### Video Transcript

Which of the following graphs
represents the equation ๐ of ๐ฅ equals negative two ๐ฅ squared plus nine ๐ฅ
minus seven?

There are two ways we can
answer this question. But the first thing that weโre
going to do is establish whether the graph of our function opens upwards or
downward. Firstly, we know that if we
have the equation ๐ of ๐ฅ equals ๐๐ฅ squared plus ๐๐ฅ plus ๐ and the value
of ๐ is positive, the graph opens upward. If itโs negative, it opens
downward.

Now, for our function, the
value of ๐ is negative two. So itโs negative. This means the function opens
downward. And we can therefore disregard
any functions that open upward. Thatโs the function represented
by graph (B) and the one represented by graph (E).

Now that weโve done this, weโre
going to plot a table of values to identify the points through which this graph
passes. Weโll need to do this for a
variety of values of ๐ฅ. Letโs start with ๐ฅ equals
zero. Thatโs ๐ of zero equals
negative two times zero squared plus nine times zero minus seven. And thatโs negative seven. So when ๐ฅ is equal to zero,
the value of the function is negative seven. And that actually means that
the ๐ฆ-intercept of our function is negative seven.

But of course the other
technique we couldโve used is to identify that for the function ๐๐ฅ squared
plus ๐๐ฅ plus ๐, the ๐ฆ-intercept has coordinates zero, ๐. Either technique here,
substituting or identifying the feature of the graph, is equally acceptable.

Weโre now going to choose some
positive and negative values of ๐ฅ to satisfy each of the graphs weโve been
given. First, letโs substitute ๐ฅ
equals negative three into the function. We get negative two times
negative three squared plus nine times negative three minus seven. Thatโs negative 52. Similarly, letโs substitute ๐ฅ
equals negative two. When we do, we find that ๐ of
negative two is negative 33. And that means our graph must
pass through the point negative two, negative 33. Substituting the remaining
values, and we see that ๐ of negative one is negative 18, ๐ of one is zero, ๐
of two is three, and ๐ of three is two. Plotting the values that we can
on our graphs, and we see we disregard option (C) completely. We notice itโs the wrong side
of the ๐ฆ-axis. And in fact the graph that does
pass through these points is graph (A).

Now, in fact, there was an
alternative method here. We began by identifying the
shape of the graph. We saw that it opened
downward. And we found the value of its
๐ฆ-intercept. We could also have set ๐ of ๐ฅ
equal to zero to find the value of the ๐ฅ-intercepts. Doing so wouldโve given us ๐ฅ
equals one and ๐ฅ equals 3.5, which once again matches function (A). So the graph that represents
the equation given is graph (A).