Solving Quadratic Equations by Graphing

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Solving Quadratic Equations by Graphing

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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.

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