In this video, we will learn how to
identify features of quadratic functions, such as the vertex, zeros, axis of
symmetry, domain, and range. We’ll look at how we can determine
these features both graphically and from the equation of the function. You should already be familiar with
the process of completing the square or writing a quadratic in completed square
form, although this will be briefly recapped in the context of examples.
We recall firstly that a quadratic
function is of the general form 𝑓 of 𝑥 equals 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐,
where 𝑎, 𝑏, and 𝑐 are constants and 𝑎 must be nonzero. An alternative form in which
quadratic functions can be represented is completed square or vertex form. 𝑓 of 𝑥 equals 𝑎 multiplied by 𝑥
plus 𝑝 all squared plus 𝑞, where 𝑎, 𝑝, and 𝑞 are constants and again 𝑎 must be
If we were to plot a graph of 𝑦
equals 𝑓 of 𝑥 for a quadratic function, then we find that all quadratic functions
share the same general shape, which is known as a parabola. The first distinction we can make
is in the type of parabola, and this is determined by the sign of the coefficient of
𝑥 squared. That’s the value of 𝑎. If 𝑎 is positive, then the
parabola will be curved upwards as in the diagram on the left, whereas if 𝑎 is
negative, the parabola will curve downwards as in the diagram on the right. This is the first key thing to look
for when determining the shape of the graph of a quadratic function.
Let’s think about some of the other
general features of quadratic functions that we need to be aware of. And we’ll do this by considering
the graph of a simple quadratic. 𝑓 of 𝑥 equals 𝑥 squared plus two
𝑥 minus three. The first thing we can determine
about this graph is its 𝑦-intercept. We recall that everywhere on the
𝑦-axis, 𝑥 is equal to zero. So by substituting zero into the
equation of 𝑓 of 𝑥, we find the value of 𝑦 when 𝑥 equals zero is negative
Now, this is the constant term in
our quadratic function, and this will always be the case. So in general, if we have a
quadratic 𝑓 of 𝑥 equals 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐, then the value of its
𝑦-intercept will be 𝑐.
The second key feature of a
quadratic function is its roots or zeroes. Now, these are the 𝑥-values at
which the graph crosses the 𝑥-axis. We know that everywhere on the
𝑥-axis, 𝑦 or 𝑓 of 𝑥 is equal to zero. So these are the solutions to the
equation 𝑓 of 𝑥 equals zero.
We can find these values by
considering the factored or factorized form of our quadratic function. In this case, our quadratic can be
factored as 𝑥 plus three multiplied by 𝑥 minus one. And we then take each of these
factors in turn, set them equal to zero, and solve the resulting linear equations,
giving 𝑥 equals negative three and 𝑥 equals one.
We now have enough information to
be able to sketch this quadratic reasonably accurately. The coefficient of 𝑥 squared is
one, so it’s positive, meaning the parabola opens upwards. We have a 𝑦-intercept of negative
three and 𝑥-intercepts of negative three and one.
The next key feature we want to
consider is the vertex or turning point of our quadratic. Now, this turning point will be a
minimum when the value of 𝑎 is positive, and it will be a maximum when the value of
𝑎 is negative. In our case, it’s a minimum
point. It’s the coordinates of this point
here. In order to find the coordinates of
this point, we consider the vertex form of our quadratic, which in this case is 𝑥
plus one all squared minus four. We’ll review how to do this in some
Now, we need to recall a general
result here, which is that for the quadratic in its general vertex form 𝑎𝑥 plus 𝑝
all squared plus 𝑞, its vertex will be at the point negative 𝑝, 𝑞. Which means for our quadratic, its
vertex will be at the point negative one, negative four. Which makes sense when we consider
the position of this point relative to the values we’ve marked on our axes.
So that’s three key features of our
quadratic functions. Let’s now consider some more.
A parabola is a smooth, symmetrical
curve, which means that every quadratic graph has an axis or line of symmetry. This will be a vertical line
passing through the vertex of our function. Vertical lines have equations of
the form 𝑥 equals constant. And the 𝑥-value through which this
line passes is the 𝑥-coordinate of the vertex. So the equation of the axis of
symmetry for this quadratic will be 𝑥 equals negative one.
The two remaining features we need
to consider are the domain and range of our quadratic. Now, we recall firstly that the
domain of a function is the set of all values on which the function acts, which we
can also think of as the input values to the function. A quadratic is just a type of
polynomial, and all polynomials can act on all 𝑥-values. This means that there are no
restrictions on the values of 𝑥 on which the function can act. So we say that the domain is the
set of all real numbers.
Finally, we consider the range of
the function, which is the set of all values the function produces. Or in the case of a graph, we can
think of it as all the values of 𝑓 of 𝑥 or 𝑦. From our graph, we can see that all
the possible 𝑦-values of the function are the values of 𝑦 from the minimum point
upwards. That’s all 𝑦-values greater than
or equal to negative four. We can either express the range as
𝑓 of 𝑥 is greater than or equal to negative four. Or we can write this using interval
notation as the interval from negative four to ∞, which is closed at the lower end
and open at the upper end.
Now that we’ve seen how to identify
the key features of quadratic functions, let’s consider some examples.
Find the coordinates of the vertex
of the graph 𝑓 of 𝑥 equals 𝑥 squared plus eight 𝑥 plus seven. State the value of the function at
the vertex and determine whether it is a minimum or maximum value.
In order to find the coordinates of
the vertex of this graph, we need to convert its equation to vertex form. 𝑓 of 𝑥 equals 𝑎 multiplied by 𝑥
plus 𝑝 all squared plus 𝑞. Now, looking at the equation of
this graph, we see that the value of 𝑎, the coefficient of 𝑥 squared, is one. So we’re actually just looking for
this quadratic in the form 𝑥 plus 𝑝 all squared plus 𝑞, which we can do by
completing the square.
Firstly, we determine the value of
𝑝 inside the parentheses. And this is always half the
coefficient of 𝑥 in the equation. Half of eight is four, so we have
𝑥 plus four all squared. Now, we want this first part of our
quadratic function to be equivalent to 𝑥 squared plus eight 𝑥. But we know that if we were to
distribute 𝑥 plus four all squared, it would give 𝑥 squared plus eight 𝑥 plus
16. So we have an extra 16 which we
need to subtract in order to ensure that these two parts of the quadratic are
𝑥 squared plus eight 𝑥 is
therefore equivalent to 𝑥 plus four all squared minus 16. And then we also have the positive
seven, which remains the same. That value of 16 that we’re
subtracting is four squared. It’s the square of our value
𝑝. Then we just need to simplify. Negative 16 plus seven is negative
nine. So we now have our quadratic in its
We then recall that for a quadratic
in its vertex form, its vertex will have the coordinates negative 𝑝, 𝑞. For our quadratic, the value of 𝑝
is four and the value of 𝑞 is negative nine. So the coordinates of the vertex
will be negative 𝑝, that’s negative four, 𝑞, which is negative nine. So we’ve found the coordinates of
the vertex of this graph.
The question also asks us to state
the value of the function at the vertex. The value of the function will be
the 𝑦-coordinate, so the value is negative nine.
Finally, we were asked to determine
whether this is a minimum or maximum value. Well, this is determined by the
coefficient of 𝑥 squared, the value of 𝑎, which in our equation is one. As 𝑎 is positive, the parabola
will open upwards, which means that the vertex will be a minimum.
So we’ve completed the problem. The coordinates of the vertex are
negative four, negative nine. The value of the function itself is
negative nine. And this is a minimum value.
In our next example, we’ll see how
to determine the domain and range of a quadratic function given in its vertex
Determine the domain and the range
of the function 𝑓 of 𝑥 equals four multiplied by 𝑥 minus four all squared minus
Firstly, we recall that the domain
is the set of all values on which the function acts, which we can also think of as
the set of input values to the function. As the function 𝑓 of 𝑥 is a
polynomial and, more specifically, a quadratic, there are no restrictions on what
values it can act on. Therefore, we say that the domain
of this function is the set of all real numbers.
The range of a function is the set
of all values the function produces, which we can think of as the set of all output
values. To determine the range of a
quadratic function, we can consider its turning point. Now, this quadratic function has
been given to us in its completed square or vertex form. 𝑓 of 𝑥 equals 𝑎 multiplied by 𝑥
plus 𝑝 all squared plus 𝑞. And we know that when a quadratic
function is given in this form, its vertex has the coordinates negative 𝑝, 𝑞. The value of 𝑝 for our quadratic
is negative four, and the value of 𝑞 is negative three. So the vertex will be at negative
negative four, that’s four, negative three.
As the value of 𝑎, the coefficient
of 𝑥 squared in our quadratic function, is four, which is positive, we know that
its graph will be a parabola which curves upwards. So this vertex of four, negative
three will be a minimum point. The possible values of 𝑓 of 𝑥
then will be all the values from this minimum value of the function negative three
We can express this either as 𝑓 of
𝑥 is greater than or equal to negative three or using interval notation as the
interval from negative three to ∞, which is closed at the lower end and open at the
upper end. We can answer the problem then by
saying that the domain of this function is the set of all real numbers and the range
is the interval from negative three to ∞, which is closed at the lower end and open
at the upper end.
In this example, we’ll see how we
can use key features of a quadratic function to identify its graph.
For the function 𝑓 of 𝑥 equals 𝑥
squared minus four 𝑥 plus three, answer the following questions. Firstly, find by factoring the
zeroes of the function. Secondly, identify the graph of
There are also two further parts to
this question. So firstly, we’re asked to find the
zeros of this function. And the method we’re told to use is
factoring. We therefore need to write our
quadratic as the product of two linear factors. As the coefficient of 𝑥 squared is
one, we know that the first term in each of our parentheses will be 𝑥. We’re then looking for two numbers
whose sum is the coefficient of 𝑥, that’s negative four, and whose product is the
constant term, that’s positive three.
Well, the two numbers that fit both
of those criteria are negative one and negative three. Negative one plus negative three is
negative four, and negative one multiplied by negative three is positive three. So our quadratic factors as 𝑥
minus one multiplied by 𝑥 minus three, which we can of course confirm by
redistributing the parentheses if we wish.
We need to use this factored form
to determine the zeroes of the function, which we recall are the 𝑥-values such that
𝑓 of 𝑥 equals zero. If we set this factored form equal
to zero, we then recall that for the product of two things to be zero, at least one
of them must themselves be zero. So we can take each factor in turn
and set it equal to zero, giving two simple linear equations. The first can be solved by adding
one to each side to give 𝑥 equals one, and the second can be solved by adding three
to each side to give 𝑥 equals three. The roots or zeros of this function
then are the values one and three.
Now, in the second part of the
question, we’re asked to identify the graph of our function 𝑓. And we can see that we’ve been
given three possibilities: a blue one, a red one, and a green one. Now, we’ve just found that our
graph has zeros at one and three. And remember, these zeros are the
values of 𝑥 at which the graph crosses the 𝑥-axis. So if our graph crosses the 𝑥-axis
at one and three, we can see from the figure that this leaves just the red and green
graphs. The blue graph crosses the 𝑥-axis
or has zeros at values of negative one and negative three.
Now, we just need to decide between
the red and green graphs, which we see are mirror images of each other. One is an upward-curving parabola,
and the other is a downward-curving parabola. We recall that the type of parabola
we have will be determined by the value of 𝑎. That’s the coefficient of 𝑥
squared. In our function, the coefficient of
𝑥 squared is one. It’s a positive value, which means
the parabola will curve upwards. That means then that the graph of
our function 𝑓 must be the red graph. It has the correct zeros and the
correct shape. We can also see that the
𝑦-intercept of this graph is three, which is indeed the constant term in our
function 𝑓 of 𝑥.
The remaining two parts of the
question, which I didn’t write down initially because they give the game away for
the previous part are. Write the equation for 𝑔, the
function that describes the blue graph. And write the equation for ℎ, the
function that describes the green graph.
Let’s look at this blue graph first
of all then. We already said that it has zeros
at negative one and negative three. This means that in its factored
form, it has factors of 𝑥 plus one and 𝑥 plus three. But there could also be a factor of
𝑎 that we multiply by. To determine whether the value of
𝑎 is one or something else, we consider the 𝑦-intercept of the graph, which we can
see is the same as the 𝑦-intercept of the red graph. It’s three. When we multiply these two factors
together, the constant term will be one multiplied by three, which is indeed
three. And so this tells us that the value
of 𝑎 is simply one. Our function 𝑔 in its factored
form then is 𝑥 plus one multiplied by 𝑥 plus three. If we distribute the parentheses,
we have 𝑔 of 𝑥 equals 𝑥 squared plus four 𝑥 plus three.
For the green graph, it has the
same zeros as our function 𝑓. So it can be written as 𝑎
multiplied by 𝑥 minus one multiplied by 𝑥 minus three. And again, we need to determine
whether the value of 𝑎 is one or something else. Well, the 𝑦-intercept for the
green graph is negative three. If we multiply together negative
one and negative three, we get a value of positive three. And so in order to ensure the
𝑦-intercept, the constant term in the expanded form of ℎ of 𝑥, is negative three,
we need the value of 𝑎 to be negative one.
The equation ℎ of 𝑥 then is
negative 𝑥 minus one multiplied by 𝑥 minus three. In fact, it is the complete
negative of our function 𝑓 of 𝑥, which we can also see because they are
reflections of one another in the 𝑥-axis. We can write the equation ℎ of 𝑥
then as the complete negative of our function 𝑓 of 𝑥. ℎ of 𝑥 is equal to negative 𝑥
squared minus four 𝑥 plus three.
Let’s now summarize some of the key
points from this video. Quadratic functions can be
expressed in their expanded form, 𝑓 of 𝑥 equals 𝑎𝑥 squared plus 𝑏𝑥 plus
𝑐. Or their completed square or vertex
form, 𝑓 of 𝑥 equals 𝑎 multiplied by 𝑥 plus 𝑝 all squared plus 𝑞, where 𝑎, 𝑏,
𝑐, 𝑝, and 𝑞 are all constants and 𝑎 must be not equal to zero. The graph of a quadratic function
is a parabola. And if 𝑎 is positive, the parabola
will open upwards, whereas if 𝑎 is negative, the parabola will open downwards.
The turning point or vertex of a
quadratic can be found from its completed square or vertex form. And in the general case, the vertex
will have coordinates negative 𝑝, 𝑞. The parabola will also have an axis
of symmetry, which is a vertical line passing through this point, with the equation
𝑥 equals negative 𝑝.
The domain of any quadratic
function is the set of all real numbers, unless specified otherwise. And the range can be found either
from the graph or the completed square form. When 𝑎 is positive, the range will
be 𝑓 of 𝑥 is greater than or equal to 𝑞. And when 𝑎 is negative, the range
will be 𝑓 of 𝑥 is less than or equal to 𝑞.
In this video then, we’ve seen how
we can use either the graph or the equation of a quadratic function to determine
these key features.