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# Intro to parabolas

Graphs of quadratic functions have a U shape called "parabola." Here, Sal graphs f(x)=-3x²+8 by creating a table of values and plotting the resulting points. Created by Sal Khan and Monterey Institute for Technology and Education.

Video transcript

We are asked to
graph the function f of x is equal to
negative 3x squared plus 8. So we'll do this by essentially
trying out different points for x, and seeing what
we get for f of x, and then graphing it. But the first question
I have for you is, just looking at
this function definition for f of x, what type
of graph will this be? Will just be a line? Will this be a parabola? Will this be something else,
a circle, something else, maybe something else
bizarre or strange? Well, this is pretty clearly
going to be a parabola here. You have the
function is defined, it's negative 3x squared, so
you have this second degree term here. You don't have any x thirds or
x to the fourths or anything else bizarre, so this is
going to be a parabola. Now, the other thing
that we could think about is whether the parabola is
going to open up like that or whether it's going
to open down like that. And just looking at this
function definition, do you have any intuition of
whether it's going to open up or it's going to open down? Well, if you look
at the coefficient on the x squared
term, the negative 3, that tells you that this
parabola is going to open down. It's going to open down. So with that
intuition now that we know it's going
to be a parabola, we know it's going to
open down, let's actually try to graph the thing. And let me draw some axes here. So let's say that this is my
x-axis, so that's my x-axis. And then let's make this right
over here, this is my y-axis. And let me make
a table of values and see what values
f of x takes on. So on one column, I'm
going to do my values for x and over on
the right I'm going to do my values for f of x. And then we can
plot these things. And actually I want to
take all of these values before I draw the
scale on these axes, so I know what might be
an appropriate scale. So I'm just going to
try a bunch of values. So let's try first what happens
when x is equal to negative 2. So when x is equal
to negative 2-- and I'm just
picking numbers that will be relatively
easy to compute. When x is equal to
negative 2 what's f of x? Well, f of x is going
to be negative 3, this negative 3,
times negative 2 squared plus 8, which is going
to be equal to, let's see. Negative 2 squared
is 4, positive 4, then we multiply that times
a negative 3, which gives us negative 12 plus 8,
gives us negative 4. Let's try another point. Let's see what happens when
x is equal to negative 1. What do we get for f of x then? Well, f of x is going to be
negative 3 times negative 1 squared plus 8. So that's going to be-- see
negative 1 squared is just 1, and then that times
negative 3 is negative 3. Negative 3 plus 8 is 5. Now, what does f of x
equal when x is equal to 0? Well, this is pretty
easy to compute. When x is equal to
0, you get negative 3 times 0 squared,
which is equal to-- and we could write that
either way-- negative 3 times 0 squared plus 8. Well, this just simplifies to 0,
and so you're just left with 8. Now, let's see what happens
when x is equal to 1. What do we get for f of x? Well, it's going to be negative
3 times 1 squared plus 8. So 1 squared is just 1,
negative 3 plus 8 is equal to 5. And then finally, what do we get
when x is equal to positive 2? What does f of x equal,
or another way of thinking about it, what is f of 2? Well, let's think about it. You get negative 3
times 2 squared plus 8. 2 squared is 4, times
negative 3 is negative 12, plus 8 is equal to negative 4. So let's see if
we can plot this. So the x values that I picked go
from negative 2 to positive 2. So let's make this
negative 2, negative 1. This is 0. This is positive 1, and
that could be positive 2. And then our f of x values,
or we are essentially graphing y is equal to f of x,
so I can even say this is going to be the graph of y
is equal to f of x. Our f of x values take on
things between negative 4 and positive 8. Let me try to draw that. So if this is positive 8, that's
positive 8, that is positive 4, and this is negative 4. This is negative 4. And if that's positive 4,
then this is positive 6, and then that right
there is 5, that is 7, this would be 2 that would be
3, and then that would be 1. Now, let's graph the points. When x is negative 2,
f of x is negative 4. And actually I
could say, this is the y is equal to f of x-axis. I'm going to plot f of x. I'm graphing, and
this is going to be the graph of y is
equal to this function. So let's graph
negative 2, negative 4. So that gets us,
when x is negative 2, f of x is negative 4. It's right over there. When x is equal to negative
1, f of x is equal to 5. And we're saying that y is
equal to f of x in this context. When x is 0, f of x or y-- I
could even write over here, I could say, y is
equal to f of x. When x is equal to
0, our f of x is 8. x is 0, f of x is 8. When x is 1, f of x is 5. When x is 1, y
equals f of x is 5. And then finally, when
x is equal to 2, f of x is equal to negative 4. So 2, negative 4,
gets us right there. And now we can connect the dots. We know this is going
to be a parabola. And I will do it in blue. So my best attempt-- I like
to draw it as a dotted line, just because it's
easier to not mess up-- so it would look
something like that. And it keeps on
going just like that, and then I can actually make the
line a little bit more solid. So we see that we
definitely got a parabola, and just as our
intuition told us, our ability to inspect
the coefficient on the x-squared term told us,
that our parabola is indeed opening downwards.