Rate of Change Slope & Point Slope Equation of Lines

Rate of Change Slope & Point Slope Equation of Lines


Hello again, I am Mr. Tarrou! Whether you
are in algebra, pre algebra, geometry, algebra 2, or even precalculus, and calculus you need
a good understanding of slope and equations of lines. We are going to start out with some
equations of lines, finding slope. We will not be doing any application problems today
but the basic structures. Let’s first start with slope. We have a line over here with
a couple of points highlighted. We are going to use these two dots to find the slope or
the direction this line travels. Before you do that though you need to know a good definition
of slope. Slope is indicated with a cursive lower case m, is rise over run. This definition
is just three words. Everybody remembers this and it is really nice if you are just going
to look at a picture and tell what the slope is. You read graphs by the way left to right.
And as I read this left to right it is rising so I know when I am done that I am going to
have a positive slope. If the graph is falling down to the right then we have a negative
slope. If my line were horizontal my slope will be zero. And if my line is vertical that
will have an undefined slope. We are going to take a look at one of those later with
some numbers. Now we are going to look at this line. You have a couple of points. Start
with the left one first. How do you get from this point to this point? Visually you just
count. How far up do you have to go? This is the rise value. The rise value is one,
two,…You start here and don’t start counting when you touch the point. One, two, three,
four, five, six, seven, eight, nine, ten. We have a rise value of ten in this graph.
How far do you run over, how far to the right do we travel? The run value starting from
this point where I left off, one, two, three, four, five. So I have a rise value of ten
and a run value of five. So let me pick up this chalk that fell. Put this back in the
equation and we have the rise of ten over the run value of five. This gives us a slope
of two over one or two. Dividing by one does not do anything so we do not have to show
that in the final answer. Well that is fine but a lot of times in textbooks they don’t
give you a graph, they give you a couple of points. Let’s make sure you know how to find
the coordinates of a point. From the origin I am moving left three places and I am going
down what looks like four places. When you move to the left on the x axis, that is a
negative movement of x. That means this is negative three. If I go down on the y axis,
down is a negative movement, so that is negative four. We are going to label this as x sub
one and y sub one. And we are just going to find the slope again even though we already
know the answer is two. This point is two, going from the origin we are right two places.
That is a positive movement on the x axis. And we are going to go up one, two, three,
four, five, six. So we have a coordinate of (2,6) I already labelled this point as the
first point, so I am going to label this one the second point. If you are just starting
off in algebra you want to get into the habit of putting this x 1 y 1 and x sub 2 and y
sub 2 into your drawing. This will make sure you don’t put the numbers in the wrong spot.
Well rise over run doesn’t work very well if you have a bunch of numbers given to you,
especially without the picture. So another way of defining slope is y sub two minus y
sub 1 over x sub two minus x sub one. Your change in y is far up you went and the change
in x is how far over to the right you went. Let’s go back to these values and see what
we have. Y 2 is six minus, now my four is negative and there is a minus sign in the
equation so make sure you are using your parenthesis when you plug in your numbers, six minus negative
four. Six for my Y2 minus my other y value of negative four. Please note the minus that
is in the equation originally is right there, this negative is coming from my negative coordinate
of negative four. All of this is over X2-X1. My x sub two is two, I have got it well labelled,
minus again my minus sign is coming from the equation and not from the coordinate and always
use parenthesis when you substitute, minus negative three. If you have this written out
correctly we will have our slope. Six minus a negative… Well negative one times another
negative number and that is going make this cancel out. Six plus four is ten. Same thing
down here, we have two negatives touching. There is no division, there is no exponents,
we just have two negatives touching and that is multiplication. Negative times negative
again cancel out and two plus three is equal to five. We are once again getting down to
ten divided by five which was up here, again getting our answer of two. Now we are all
really happy with the fact that this line has a slope of two, we showed that twice.
What if you were told that you had a problem whose slope, this is going to be another example,
whose slope was equal to two and that goes through the point of….. I am going to come
back over here and I am going to pretend that I don’t know that this y value is negative
four. Again we are going to another example off of this same picture. So you have a slope
of two and point of negative three and this unknown value of y. We know it is really negative
four:) We have another point this line goes through of two six. So I gave you a slope
and I am giving you a question of, “What is the y value of this coordinate?” In all math
classes if they give you distance you are going to use the distance formula. If you
are given an area you will use that area formula. If they gave you slope and then asked you
a question, guess what, you are using the slope formula! So, I am going to walk through
this again. I am going to keep my labels of x sub 1 y sub 1 and x sub 2 and y sub 2. We
are going to walk through the slope formula just like we did a second ago. That is going
to be the slope is equal to two equals Y2 is 6 minus y over X1 of 2 minus don’t forget
the parenthesis and negative three. We are going to find that y value of negative four
in a minute. What do we do from here? Well, we have two equals six minus y over two, the
negatives are touching there is no exponent or weird stuff going on, just subtracting
a negative three so that is just two plus three is five. And we have the y here on the
right hand side of the equation, I want that alone. I want to find that is negative four
in a minute. With that y there is a division of five and I want to get rid of that division
of five. If you have a division of five and you want to get rid of it, if you are solving
equations you want things to go to the other side of the equation….I want that five to
go from the right to the left hand side, you must do the opposite of how it is connected
to the variable. The five is connected with division so I am going to get a different
color chalk here and we are going to multiply both sides of the equation by five. I am going
to get an answer of five times two is ten equals six minus y. I am almost running out
of room here. I am going to have erase a little bit down here so I can keep going. Now I have
got the y and a six. There is a positive six here, the minus sign is for the y, and this
six is positive. I want to get rid of the positive six with the x so I am going to do
that by subtracting both side by six, and giving my self a little more room to write
up here, ten minus six is equal to four and the six is now gone because I subtracted it
so negative y. I still have not figured out that y is negative four yet because y is not
yet by itself, it has a negative sign. So we are going to divide both sides by negative
one to get rid of that and for the third time we know the y value of that point that I just
erased, that you can back up the tape and look at, is negative four. So if a textbook
gives you a slope and a couple of points but you are missing one of those points, use the
slope formula and you will get your answer. Ok, now with slope, a lot of times you finding
the equations of lines. So let’s do one of those problems. Let’s say we want to find
the equation of a line that is passing through the point four seven and eight ten, and I
want the equation again. This is going to be pretty hard if you do not know don’t know
the three or four standard forms of a line. Right now I am going to bring up three and
uhh… let’s just say there is three. We have slope intercept form which is y equals mx
plus b. We have got Point Slope which I will be using in a minute which is y minus y sub
one, this is where we are going to put a number in a minute, equals m times x minus x sub
one. Again we are going to put some numbers in here in a minute for slope and these values
of x and y that we are given in the point. Another form or equation of lines is Standard
Form. This is Ax plus By plus c equals zero. A, B, C, these all have to be integers and
A should be positive. We are not going to work with this right now though. We are mainly
going to focus on this Point Slope Form. The name of the equation tells you what you need.
If you want the equation of a line you need a point and a slope. I have two points but
I do not have a slope. So let’s figure out what that is. X1 and Y1. X sub 2 and Y sub
2. Slope is Y sub two minus y sub one over x sub two minus x sub one. That is three.
That is four, and that is my slope. Now we are going to use that slope and one of these
points, and it makes absolutely no difference which one I pick. So I am going to pick the
first point just because it is the first thing I looked at. There is my point and there is
my slope and there is point slope form of a line. We are going to find the equation
of this line passing through this point using this slope. Y minus Y1 and the y of my point
is seven equals m, that is where this three fourths goes times x minus X1… my X1 equals
four. Now we have an equation with a fraction in it and a lot of people don’t like those
fractions. You can deal with it. This has a division of four and this has a value of
four so really it is not going to be a big problem at all. If however you are one of
those students that really really really do not like fractions you can get rid of it right
from the get go. This is probably not a good idea because this will cancel out, but lets
go ahead and multiply both sides by four. Now what is the purpose of this. A fraction
bar is division. If you have division that you don’t want, multiply it out. These parenthesis
are locking all of this together as one term so I only have to multiply by four once. When
I multiply by four these fours just cancel out. Over here the four is going to be distributed
through the parenthesis. We end up with four y minus, four times negative seven is negative
twenty-eight equals three times x minus four. I am going to run out of space again so I
am going to erase these standard forms here for you…for myself so I can write this out.
I am going to take the three and multiply it through the parenthesis. We get four y
minus twenty-eight equals three x minus twelve. I want to get y by itself. That is easy to
graph when you are in slope intercept form. Which I will have to do in another video.
This y I am going to isolate so I need to move this twenty-eight. It is minus twenty-eight
and whenever you want something to go over the equal sign you need to apply the inverse
operation, so I am going to add twenty-eight. You get four y equals three x plus sixteen.
And now I have a four that I do not want. So again if I want to graph this, if i want
to get this in slope intercept form, which is when y is by itself y=mx+b… so I am going
to divide everything by four. Y is equal to 3/4, there is the fraction that we got rid
of in the first place… it is coming back again, but for slope this is rise over run
and we want this fraction plus sixteen divided by four is four. This is our equation. We
started off with point slope form and what you just saw me do was convert this into slope
intercept form. This is very nice for graphing because we have the slope and the y intercept.
I think I am out of time. So, I will be back with you in a minute.

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