### Video Transcript

Which of the following graphs
represents π of π₯ equals negative one times π₯ minus one squared plus two? Graphs (a), (b), (c), and (d).

In this question, we are given a
quadratic function π of π₯ and asked to determine which of four given graphs
represents the graph of this function. There are many different ways we
can answer this question; we will go through two of these. Letβs start by determining
information about the graph of π of π₯ from its definition.

We can start by noting that π of
π₯ is a quadratic function, so we know the shape of its graph will be a
parabola. We can also note that if we
expanded the expression, we would find that the coefficient of π₯ squared is
negative, so the parabola must open downwards. This allows us to eliminate graphs
(a) and (d) since they open upwards.

We can determine which of the
graphs (b) or (c) is correct by finding the coordinates of a point on the graph. We will check when π₯ equals zero,
since we can see that both of these graphs have different π¦-intercepts. We substitute π₯ equals zero into
the function π of π₯ to obtain π of zero equals negative one times zero minus one
squared plus two. If we evaluate this expression,
then we get one. This means that the π¦-intercept of
the graph is at one. So, we can say that graph (c) is
not possible since its π¦-intercept is at negative three. Hence, the correct graph must be
graph (b).

This method requires us to be given
options to eliminate in order to identify the correct graph. We can also sketch this graph by
constructing a table of values. We have already calculated that π
evaluated at zero is equal to one. If we evaluate π at one by
substituting π₯ equals one into the function, we obtain two. Similarly, when π₯ equals two, we
can calculate that π of π₯ is equal to one.

We can follow this same process to
fill in the rest of the values in this table. This gives us the coordinates of
five points on the graph. We can then notice something
interesting about the table of values we have constructed: the outputs of the
function when π₯ equals zero and π₯ equals two are the same.

We can then recall that all
parabolas have a line of symmetry through a point called their vertex. We can note that the line of
symmetry will be halfway between these two points with the same output. This is when π₯ equals one. We can then use this to sketch the
parabola passing through these points with vertex at the point one, two. We can then see that this matches
graph (b).