Video Transcript
Which of the following graphs
represents the equation 𝑓 of 𝑥 equals negative two 𝑥 squared plus nine 𝑥
minus seven?
There are two ways we can
answer this question. But the first thing that we’re
going to do is establish whether the graph of our function opens upwards or
downward. Firstly, we know that if we
have the equation 𝑓 of 𝑥 equals 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 and the value
of 𝑎 is positive, the graph opens upward. If it’s negative, it opens
downward.
Now, for our function, the
value of 𝑎 is negative two. So it’s negative. This means the function opens
downward. And we can therefore disregard
any functions that open upward. That’s the function represented
by graph (B) and the one represented by graph (E).
Now that we’ve done this, we’re
going to plot a table of values to identify the points through which this graph
passes. We’ll need to do this for a
variety of values of 𝑥. Let’s start with 𝑥 equals
zero. That’s 𝑓 of zero equals
negative two times zero squared plus nine times zero minus seven. And that’s negative seven. So when 𝑥 is equal to zero,
the value of the function is negative seven. And that actually means that
the 𝑦-intercept of our function is negative seven.
But of course the other
technique we could’ve used is to identify that for the function 𝑎𝑥 squared
plus 𝑏𝑥 plus 𝑐, the 𝑦-intercept has coordinates zero, 𝑐. Either technique here,
substituting or identifying the feature of the graph, is equally acceptable.
We’re now going to choose some
positive and negative values of 𝑥 to satisfy each of the graphs we’ve been
given. First, let’s substitute 𝑥
equals negative three into the function. We get negative two times
negative three squared plus nine times negative three minus seven. That’s negative 52. Similarly, let’s substitute 𝑥
equals negative two. When we do, we find that 𝑓 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 𝑓 of negative one is negative 18, 𝑓 of one is zero, 𝑓
of two is three, and 𝑓 of three is two. Plotting the values that we can
on our graphs, and we see we disregard option (C) completely. We notice it’s the wrong side
of the 𝑦-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
𝑦-intercept. We could also have set 𝑓 of 𝑥
equal to zero to find the value of the 𝑥-intercepts. Doing so would’ve given us 𝑥
equals one and 𝑥 equals 3.5, which once again matches function (A). So the graph that represents
the equation given is graph (A).
