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CCSS.Math: , , , ,

We're asked to graph
the following equation y equals 5x squared
minus 20x plus 15. So let me get my
little scratch pad out. So it's y is equal to 5x
squared minus 20x plus 15. Now there's many
ways to graph this. You can just take three
values for x and figure out what the
corresponding values for y are and just graph
those three points. And three points actually
will determine a parabola. But I want to do something a
little bit more interesting. I want to find the places. So if we imagine our axes. This is my x-axis. That's my y-axis. And this is our curve. So the parabola might
look something like this. I want to first figure out where
does this parabola intersect the x-axis. And as we have already
seen, intersecting the x-axis is the same thing
as saying when it does this when does y equal
0 for this problem? Or another way of
saying it, when does this 5x squared
minus 20x plus 15, when does this equal 0? So I want to figure
out those points. And then I also want to
figure out the point exactly in between, which is the vertex. And if I can graph
those three points then I should be all set with
graphing this parabola. So as I just said,
we're going to try to solve the equation 5x
squared minus 20x plus 15 is equal to 0. Now the first thing
I like to do whenever I see a coefficient out here on
the x squared term that's not a 1, is to see if I can
divide everything by that term to try to simplify
this a little bit. And maybe this will get us
into a factor-able form. And it does look like every
term here is divisible by 5. So I will divide by 5. So I'll divide both sides
of this equation by 5. And so that will give
me-- these cancel out and I'm left with x squared
minus 20 over 5 is 4x. Plus 15 over 5 is 3 is
equal to 0 over 5 is just 0. And now we can attempt to
factor this left-hand side. We say are there two numbers
whose product is positive 3? The fact that their
product is positive tells you they both
must be positive. And whose sum is negative
4, which tells you well they both must be negative. If we're getting a
negative sum here. And the one that
probably jumps out of your mind--
and you might want to review the videos
on factoring quadratics if this is not so fresh-- is
a negative 3 and negative 1 seem to work. Negative 3 times negative 1. Negative 3 times
negative 1 is 3. Negative 3 plus negative
1 is negative 4. So this will factor out as
x minus 3 times x minus 1. And on the right-hand
side, we still have that being equal to 0. And now we can think about what
x's will make this expression 0, and if they make
this expression 0, well they're going to
make this expression 0. Which is going to make
this expression equal to 0. And so this will be true if
either one of these is 0. So x minus 3 is equal to 0. Or x minus 1 is equal to 0. This is true, and you can
add 3 to both sides of this. This is true when
x is equal to 3. This is true when
x is equal to 1. So we were able to figure
out these two points right over here. This is x is equal to 1. This is x is equal to 3. So this is the point 1 comma 0. This is the point 3 comma 0. And so the last one
I want to figure out, is this point right
over here, the vertex. Now the vertex always
sits exactly smack dab between the roots,
when you do have roots. Sometimes you might not
intersect the x-axis. So we already know what its
x-coordinate is going to be. It's going to be 2. And now we just have
to substitute back in to figure out its y-coordinate. When x equals 2, y is going to
be equal to 5 times 2 squared minus 20 times 2 plus 15,
which is equal to-- let's see, this is equal to 2 squared is 4. This is 20 minus 40 plus 15. So this is going
to be negative 20 plus 15, which is
equal to negative 5. So this is the point
2 comma negative 5. And so now we can go
back to the exercise and actually plot
these three points. 1 comma 0, 2 comma
negative 5, 3 comma 0. So let's do that. So first I'll do the vertex
at 2 comma negative 5, which is right there. And now we also know
one of the times it intersects the
x-axis is at 1 comma 0. And the other time
is at 3 comma 0. And now we can check our answer. And we got it right.