Video Transcript
Which of the following functions is
graphed in the given figure? Option (A), (B), (C), (D), or
(E).
We’re given a graph, and we need to
determine which of five options is this a graph of. And we can see that all five of our
options are polynomial functions. So we can use what we know about
graphing polynomials to help us answer this question.
The first thing we can notice about
our curve is its 𝑦-intercept is positive. It’s above the 𝑥-axis. And if we say that this is a curve
𝑦 is equal to some function 𝑓 of 𝑥, then the 𝑦-intercept happens when 𝑥 is
equal to zero. In other words, we know that 𝑓
evaluated at zero is positive. So we could substitute 𝑥 is equal
to zero into all five of our expressions to see which ones give us positive
outputs.
For now though, let’s just keep in
mind our 𝑦-intercept is positive. Instead, we’ll notice that there
are two 𝑥-intercepts: one at negative five and one at 𝑥 is equal to five. And remember, at 𝑥-intercepts, our
function is outputting zero. And by using the remainder theorem,
if our polynomial is outputting zero when 𝑥 is equal to five, it must have a factor
of 𝑥 minus five. In fact, it could have
multiple. And the exact same will be true at
negative five. Our function must have some factors
of 𝑥 plus five. And these are the only
𝑥-intercepts. There’s no more linear factors.
However, it is worth pointing out
there could’ve been some extra polynomial expression with no roots. And this on its own doesn’t help us
answer this question because we can see all five of our options are in this
form. However, we can combine this with
what we know about the multiplicity of our function near its roots.
Let’s start by looking at the shape
of our function near 𝑥 is equal to five. If we look closely, we can see this
is very similar to a straight line. We could say that this shape is of
the form 𝑦 is equal to negative 𝐴 times 𝑥 minus five to the first power or just
𝑥 minus five. The important thing to notice here
is, though, that this is a straight line. And the shape of the graph for
these roots must match the multiplicity of our function. In other words, because we have a
straight line when 𝑥 is equal to five, the multiplicity of 𝑥 minus five must be
one.
So let’s take a look at our five
options. We can remove all of the options
where the multiplicity of 𝑥 minus five is one. In option (A), the multiplicity of
𝑥 minus five is three. So option (A) can’t be the correct
answer. Both options (B) and (C) have a
multiplicity of one for 𝑥 minus five. So these could be correct. However, in option (D), its
multiplicity is three. And in option (E), its multiplicity
is two. So neither of these can be
correct.
Now we’re ready to use our
information about the 𝑦-intercept being positive. We’ll just substitute 𝑥 is equal
to zero into both option (B) and option (C). Substituting 𝑥 is equal to zero
into option (B), we get zero plus five all cubed times zero minus five. And if we were to evaluate this, we
would get negative five to the fourth power or negative 625. And this is a negative answer, so
it should appear below the 𝑥-axis on our diagram. Therefore, option (B) can’t be the
correct option.
And for due diligence, let’s check
option (C) in its entirety. We can substitute 𝑥 is equal to
zero into the function in option (C) to find its 𝑦-intercept. If we do this, we get negative one
multiplied by zero minus five all cubed times zero plus five. And if we evaluate this, we get
five to the fourth power or 625.
And therefore, of the options
listed, this can only be a sketch of the graph of the curve 𝑦 is equal to negative
one times 𝑥 plus five all cubed times 𝑥 minus five.
