What I want to do in this video
is to actually use a
graphing calculator, get some
experience on the calculator
itself, for graphing some of
these quadratics and actually
figuring out the roots
visually by
looking at those graphs.
So the first equation I want to
find the x-intercept for is
y is equal to negative 3x
squared plus 4x minus 1.
So we're just going to go
straight to the calculator.
We'll get our TI-85 out here.
And there are obviously better
things you can use, especially
on your computer, that are going
to be much more visually
interesting and in color and
even faster, but many of us
have a TI-85 at our disposal.
The TI-85 hasn't changed much
actually since I was in high
school, or the TI-83 or whatever
else there is out
there, TI-92.
And this at least will give you
a sense of how to use it,
because this is what you have,
and how to analyze the graph
once you are able to graph it.
So let's-- I already turned it
on-- so let's input this graph
right here.
So we click on GRAPH and then
we'll click on-- this F1 tells
us to pick this option right
here, this y of x, right?
So they're saying y is a
function of x, y is indeed a
function of x.
So let's click there, and then
that's going to be negative
3x-- let's see where
is the variable--
there's the x variable.
x squared plus 4x, minus 1.
There is our function.
Now let's graph it.
So we want to pick this
choice up here.
So to pick this second choice--
so notice, if you
want to pick any of these
choices on the bottom row, you
can just punch in
these numbers.
But if you want to pick the
choice that's above, you could
press 2nd, and then press that,
and then 2nd tells you
to pick the thing that's
right above that.
so I'll pick the thing in
the yellow, essentially.
So I'll do 2nd.
And I'm selecting the
GRAPH option.
When I was in high school, this
was all very fancy stuff,
but now compared to pretty much
most cell phones, this is
pretty archaic.
But it gets the job done.
Now, let's zoom in, because
I really can't see
the 0's that well.
So let me zoom here.
I like to use the box, we zoom
in on a certain area that I
can define.
So, I'll select the box, and
then I select where I want to
start the box for using
the arrow keys.
Press ENTER, that tells it to
start the box, and then I can
zoom in using the box.
I define the box, and
when I have a box I
like, I press ENTER.
And it'll redefine the range
of x and y values that it
calculates the graphing
function.
Now, I want to figure out what
these x-intercepts are.
So I'm going to use the
trace function.
And the trace function
is just going to
walk along this graph.
So let me hit 2nd, trace.
And now I'm walking along this
graph as I go to the right.
It just increments
the x-value.
We're getting closer and closer
to y is equal to 0.
And it looks like it happens
right at about 1.
Right as I cross 1, I'm crossing
y is equal to 0.
y is a very small positive
number here, and then it goes
to becoming a less small, but
still a small negative number,
right as we cross 1.
I'm going to evaluate it
directly at x is equal to 1.
But, I think that's a pretty
good approximation, and we can
even try it out in
our equation.
Let's try it out.
If x is equal to 1-- let
me write it this way.
y of 1, if we write y is a
function of 1, this is going
to be equal to negative 3 times
1 squared, plus 4 times
1, minus 1.
So this is equal to negative
3 plus 4, minus 1,
which is indeed 0.
So the point x is equal to 1, y
is equal to 0 is definitely
on the graph, and this is
one of our x-intercepts.
Let's see if we can figure out
the other one using our
calculator.
So let's see, we need to figure
out this other point
right here.
Let's get the zoom box going.
Actually, let's try just
to trace there.
Scrolling all the way.
And it looks like it's
around 0.33.
I suspect it's close to 1/3,
but let me zoom in a little
bit better.
So let me click GRAPH again,
and click on zoom.
I'm going to use
the box again.
And then let me do the top left
corner of the bottom, and
zoom in really narrowly on
that point right there.
Really, really zoom in.
So it's around 0.33 that
I want to zoom in.
So that's going to be the
beginning of my box, and let
me go right below the y-axis and
then go right above 0.33.
And so it's really
zoomed in on that
y-intercept right there.
So now if I trace it, I really
should be able to figure out
what that value is.
So let me do 2nd, and trace,
and keep scrolling.
So it's a little bit
more than 0.33.
It looks like it's really
approaching
0.3333333, which is 1/3.
It looks like it's
0.3 repeating.
Then we're getting to very,
very small y-values.
So it looks like 1/3, so let's
try out 1/3 and see if we got
the right answer.
So, if we have y of 1/3 is equal
to negative 3 times 1/3
squared, plus 4, times
1/3, minus 1.
This is equal to-- negative 3
times 1/9 is negative 1/3.
This is plus 4/3, and then
minus 1 or negative 1,
depending on how you view it,
it's the same thing as
minus 3 over 3.
So this is going to be negative
1 plus 4, minus 3,
all of that over 3,
which is indeed 0.
So we also have the
x-intercept.
x is equal to 1/3,
y is equal to 0.
Let's do another one.
Let's say we have y is
equal to 1/2x squared
minus 2x, plus 3.
Let's see if we can get its
roots, or the 0's, of this
equation, where it intersects
the x-axis.
So let's get our calculator
back.
And we're going to have to
zoom out of this thing.
So let me do zoom.
I think I can just use
the previous zoom.
Well, the previous zoom
is still a little bit
not too zoomed in.
Let me just do the previous
zoom before that.
And then, I'll do another
zoom-- oh, it just keeps
alternating between the zooms,
so maybe I have to clear out.
I am not an expert at this.
All right.
So let me just input
my new graph.
So it's 1/2x squared.
I could write that as 0.5x
squared minus 2x, plus 3.
And then I want to graph this.
Oh, so it's on that super-zoomed
in version, so
let me zoom out.
There you go, zoom standard,
that resets it.
There you go.
That's all I had to do,
the standard zoom.
So you see.
Let's see, where does it
intersect the x-axis?
Well, it doesn't intersect
the x-axis.
It just goes down here,
then keeps going up.
So this quadratic function
right here will have no
x-intercepts.
I mean, you can see it
graphically here.
It does not intersect
the x-axis.
So, if you were trying to
solve the equation 1/2x
squared minus 2x, plus 3 is
equal to 0, what you're
essentially trying to do is say,
what x-values give me y
is equal to 0 here?
We just graphed it, it never
intersected the x-axis.
It just did something
like this.
It never equaled y
is equal to 0.
So this right here will have
no real solutions.
And I use the word real in
the mathematical sense.
There no real number
solutions.
We'll see, not too far off in
the future, that there are
things called complex numbers,
and those can be solutions to
quadratics that don't intersect
the x-axis, at least
in kind of the real domain.
Now let's do one
more of these.
Let's say we have x squared plus
6x, plus 9 is equal to 0.
And we should actually be able
to factor this in our head.
What two numbers add up to
6 and when you multiply
them you get 9?
It probably jumps out
of your head.
It's 3 and 3.
So this is x plus 3, times
x plus 3 is equal to 0.
Or-- and we just have to use
one of these-- x plus 3 is
equal to 0.
Subtract 3 from both sides.
x is equal to negative 3.
So this has one 0.
So how do you think this is
going to look if it only
intersects the x-axis at
exactly one point?
Well, the best guess is is that
that one point is going
to be the vertex.
So let's graph it just to
verify for ourselves.
So let me to clear this one
out, and we have x squared
plus 6x, plus 9.
And we want to graph it.
And there you go.
The vertex is right at x is
equal to negative 3, y is
equal to 0.
It just hits it right
over there.
So our intuition was correct.
And just as a bit of review--
and I should have pointed this
out when I was showing you the
other graphs-- notice, this
has a positive coefficient
on x-squared.
It is upward-opening.
The second problem we did also
had a positive coefficient.
It didn't intersect the x-axis,
but it was also
upward-opening.
And if you rewind this video to
the very first problem we
did, you saw it had a negative
coefficient, and it also had a
downward-opening parabola, or
downward-opening u-shape.
Well, anyway, hopefully you
found this little practice
with the graphing calculator
helpful.
