NARRATOR: Exercise 10.
Sketch the graph of y = x to the power of 4 – x squared.
I'm first going to calculate the intercepts.
So we'll start with the y-intercept,
which is obtained by making x equal to 0,
so we end up with y = 0.
So we have a y-intercept at (0, 0).
Then we'll consider the x-intercepts.
And we're expecting to get one x-intercept at 0,
given that we know the graph goes through (0, 0) already.
So x-intercepts are obtained by making y equal to 0.
So 0 = x to the power of 4 minus x squared.
x squared is a common factor there
so we have x squared (x squared – 1).
And then using the null factor law, we know that x squared = 0, so x = 0,
and we also know that x squared – 1 equals 0
so x squared = 1.
So x = positive or negative square root of 1,
which is positive or negative 1.
So we know we have x-intercepts at
(0, 0), (1, 0) and (-1, 0).
We also know due to
the repeated factor of x squared
that we have a turning point
at this intercept.
So we're starting to get a sense of the shape of the graph.
We know that it has x-intercepts
at –1, at 0 and at 1.
We also know that it's a positive quartic
due to the + x to the power of four term in the equation,
which means that the graph is going to start up and also finish up.
So we're getting this sort of a shape occurring.
So something like that.
So now we need to also determine
the coordinates of these two stationary points.
We already know that we have the stationary point here at (0, 0),
so we'll expect to find that, but we need to find the other two as well.
So for the stationary points,
we know that stationary points occur where the derivative is 0.
So we first need to find the derivative of this function.
So dy on dx is equal to 4x cubed – 2x.
And for stationary points, we need that derivative to equal 0.
So we have a common factor of 2 here
that we're going to divide through by,
And then we have a common factor of x that we can factor out.
So now using the null factor law,
we know that either x = 0 or 2x squared – 1 = 0.
So 2x squared = 1.
x squared equals a half.
So we know that x is equal to
the positive or negative square root of a half,
which is positive or negative 1 on root 2,
which is equivalent to positive or negative root 2 on 2.
So we need to work out the corresponding y-coordinates.
We already know that the turning point at 0
has coordinates of (0, 0),
but let's think about these other two.
So we know that when x =...
And I'm gonna substitute square root of a half
because that's gonna make the simplification easier.
When x = square root of a half,
y is equal to square root of half to the power of four
minus square root of half squared.
So that's half squared minus half.
So a quarter minus a half,
which means we have negative a quarter.
Which matches the picture that we saw earlier as well.
So we also know we need to think about
substituting negative square root of a half,
but we're not actually going to get any different result.
So I know if I had negative square root of a half here,
that would give me
negative square root of a half and negative square root of a half there.
And then because of these even powers - power of 4 and power of 2 -
those negatives aren't going to make any difference
and we'll still end up with a y-coordinate at negative a quarter.
So that means that we have turning points with coordinates
(root 2 on 2, negative a quarter)
and also (negative root 2 on 2, negative one quarter).
So we have three stationary points.
So now we can finalise our graph.
So here's our graph with our intercepts at –1, 0 and 1.
We now also have coordinates for these two stationary points.
This is (root 2 on 2, negative one quarter).
And this is (negative root 2 on 2, negative one quarter).
And we can fill this one in as well with coordinates of (0, 0).
And there is the graph of y = x to the power of 4 – x squared.