Hello wonderful mathematics people. I'm Anna Cox from Kella Community College. To find intercepts of a graph. To find the X intercepts, the intercept is going to be in the form a comma 0. So we're going to set Y equal to 0 and solve for X. For AY intercept 0B, we're going to set X equal to 0 and solve for Y. So if we look in an equation such as Y equal X ^2 - 9, if we wanted to find the X intercepts, we're going to set Y equal to 0. So y = 0, X squared -9 is the difference of squares, so X - 3 * X + 3. So we know that X - 3 = 0 and also X + 3 = 0, X equal 3, X equal -3 so when Y is 0X is 3 and also -3. So if we come over here to our graph, there's 30 here's -3 zero. We want to figure out what our Y intercept is. For our Y intercept, we're going to put zero in for our X so that when 0 is our X -9 comes out for our Y. Now if we want this to be a little more accurate, we can always plot a few more points. Say what happens if we have F of one? Well, we put one in for our X and we get out -8. So this is really a point. This is the point when X IS1Y is -8. Now, because of symmetry and because we're squaring this, we also know that -1 will give us that -8. What happens for F of two? Well, 2 ^2 - 9 four -9 negative 5. It's a 2 -5, so 2 negative 512345. We'd also have -2 negative 5 due to symmetry. So there's your X intercepts and your Y intercepts. We're going to look at symmetry about the X axis. Symmetry about the X axis means that if we put a point up here, say 3-2, then the corresponding point that's a reflection through the X axis three -2 would also have to be on the graph. So if we thought about this point being X, Y instead of 32, if we look we've gone the same quantity over, so we've still gone over X, but instead of going up Y, we've now gone down Y or negative Y. So for symmetry about the X axis, the X's have to be the same, but the YS are going to be the opposite value. So if we thought about looking at a graph, say something like this, this would have symmetry about the X axis, because any point that I chose here would also have a corresponding point below the Y axis. The same kind of concept here. If we have a point on the graph, we need to have the point that's reflected through the Y axis also being on the graph. So if we thought about this point as XY, now the X has gone the opposite direction, the same quantity, but the Y has still gone up the same amount. So for about the Y axis, we're going to have XY and negative XY. The origin is this point here, and the origin is special. We use this point, and what we think about is if we pick the graph up and turned it 180°, we'd have the corresponding point on. So right here, if we look at these two points, once again, let's call this original 1 XY. Instead of going positive direction, we're now going the opposite direction, hence the opposite of X. And instead of going up Y, we're going down Y, hence negative Y. Now this first one can't be a function, it doesn't pass the vertical line test, but the Y axis and the origin are possible functions. If we thought about just putting in some kind of a parabola here, this would show us that for the Y axis, F of X would have to equal F of negative X. Because when I stick an XI really get out Y, and when I stick out negative XI also get out the same Y value. So when I put an X to an equation, if I put a negative X, the two are equal. They're symmetry about the Y axis for the origin. F of X is not going to be the same thing as F of negative X because the Y values aren't the same. the Y values are what they're opposite of each other, so I can put a negative on either side or the opposite and get a new equivalency for if it has symmetry about the origin. A circle is the set of all points and a Cartesian coordinate system that are an equal distance, called the radius, from a fixed point called the center. Let's look at that graphically. If we have a center point, let's call it H, K, and every single point on the circle is an equal distance away, called the radius. So any point that I choose, let's call this point X, Y, every single point should be this distance away. If we thought about dropping and making a right triangle by dropping a vertical line and a horizontal line, this new point went over the same distance as that XY. So this new point would be called X and we went up the same amount as our center or K. Now if we use the distance formula, the distance formula says that we are going to subtract the X coordinates here. So this distance here, because it's a horizontal line, is just going to be my X minus my H We don't worry about the YS in that particular case because the YS were the same. If we look at this vertical line, it's going to be the upper Y minus the lower Y, or in this case, y -, K, and we square them both. So this is really the distance formula. We're finding the distance between two points. Well, it actually is also the formula for a circle. If I square each side and change this D to R for radius, R-squared equals the quantity X -, H ^2 + y -, K ^2, where H, K is the center of the circle and R is the radius. Now that's a very important form of a circle, but there's another one that's also important. So the standard form is the one we just looked at. X -, H quantity squared plus y -, K quantity squared equal R-squared. If we took this and used our wonderful algebra skills and foiled it out, we'd get a general form that says X ^2 + y ^2 plus CX plus D y + e = 0. Now eventually with quadratics, we actually put coefficients here. So we'd have AX squared plus B y ^2 plus CX plus D y + e = 0. And for this second more generic equation to be a circle, A has to equal B. Unit circle means that our radius is 1. So in a unit circle we have a radius of 1 and the center is at the origin. So if we have the center at the origin, the origin is the .00 X -0 ^2 + y - 0 ^2 equal 1 ^2. So a unit circle is just X ^2 + y ^2 equal 1. Thank you and have a wonderful day. This is Anna Cox.