HELLO, Mr. Tarrou
So, I have a lot of students that are struggling with graphing linear equations.
I want to do some of those examples with you today and make sure that next time it is not an issue.
Linear equations, just simple straight lines.
Now, it is not so simple if you are in Algebra 1 and you are watching this video.
So, lets take a look at
Graphing equations of lines in the most common form that we use which is called Slope Intercept Form.
Alright, so we have slope-intercept form
Now slope intercept form, you have y by itselft.
That is the biggie.
Almost every time you have an equation where y is alone
you are probably going to be in slope-intercept form
and that is y=mx+b
Note that there are no parenthesis in here.
So, if you have an equation where y is by itself but there are parenthesis in there
you will need to get rid of those () before you try and graph it.
We really like graphing in this form, because when y is alone
you are looking at b the constant.. the number that does not have or is not with the variable
this is your y intercept.
And m, or the number immediately in front of the x is your slope.
So slope is rise over run
There is also an equation attached to slope which we are not going to be using today.
But it is, y sub 2 minus y sub 1 over x sub 2 minus x sub 1
ok, so slope is rise over run. Your y axis is your up and down movement.
So when you subtract you will find your rise and your run value.
I like to teach graphing and teach slope as if you are reading a book.
So if your slope is positive,
you are going to be going up
for a positive slope
and down for a negative slope.
Now this is not an absolute requirement (reading left to right) I just think it makes this a little bit easier.
If you think of reading slope as left to right,
that will not leave any doubt that your run is always to the right.
We are going to be graphing some lines and when the slope is negative am going to be going down
when this slope is positive I am going up
but the bottom number I am going to hold as a consistent direction to the right.
The is the teaching method for graphing that I have found to be the easiest and clearest.
Lets do a couple of examples that are already set up solved for y.
or are y in terms of x.
Lets do a blue one here first.
y equals negative x plus 7.
Ok, this has a slope of negative 1 over 1.
Now I don't see a fraction there so I am going to write it as -1/1.
A lot of students are nervous with fractions
but with slope you need a rise and a run value.
You need to write slope as a fraction even it does not look like on in your problem.
Because the slope is negative
also note, if you did not know where the 1 came from in the first place,
If there is nothing written in front of an x or any variable for that matter, you always want to know there is a 1 there.
So this negative x or negative 1 x shows we have a slope of down 1 and over 1.
always to the right. And your y-intercept is b which is 7.
Now something else to think about in slope intercept form, when you are graphing
and we are doing these without the help of t-tables
is you always want to do
the y-intercept first and the slope second.
This line is going to cross the y axis at 7
1,2,3,4,5,6,7 mark a point. From there
because there is a negative in front of the x
you are going to go down 1 and over 1
down 1 and over 1, down 1 and over 1, this repetition is not needed
I am just trying to make sure my line stays relatively straight.
I am going down 1 and over 1 because of the slope of -1/1.
So here is my line.
GOOD!
Now what if the equation
did not already have y by itself?
Maybe it was 2y is equal to 3x minus 10.
This is close to having y by itself but there is a 2 there.
so the y is not completely isolated.
So you want to divide everything by 2
and get y is equal to 3 over 2 x and negative 10 divided by 2 is negative 5
Because I am running out of space I am going to erase this as I do the rest of my examples.
My slope
is in front of the x, always in front of the x, the slope is 3 over 2
NOT 3 over 2 x!!!
The slope is positive. See there is nothing out front so it is like a plus sign.
That means we are going to up 3 and over 2.
The y-intercept of -5 is what we are going to plot first.
Again, always the last number or the constant is your y-intercept
Always the number in front of x is your slope
and you do the y-intercept first and then the slope
You cannot graph a line by just looking at the equation
unless the y variable is already by itself... one more time.
This is crossing at a y value of negative five
1,2,3,4,5
It does have a slope of positive 3 over 2
so from this point, not the origin but from the y intercept
we are going to go up three and over two and make a new point.
and VIOLA!
We have a line.
Now if you are doing a system of equations
the system to that systems of equations would be that point where the two lines cross.
We are not, just thought I would throw that in for you.
OK, one more example.
A line where the equation has both an X and Y value on the same side.
How about negative 3y minus 4x equals negative 12.
Everything is negative!
Alright, so this is not going to be graphed without at least a little bit of scratch work.
This is equation is almost in Standard Form
and it would be good to graph this using x and y intercepts which I will do in a minute.
But I want to first do another example with slope intercept form.
This means the y value / variable needs to be by itself.
I am going to bring the 4x over with addition.
and get negative 3y equals 4x minus 12.
Now for you Algebra 1 students
remember, or please note, I am not doing negative 12 plus four and getting negative 8.
That is because this does not have a x and this term does.
These are not like terms.
So I can show these terms together on the right hand side of the equation
but I cannot actually put them together.
Divide everything by negative 3
and we get y is equal to -4/3 x plus 4.
Now you may note this negative here, when I divided everything by negative 3
the negative is in the bottom (denominator) of the fraction
and now I have written it on the top.
That is because it does not matter where you put the negative sign
you can put it anywhere you want as long as there is only one negative sign in the fraction.
Ok, so now y is by itself.
Now I can find the slope
which is in front of the x.
and that slope is negative 4/3
so we are going to go down 4 and over 3.
and the y intercept is always the constant usually at the end of your equation.
b, the intercept is four
And again, you want to do the y intercept first
and the slope second.
This is going to be a busy graph over here so I will be erasing it in a second.
But I think I can put one more on here.
This line crosses the y axis at four.
This line has a slope of - 4/3
so again rise over run means down four and the bottom should always be to the right.
so down 4 and over 3
so 1,2,3
and we have
this line line. Looks like it almost the same slope but they are not.
So that is 3 examples of how you graph lines in slope intercept form.
HA, I have a few minutes left.
Lets talk about what you do
...how you graph a line when the equation does not have both an x and a y variable.
Like how do you graph the equation x equals 4.
It is like there is not even enough there to do.
These questions as math teachers
we kind of think they are the easy ones, but
there is so little going on in the equation that they are missed more often.
X equals four. There is no x and y.
You are not sure what this looks like.
If you are graphing lines, just think of a couple of points that both have an x coordinate of four.
Like (4,0)
So, 4...0
That has an x coordinate of 4.
Think of another point with an x coordinate of 4 like (4,2)
4....2
That has an x coordinate of four.
How about (4,5)
4...2..3..4..5
See what I am doing is coming with points that all have an x coordinate of 4.
And, maybe I forget that x equals a number is a vertical line.
Maybe I forgot that x equals a number line, that is a vertical line, has a slope that is undefined.
Now the slope of zero is not undefined. zero is a value that actually exists.
Undefined slope is a number that does not exist.
There is a rise, but the run value is zero and you cannot divide by zero.
There is no slope. It is undefined.
What if you want to graph the line y equals 2.
Well, think of...lets just use the same graph again...
Think of a couple of coordinates that have a y value of 2 like (0,2)
Right, when you make a coordinate, coordinate have the values of (x,y)
When you make a coordinate with a y value of 2, make sure 2 is the second number.
So (0,2)
How about (2,2)
Over 2 and up 2.
What about 4 comma 2
(4,2)
So see, if you have forgotten that
Y equals a number is a horizontal line.
If you have forgotten that horizontal lines have a slope of zero,
Then you can come up with a couple of points, or a few points, that have a y coordinate of 2
and just connect the dots and get the right answer anyway.
So, there is graphing using slope intercept form,
y=mx+b
Do the slope and y intercept, make sure you do the y intercept first.
A reminder of what vertical lines are.
X equals a number and they have an undefined slope.
And horizontal lines with a slope of zero. So there you go...badda bing, badda boom.
That is graphing lines in slope intercept form.
The next lesson will be for x and y intercepts.
GO DO YOUR HOMEWORK!!!