### Video Transcript

Which of the following is the
equation of the rational function graph below? Is it (A) π of π₯ is π₯ plus four
times π₯ plus one times π₯ minus one times π₯ minus three? (B) π of π₯ is one fourteenth π₯
plus four times π₯ plus one times π₯ minus one times π₯ minus three plus 0.5. (C) π of π₯ is π₯ plus four times
π₯ plus one times π₯ minus one plus 0.5. Or (D) π of π₯ is π₯ minus four
times π₯ plus four times π₯ plus one times π₯ minus one times π₯ minus three plus
0.5.

Weβve been given the graph of a
rational function π of π₯. Before looking at our options for
the equation of this graph, letβs just think what we do know about it. Itβs a sort of W shape. It has one, two, three turning
points. Now, one thing we know about
polynomial functions is that the maximum number of turning points of a polynomial
function is always one less than the degree of the function. And this means the degree of our
function is at least four. Itβs four and above.

Now, the degree of the function
also tells us the general shape. Weβve already said that the degree
is four or above. Now, we know that a function with a
degree of four has either a W or an M shape. Now, the sign of the coefficient of
the leading power of π₯ will tell us if itβs a W or an M. And the shape of a graph with a
degree of five is as shown. Once again, this very much depends
on the sign of the highest power of π₯ . Now, we said our graph has a W shape, so
its degree must indeed be equal to four.

Weβre now going to go back to the
equations that weβve been given in this question and see if we can eliminate
any. Letβs look at (A). π of π₯ is π₯ plus four times π₯
plus one times π₯ minus one times π₯ minus three. Now, letβs imagine we were to
distribute these parentheses or expand our brackets. Eventually, we would end up
multiplying π₯ by π₯ by π₯ by π₯, giving us π₯ to the fourth power. So the function could indeed be
function (A).

Similarly, distributing the
parentheses for function (B) also gives us π₯ to the fourth power as the highest
power of π₯. So it could be function (B). If we were to distribute the
parentheses for function (C), however, weβd be multiplying eventually π₯ by π₯ by
π₯. Now, that actually gives us π₯
cubed. And so we can eliminate function
(C) from our question.

Finally, if we were to distribute
the parentheses from function (D), weβd be multiplying π₯ by π₯ by π₯ by π₯ and then
by π₯ again, giving us π₯ to the fifth power. And so weβre going to eliminate
function (D).

Now, weβre simply left with
choosing between function (A) and function (B). And so next, we could consider the
roots or the zeros of our equation. The roots of our equation
correspond to the π₯-intercepts of our graph. Now, actually, these arenβt
particularly nice to define from the graph. If we try to read the π₯-intercepts
off of our graph, we get a roughly π₯ is equal to negative 3.9, negative 1.3, 1.3,
and roughly 2.8. Letβs compare these to our final
two equations.

To find the roots or the zeros of
our equation, we set the equation equal to zero and solve for π₯. So for equation (A), thatβs π₯ plus
four times π₯ plus one times π₯ minus one times π₯ minus three equals zero. Now, each binomial, each expression
inside our parentheses, is itself a number. And when we multiply these four
numbers together, we get zero. So what does that tell us about any
one of these numbers? Well, it tells us that at least one
of these numbers must itself be equal to zero. So we could say that either π₯ plus
four is equal to zero, π₯ plus one is equal to zero, π₯ minus one is equal to zero,
or π₯ minus three is equal to zero.

Solving for π₯ in our first
equation, we subtract four, in our second, we subtract one, and so on. And we find the zeros of this
equation to be π₯ equals negative four, π₯ equals negative one, π₯ equals one, and
π₯ equals three. Now, we actually see that these do
not correspond to the π₯-intercepts that weβve read from our graph. And so (A) cannot be the correct
graph. And it therefore must be (B).

Now, we would have a little more of
a job trying to solve the equation of fourteenth π₯ plus four times π₯ plus one
times π₯ minus one times π₯ minus three plus 0.5 equals zero. Weβd have to distribute our
parentheses, refactor, and then solve for π₯. But there is one other hint that
can tell us that graph (B) is indeed the correct graph. We know that, for a polynomial
function, the constant value will tell us the value of the π¦-intercept. So letβs imagine we were
distributing our parentheses. What would our constant be?

Well, if we were just multiplying
the constant inside our parentheses, we get 12. We then see that we multiply that
by a fourteenth and add 0.5. That gives us 1.357 and so on. And whilst itβs difficult to read
from the graph, we can see that this is indeed the value of our π¦-intercept. And so the equation of the rational
function weβve been given is (B). Itβs π of π₯ equals a fourteenth
times π₯ plus four π₯ plus one times π₯ minus one times π₯ minus three plus 0.5.