2 is also two. And we stick a -2 + 3 squared -2 we're going to get 1 - 2. When we simplify our -1 negative 4 is going to also give us -1 so -1 positive two -1 positive 2 -2, negative one, negative three -2 negative four -1, et cetera. We put in our dotted line, connect our points, and that's our parabola, OK? This one differs just because of the coefficient negative in front. So it's going to be going down, which will make it a maximum, and it's a + 4 at the end. So instead of moving the whole parabola down, it's going to move it up X + 1 = 0 X equal -1 so our vertex is -1 four X equal -1 a maximum at 4:00. If we put in our T chart, we have -1 four. We're going to go 10 negative one -2 negative 3. Putting in one 1 + 1 is 22 squared is 4 negative 2 * 4 is -8 negative 8 + 4 is -4 so -2 let's just write down the -3 one negative 3 + 1 ^2 + 4. So order of operation says we do that parenthesis first, then we do the exponent, then we do multiplication, and finally addition. So we would expect the one and the -3 to give us the same Y values, and they do. If we put in zero 0 + 1 ^2 + 4, we do the parentheses first, then the exponents, then the multiplication, and then the addition. So we get 2. You can check -2 it will also give us out 2. So +1 gives us negative, 412340 gives us out, two -1 gives us out, four -2 gives us out, two -3 gives us out -4 Connecting those points again, putting in our axis asymmetry. Seven and eight are going to be just the same. I'm going to let you do those on your own. Remember that 8 hasn't understood what in front of that parenthesis. If there's no number, that's really the same thing as a -1. If we continue on, that's the rest of the handout. So do 7:00 and 8:00. Thank you and have a wonderful day.