BAM! Mr. Tarrou We are going to continue our discussion about graphing linear equations,
linear functions. We are going to do this now instead of using slope intercept from
of y=mx+b, which is a really nice form because we can draw a graph just by looking at the
equation, but what if we have an equation in Standard Form and we don't feel like setting
it equal to y or our your teacher says that they want you to graph using x and y intercepts.
Let's take this equation 3x plus 7y equals 21. That equation is in Standard Form. Now
what Standard From, I will erase this in a minute, but Standard Form is when an equation
has.... We are talking about linear equations where all the exponents are equal to one.
Standard Form is Ax + By = C. A, B, and C are all integers. Some textbooks will even
tell you that the coefficient of x will have to be positive. Most textbooks that I have
seen have that condition in there about A being positive. The last book I looked at
didn't. Standard Form is when you have the equation, here is the equal sign, the x and
the y's are on the left hand side and the coefficients that you have in here, almost
every textbook that I have ever seen will say that those must be integers and no decimals
or fractions are allowed. Here we have an equation in Standard Form and really for graphing,
I don't really care so much that it is standard form, I just know that the y is not alone.
If y was isolated, then we could just use the slope and the y intercept because it would
be in slope intercept form or very close. But, this one is not. How do you find an x
intercept? Think of that is where the graph crosses the x axis. If you touch a point on
the x axis, name off the coordinate. That is (3,0), here is a point on the x axis and
that is (-5,0). Here is another point. I don't know what the x coordinate is of that point,
but I know that y coordinate is zero. If you have a line coming through and crossing the
x axis, that is an x intercept. If you are on the x axis, the y has to be zero. So, if
you want an x intercept just take your equation even if it is in slope intercept form, and
it does not even matter what kind of an equation it is...you might be graphing something to
the fourth degree...if you plug in y equals zero you will find an x intercept. These are
just linear functions though, so we have 3x plus 7 times 0 equals 21. Seven times zero
is zero. When you add by zero nothing changes so now I don't need to write that. 3x equals
21 and when I divide by three, I get x is equal to seven. So this line has an x intercept
at seven, so 1, 2, 3, 4, 5, 6, 7. Now, let's take a look at finding the y intercept. Well,
if you don't already know how to find the y intercept just think about it. If the line
is coming through here, and I don't know if the line is going to go up or down, but you
have line coming through and you are crossing the y axis. Well, if you are on the y axis,
what is the x coordinate have to be? This coordinate is (0,4), this coordinate is (0,1),
this coordinate is (0,-3), (0,-5), (0,-9). If I am touching the y axis, there has been
no left or right movement from the origin so the x has to be zero. So, for an x intercept
you let, excuse me... For a y intercept you let x equal zero. Plug it into your equation.
Three times zero plus seven y equals 21. Now, 7y equals 21. Divide both sides by 7 and you
get y equals 3. What is that?! Anyway, it looks like a muffin. This equation is going
to cross the x axis at seven, this equation, this linear equation with exponents of one,
is going to cross the y axis at three. There we go. There is our line. Not too bad:) I
was going to say something else but forgot. Ok, that is the line...OHH. You only need
two points to define a line. A lines slope never changes, so two points are sufficient
for graphing a line and that is what we have in the x and y intercepts. One more example.
Let's take a look at negative five x plus two y equals twelve. Alright, x intercept.
Remember what that is? Let y equal zero. For an x intercept, we are going to let y equal
zero and get -5x plus 2 times 0, see I am taking the y out and replacing it with zero,
equals 12. We get -5x after that cancels out equals 12. We divide both sides by a negative
five and x is equal to.... Ok, how many fives are in twelve? That is five times two is ten,
so it is negative two and twelve minus ten is two, so the remainder is two and we are
dividing by five...so this is negative two and two-fifths. That is great if your teacher
is reinforcing fractions, but that is not so great for graphing. Where is 2/5? Well,
two divided by five is .4 so x equals -2.4. That is going to be our x intercept. Zero,
negative one, negative two, and I don't want to go to negative three, the point is near
the middle of -2 and -3. There is that x intercept. What about the y intercept? If you set y equal
to zero for the x intercept, you let x equal zero for the y intercept. The y intercept
is when x equals zero. That is negative five times zero for x plus two y equals twelve.
Now 2y = 12 because multiplying negative five by zero gives you zero, and again when you
add by zero nothing happens so we don't need to write it. You undo multiplication with
division. We are trying to solve for y. We are trying to get two away from y and get
it to the other side. The inverse of multiplication is division so we divide both sides by two
and y equals six. Ok, so this now, this blue equation is crossing the x axis at negative
two, it is crossing the y axis at six, 1, 2, 3, 4, 5, 6. Voila, oops the point a little
bit there, we have another line. If your were solving linear system of equations, that might
be coming up in your algebra book, solving a system of equations is finding where those
two lines cross. It looks like here it as (-1,3.5), but that is just an estimate and
I am sure it is not exactly right. Even though I said this section was graphing by using
x and y intercepts, let me just remind you that if you are solving an equation that has,
excuse me, not solving but graphing an equation that only has one variable such as x = -2.
x equals negative two, I just said in the last lesson, if you don't remember pick a
couple of points that have x values of negative two and see that it is a vertical line through
x equals negative two. If you are asked to graph the line y = -4, y equals a number is
a horizontal line. So, just go to where y equals negative four, 1, 2, 3, 4, and draw
that horizontal line. One more time, if you don't remember that y equals a number is horizontal,
just make a coupe of points that have y values of negative four and you will will have your
graph. Alright. So that is graphing linear functions by using x and y intercepts. WHOOO!
Go Do Your Homework!