Video Transcript
In this video, we’ll learn how to
graph and interpret quadratic functions of the form 𝑦 equals 𝑘𝑥 squared plus
𝑐.
The word “quadratus” is Latin, and
it means to square. In mathematics, the term quadratic
comes from this and describes something that relates to squares, squaring, or
equations that involve terms where the variable is raised to the power of two. In particular, a quadratic function
is one of the form 𝑓 of 𝑥 equals 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐, where 𝑎, 𝑏, and
𝑐 are real numbers. And of course, 𝑎 is not equal to
zero. In this lesson, we’ll primarily be
focusing on functions in which 𝑏 is equal to zero, so our functions are going to be
of the form 𝑓 of 𝑥 equals 𝑘𝑥 squared plus 𝑐. With this in mind, let’s recap how
we work with functions.
When a relation assigns exactly one
output for some given input, it’s called a function. Since 𝑥 is generally the input to
the function, the value of the function for some number can be found by substituting
that number for the variable 𝑥 in the function equation. This process can be repeated any
number of times and organized in a function table. We say that the set of values we
input into the function is called its domain, and the set of values that result from
substituting these values is called the range.
So, let’s take some function 𝑓 of
𝑥 equals 𝑘𝑥 squared plus 𝑐, which can be equivalently written as 𝑦 equals 𝑘𝑥
squared plus 𝑐. To create a table of values for a
function in this form, we substitute various values of 𝑥 into the function equation
and fully simplify the result. We can then plot the resulting
ordered pairs on a coordinate plane. Let’s demonstrate this in a little
more detail.
This is a table for 𝑓 of 𝑥 equals
𝑥 squared plus two. Complete it by finding the values
of 𝑎, 𝑏, and 𝑐.
Remember to complete a table of
values for a function of the form 𝑓 of 𝑥 equals 𝑘𝑥 squared plus 𝑐, we
substitute each value of 𝑥 into the function. So, to find the value of 𝑎, we’ll
substitute 𝑥 equals negative two into the function 𝑓 of 𝑥 equals 𝑥 squared plus
two. In other words, 𝑎 is 𝑓 of
negative two. Since our function is 𝑥 squared
plus two, that’s negative two squared plus two. And of course, negative two squared
is four. So, this is four plus two, which is
equal to six. We now see that we’re not
interested in the value of the function when 𝑥 equals negative one. And we’re told that when 𝑥 is
equal to zero, 𝑓 of 𝑥 is two.
So, in order to find 𝑏, we’re
going to let 𝑥 be equal to one. This means 𝑏 is the value of the
function at this point; it’s 𝑓 of one. That’s one squared plus two, which
is one plus two, which, of course, equals three. Finally, we find the value of 𝑐 by
substituting 𝑥 equals two into our function. This means 𝑐 is equal to 𝑓 of
two, which is two squared plus two. Once again, that’s four plus two,
which is equal to six.
Let’s check our method by
calculating 𝑓 of zero and checking that it gives the correct output of two in the
table. 𝑓 of zero is zero squared plus
two. That’s zero plus two or two as we
expected. Since this value of 𝑓 of zero
corresponds to the value given in our table, we can be fairly confident in our
method. So, 𝑎 is equal to six, 𝑏 is equal
to three, and 𝑐 is equal to six.
Now, in this example, we could have
also calculated the value of 𝑓 of negative one. We substitute 𝑥 equals negative
one into the function, and we get negative one squared plus two, which is equal to
three. Adding these values to our table
and we now might notice that there’s a symmetry to our values of 𝑓 of 𝑥. This is not accidental. The graphs of quadratic functions
are symmetrical about a vertical line, as demonstrated in the diagram. For very simple quadratic
functions, as in the one in this question, this can be observed in the table of
values and that gives us a helpful way of checking our results.
We also notice that whilst there’s
reflectional symmetry between the coordinates generated, the values of the function
don’t increase linearly. And this means that we have to join
the coordinates with a smooth curve instead of a straight line.
In our next example, we’ll
demonstrate how to complete a table of values for a quadratic function and then
sketch its graph.
Complete the following table for
the graph of 𝑓 of 𝑥 equals two minus two 𝑥 squared by finding the values of 𝑎,
𝑏, 𝑐, and 𝑑. Which figure represents the graph
of the function 𝑓 of 𝑥?
Remember, to complete a table of
values for some function 𝑓 of 𝑥, we substitute each value of 𝑥 into the
function. Now, whilst it doesn’t look like
it, our equation 𝑓 of 𝑥 equals two minus two 𝑥 squared is of the form 𝑓 of 𝑥
equals 𝑘𝑥 squared plus 𝑐. It’s just got the terms in a
different order. And so, we are expecting the graph
of a simple quadratic function.
Let’s begin by finding the value of
𝑎. We need to substitute 𝑥 equals
negative two into our function, remembering to apply the order of operations. It’s two minus two times negative
two squared. We calculate negative two squared,
which is four. And then, we’re going to multiply
this by two. So, our expression becomes two
minus eight, which is, of course, negative six. And so, our value for 𝑎 is
negative six. Next, to find the value of 𝑏, we
let 𝑥 be equal to negative one, so it’s two minus two times negative one
squared. That’s two minus two times one,
which is two minus two or zero. And so, we see our value for 𝑏 in
our table is zero.
Next, let’s calculate the value of
𝑐 by calculating 𝑓 of zero. It’s two minus two times zero
squared. That’s two minus zero, which is
simply two. So, 𝑐 is equal to two. In a similar way, 𝑑 is the value
of 𝑓 of one, so it’s two minus two times one squared, which, once again, is two
minus two or zero.
So now that we have the values for
𝑎, 𝑏, 𝑐, and 𝑑, let’s replace these in our table. And so, we have a completed table
of values for the function 𝑓 of 𝑥 equals two minus two 𝑥 squared. We’re going to use this to find the
graph of the function 𝑓 of 𝑥. And to do so, we’ll write the
values from our table as a list of ordered pairs in the form 𝑥, 𝑓 of 𝑥. The first is negative two, negative
six. We have negative one, zero, the
third ordered pair zero, two, and our final two ordered pairs are one, zero and two,
negative six. If we plot them on one of our
coordinate planes, we see that graph (A) passes through all of these points. In fact, the graph of our function
is a smooth curve that passes through all of them. And so, the answer is option
(A).
Now, in this example, our function
𝑓 of 𝑥 equals two minus two 𝑥 squared generated this inverted or upside-down
parabola. Let’s compare this with the graph
of the function two plus two 𝑥 squared. Whilst the graphs of these two
functions share the same 𝑦-intercept, the parabola is n shaped when the coefficient
of 𝑥 squared is negative and U shaped when the coefficient of 𝑥 squared is
positive.
We can generalize this and say that
for a function of the form 𝑓 of 𝑥 equals 𝑘𝑥 squared plus 𝑐, if 𝑘 is greater
than zero, if it’s positive, we get a U-shaped parabola, and if it’s negative, we
get an n-shaped parabola. In fact, this also means that the
vertex or the turning point of the function is an absolute maximum when the
coefficient of 𝑥 squared is negative, and it’s an absolute minimum when the
coefficient of 𝑥 squared is positive. Let’s demonstrate these features in
another example.
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.
In our final example, we’ll
demonstrate how to use everything that we’ve seen so far to identify the equation of
a quadratic function 𝑦 equals 𝑘𝑥 squared plus 𝑐 given the graph of that
function.
Write the quadratic equation
represented by the graph shown.
We’re told that this is going to be
a quadratic equation. And in fact, observing the general
shape of the graph, it’s a parabola. So, we know it’s definitely the
graph of a quadratic function. This is of the form 𝑓 of 𝑥 equals
𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐, where 𝑎, 𝑏, and 𝑐 are real numbers and 𝑎 is not
equal to zero.
Now, in fact, our job is to
identify the values of 𝑎, 𝑏, and 𝑐. And we can quite quickly determine
the value of 𝑏. The function of the form 𝑎𝑥
squared plus 𝑐, or indeed 𝑘𝑥 squared plus 𝑐, has a line of symmetry 𝑥 equals
zero; it’s symmetrical about the 𝑦-axis. We see our graph is indeed
symmetrical about the 𝑦-axis, so we can deduce that the value of 𝑏 must be equal
to zero. And so, our function is of the form
𝑓 of 𝑥 equals 𝑎𝑥 squared plus 𝑐.
And next, we can use information
about its 𝑦-intercept to work out the value of 𝑐. Given a function of the form 𝑓 of
𝑥 equals 𝑎𝑥 squared plus 𝑐, its 𝑦-intercept is zero, 𝑐. Now, our 𝑦-intercept is zero,
two. It passes through the 𝑦-axis at
two. And so, 𝑐 must be equal to two,
meaning we can write our function as 𝑓 of 𝑥 equals 𝑎𝑥 squared plus two.
But how do we find the value of
𝑎? Well, we can choose a point that
lies on the given curve and substitute the 𝑥- and 𝑦-values of this coordinate into
the function. Let’s choose the point with
coordinates two, negative two. As an ordered pair, this is 𝑥, 𝑦
or 𝑥, 𝑓 of 𝑥. And so, 𝑓 of 𝑥 equals 𝑎𝑥
squared plus two becomes negative two equals 𝑎 times two squared plus two. Two squared is four. So, the equation becomes negative
two equals four 𝑎 plus two. Then subtracting two from both
sides of this equation, we get negative four equals four 𝑎. And then we divide through by four,
giving us 𝑎 is equal to negative one.
And so, we could write our function
as 𝑓 of 𝑥 equals negative one 𝑥 squared plus two, or we could write it as 𝑦
equals negative 𝑥 squared plus two. And in fact, since the parabola in
our diagram is n shaped, we were expecting the coefficient of 𝑥 squared to be
negative. The equation of the curve is 𝑦
equals negative 𝑥 squared plus two.
In this video, we’ve learned how to
create a table of values for a quadratic function. We also demonstrated some of the
key properties of graphs of simple quadratic equations and used these properties to
identify the equation of a given graph.
Let’s now recap the key points. In this lesson, we learned that the
graphs of quadratic functions are symmetrical about a vertical line that passes
through their vertex. We saw that, for quadratic
functions, if the coefficient of 𝑥 squared is positive, we get a U-shaped graph,
and if the coefficient of 𝑥 squared is negative, we get an n-shaped graph. Finally, we saw the graphs of the
form 𝑓 of 𝑥 equals 𝑘𝑥 squared plus 𝑐 are symmetric parabolas with a line of
symmetry 𝑥 equals zero or the 𝑦- axis and their 𝑦-intercepts are located at their
vertex and have coordinates zero, 𝑐.
