Video Transcript
Which of the following graphs
represents the quadratic function 𝑓 of 𝑥 equals 𝑥 squared plus two on the closed
interval negative 2.3 to 2.3?
Remember to draw the graph of a
function, we can begin by constructing a table listing values of 𝑥 and 𝑓 of
𝑥. Now, before we do, we can actually
disregard one of our graphs straightaway. We know that the graph of a
function 𝑓 of 𝑥 equals 𝑘𝑥 squared plus 𝑐 is a parabola. If the value of 𝑘 is positive,
then we get that U-shaped parabola and if it is negative, we get the n-shaped
parabola. We might notice that the graph (A)
doesn’t look like either of these. In fact, it looks like a c shape,
so we’re going to disregard this graph straightaway.
Next, let’s construct our table of
values. Each graph is given over the closed
interval negative 2.3 to 2.3. So, we’ll calculate the values of
𝑓 of 𝑥 from 𝑥 is equal to negative two up to 𝑥 equals two. Then, to find the first entry, in
our second row, we find 𝑓 of negative two. It’s negative two squared plus
two. That’s four plus two, which is, of
course, equal to six. In a similar way, we can calculate
the second value in our table by finding 𝑓 of negative one. That’s negative one squared plus
two, and that’s equal to three. So, when 𝑥 is equal to negative
one, the output for our function is three. In a similar way, we calculate 𝑓
of zero. That’s zero squared plus two, which
is equal to two. Then, 𝑓 of one is one squared plus
two, which is three. And finally, 𝑓 of two is two
squared plus two, which is equal to six.
And so, we have the completed table
of values for 𝑓 of 𝑥 equals 𝑥 squared plus two. The ordered pairs that we’re going
to plot on our axes are negative two, six; negative one, three; zero, two; one,
three; and two, six. And since we’re drawing a quadratic
function, we join them with a smooth curve. Adding these points and the curve
to each of our remaining graphs, and we see that the only one that satisfies these
points is graph (B).
So far, we’ve graphed functions of
the form 𝑓 of 𝑥 equals 𝑘𝑥 squared plus 𝑐 by using a table of values to create
ordered pairs. And we’ve seen the graphs of this
form are symmetric parabolas. Now, they have a vertical line of
symmetry which in fact passes through their vertex. In the special case of quadratic
functions of the form 𝑓 of 𝑥 equals 𝑘𝑥 squared plus 𝑐, the line of symmetry is,
in fact, the 𝑦-axis or the line 𝑥 equals zero. Now, we can deduce one further
property of these functions.
Remember the 𝑦-intercept of a
function 𝑦 is equal to 𝑓 of 𝑥 is found by substituting 𝑥 equals zero. In other words, it’s the value of
𝑓 of zero. And of course, in the case of the
simple quadratic function, when 𝑥 is equal to zero, that’s 𝑘 times zero squared
plus 𝑐. But 𝑘 times zero squared is simply
zero, and so 𝑓 of zero is equal to 𝑐. And so, the 𝑦-intercept for the
graph of 𝑓 of 𝑥 equals 𝑘𝑥 squared plus 𝑐 is 𝑐. It has coordinates zero, 𝑐.
