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.