Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. A very useful form of a line is point slope. Point slope is the form y -, y one equal M times the quantity X -, X one, where M is the slope and X1Y1 is any point on the line. Now this actually comes from just solving for the slope formula. We know that the slope formula was the change of YS over the change of X's. If we thought about cross multiplying to get rid of the denominator, we could see MX 2 -, X one would equal Y 2 -, y one. And remember the twos and the ones just say that they're from the same point. So instead of an X2 I could just call that an X and instead of AY 2 I could call it Y because then that plain X&Y would be from the same point and the X1Y1 would be from the same point. So really, point slope form is just a different way of writing the slope. Parallel lines have the same slope and perpendicular lines have slopes that are negative reciprocals of each other. So the first thing we're going to do is we're going to rewrite in slope intercept form. So slope intercept form was Y equal MX plus B. So now we have point slope form and we're dealing with slope intercept form. So we're going to start by distributing the six and then taking the five to the other side in order to get the Y by itself. So here we can see the slope is 6 and the Y intercept is 0, negative 53. Well, if I was asked to graph that -53 is pretty far down on the axis. A different way to do this would be to realize that the slope is 6 and a point on this graph would be 8 negative 5. So we could actually go to the .8 -5 somewhere down here and then we could do our slope of six and our slope of six would say up 6 and over one. And if we connected those very rough, rough sketch, but that would be the line. And if we kept going and going and going, eventually we would get to 0 negative 53. So we're going to give you 1 to try. Then the next one we're going to look at is rewrite and slope form. So Y equal or Y minus the Y value equal the slope times X minus the X value. We want to rewrite it in point slope form. So y + 4 equal 4 sevens X -, 3. This is point slope form, and we're done. If the directions had just said write it in slope intercept, we'd then solve for Y. So we'd multiply the four sevenths. Let's go ahead and do that just for some practice. We'd have 4 sevenths X -, 12 sevenths. Remember that 3 is 3 / 1, and then we would subtract the four. So 4 sevenths, X -, 12 sevenths -4 is 4 * 728 sevenths. So we would have Y equaling 4 sevenths, X -, 40 sevenths. So slope 4 sevenths, the .0 negative 40 sevenths. We could graph it from that, but it'd be much easier just to use the .3 negative 4 and then the slope up four and over 7 to graph that rewrite and point slope form. Well, in this case they gave us two points, so the very first thing we have to do is we have to find what slope is. So slope is the change of the YS over the change of the X's. Remember, it doesn't matter which X&Y come first, as long as the -2 negative fours from one point and the one negative sevens from the other. So -4 -, a negative 7, positive 3 / -3. So the slope here is -1. Now I can use either point, so y - -4 equaling -1 * X - -2 so y + 4 equal -1 * X + 2. Or I could have used y - -7 equaling -1 X -1, so y + 7 equaling -1 X -1. Now you might say those don't look at all to be the same. Well, if we solve for Y, if we put them in slope intercept form, this top one would turn into Y equaling 4. Actually, let's put another step. Y + 4 equaling negative X - 2 or Y equaling negative X - 6. If we look at this other one, y + 7 equal negative X + 1, Y equal negative X -, 6, they really are the same equation even though they looked different. So there's more than one way to do math, and sometimes there's more than one answer, even though they don't look quite exactly the same. Parallel and perpendicular to a point. So if we're given the .35 and the slope of -1, if we want it parallel, we'd have Actually, this isn't really right. Parallel and perpendicular point. How about just find the equation for the line? So we have y -, 5 equaling -1 X -3. This is point slope form. Then if I wanted to have it perp and or if I wanted to have it in slope intercept, we would distribute the -1 and we would take the five to the other side. So this is point slope. No, sorry. This is slope intercept. So this is if I have a point and a slope making the equation. Now let's look at making them perpendicular and parallel to a given line. So if I start with five X -, 7 Y equal 12 and two -3 the first thing we're going to do is we're going to find the slope of this given line. So we're going to solve for the Y and we get Y equal 5 sevenths X -, 12 sevenths. So the slope here is 5 sevenths. So if I wanted parallel, parallel would tell me it's the same slope. So parallel would be y - -3 equaling 5 sevenths X - 2 or y + 3 equaling 5 sevenths times the quantity X -, 2. Now if we wanted perpendicular, we're going to do the exact same steps, but we're going to use the negative reciprocal. So the perpendicular slope instead of 5 sevenths is -7 fifths, so y - -3 equaling -7 fifths X - 2. So y + 3 equaling -7 fifths X - 2. Now both of those are in point slope form. If it asks for Y intercept form, we would just solve for Y. What if it was X equal 4 and the .73? Well, if it's X equal 4, we know that X equal 4 is really just a vertical line. So if we wanted a parallel to a vertical line, it's going to be another vertical line. Parallel would just give me X = 7 perpendicular. Well, perpendicular to a vertical line is a horizontal line, so Y equal 3. So if you thought about looking at this as a picture, X equal 4 out here somewhere and we want to go through the .73, the parallel would be X = 7, the perpendicular would be Y equal 3, interpolation and X terpolation. This is if we're given some kind of data. Say we're given a bunch of points, just information point here, a point here, a point here, a point here. So this I drew, drew my point so that they weren't exactly aligned, but they were close to a line. If I connected these in the best fit, I might get something that look, I don't know, like that interpolation would be an estimation of what might happen in between given data. So at the green location, it would be about here. So interpolation is inside given data, extirpolation is if it was on the outside, what if it said what happened back here at zero? Well, we don't have that data. What we do is we think about, well, what if we extended the line, what would happen? And so the extirpolation is the outside of the given information. Interpolation is inside. Thank you and have a wonderful day.