### Video Transcript

For the function π of π₯ equals π₯
squared minus four π₯ plus three, answer the following questions. Firstly, find by factoring the
zeroes of the function. Secondly, identify the graph of
π.

There are also two further parts to
this question. So firstly, weβre asked to find the
zeros of this function. And the method weβre told to use is
factoring. We therefore need to write our
quadratic as the product of two linear factors. As the coefficient of π₯ squared is
one, we know that the first term in each of our parentheses will be π₯. Weβre then looking for two numbers
whose sum is the coefficient of π₯, thatβs negative four, and whose product is the
constant term, thatβs positive three.

Well, the two numbers that fit both
of those criteria are negative one and negative three. Negative one plus negative three is
negative four, and negative one multiplied by negative three is positive three. So our quadratic factors as π₯
minus one multiplied by π₯ minus three, which we can of course confirm by
redistributing the parentheses if we wish.

We need to use this factored form
to determine the zeroes of the function, which we recall are the π₯-values such that
π of π₯ equals zero. If we set this factored form equal
to zero, we then recall that for the product of two things to be zero, at least one
of them must themselves be zero. So we can take each factor in turn
and set it equal to zero, giving two simple linear equations. The first can be solved by adding
one to each side to give π₯ equals one, and the second can be solved by adding three
to each side to give π₯ equals three. The roots or zeros of this function
then are the values one and three.

Now, in the second part of the
question, weβre asked to identify the graph of our function π. And we can see that weβve been
given three possibilities: a blue one, a red one, and a green one. Now, weβve just found that our
graph has zeros at one and three. And remember, these zeros are the
values of π₯ at which the graph crosses the π₯-axis. So if our graph crosses the π₯-axis
at one and three, we can see from the figure that this leaves just the red and green
graphs. The blue graph crosses the π₯-axis
or has zeros at values of negative one and negative three.

Now, we just need to decide between
the red and green graphs, which we see are mirror images of each other. One is an upward-curving parabola,
and the other is a downward-curving parabola. We recall that the type of parabola
we have will be determined by the value of π. Thatβs the coefficient of π₯
squared. In our function, the coefficient of
π₯ squared is one. Itβs a positive value, which means
the parabola will curve upwards. That means then that the graph of
our function π must be the red graph. It has the correct zeros and the
correct shape. We can also see that the
π¦-intercept of this graph is three, which is indeed the constant term in our
function π of π₯.

The remaining two parts of the
question, which I didnβt write down initially because they give the game away for
the previous part are. Write the equation for π, the
function that describes the blue graph. And write the equation for β, the
function that describes the green graph.

Letβs look at this blue graph first
of all then. We already said that it has zeros
at negative one and negative three. This means that in its factored
form, it has factors of π₯ plus one and π₯ plus three. But there could also be a factor of
π that we multiply by. To determine whether the value of
π is one or something else, we consider the π¦-intercept of the graph, which we can
see is the same as the π¦-intercept of the red graph. Itβs three. When we multiply these two factors
together, the constant term will be one multiplied by three, which is indeed
three. And so this tells us that the value
of π is simply one. Our function π in its factored
form then is π₯ plus one multiplied by π₯ plus three. If we distribute the parentheses,
we have π of π₯ equals π₯ squared plus four π₯ plus three.

For the green graph, it has the
same zeros as our function π. So it can be written as π
multiplied by π₯ minus one multiplied by π₯ minus three. And again, we need to determine
whether the value of π is one or something else. Well, the π¦-intercept for the
green graph is negative three. If we multiply together negative
one and negative three, we get a value of positive three. And so in order to ensure the
π¦-intercept, the constant term in the expanded form of β of π₯, is negative three,
we need the value of π to be negative one.

The equation β of π₯ then is
negative π₯ minus one multiplied by π₯ minus three. In fact, it is the complete
negative of our function π of π₯, which we can also see because they are
reflections of one another in the π₯-axis. We can write the equation β of π₯
then as the complete negative of our function π of π₯. β of π₯ is equal to negative π₯
squared minus four π₯ plus three.