Video Transcript
In this video, we will learn how to
graph cubic functions, write their rules from their graphs, and identify their
features. So let’s begin by looking at the
standard cubic function, which is 𝑓 of 𝑥 is equal to 𝑥 cubed.
It’s always good to remember that
we can graph any function by finding some coordinates that lie on the function. We can choose a range of input or
𝑥-values and work out their corresponding output or 𝑓 of 𝑥 values. Once we find these coordinates or
ordered pairs, we can plot them and join them with a curve.
So let’s have a look at the
characteristics of the standard cubic function. Firstly, we can say that the value
of the function is positive when 𝑥 is positive, negative when 𝑥 is negative, and
zero when 𝑥 is equal to zero. Secondly, as a polynomial with an
odd degree of three, it has opposite-end behaviors. This end behavior is such that as
𝑥 increases to ∞, then 𝑓 of 𝑥 also increases to ∞. And as 𝑥 decreases, then 𝑓 of 𝑥
also decreases to negative ∞. Finally, we can also say that it is
an odd function, so 𝑓 of negative 𝑥 is equal to negative 𝑓 of 𝑥 for all values
of 𝑥 in the domain of 𝑓. We also note that when we transform
this function, the definition of the curve is maintained.
So now let’s take a look at some of
the different transformations. We can categorize the different
transformations into two main types, either changing the input or changing the
output. Furthermore, these changes to the
input or the output can be classified as either addition, multiplication, or
negation.
We’ll start by looking at how
addition changes how the function looks, beginning with changes to the output.
Let’s take the standard cubic
function 𝑓 of 𝑥 is equal to 𝑥 cubed and add in two other functions to consider:
𝑔 of 𝑥 is equal to 𝑥 cubed plus two and ℎ of 𝑥 is equal to 𝑥 cubed minus
one. Notice how the output has changed
in these two functions. It isn’t just 𝑥 cubed; it’s 𝑥
cubed plus two and 𝑥 cubed minus one. We could use a table of values to
help us and then plot these three functions together on the same graph.
We can observe that the standard
cubic function passes through zero, zero, but 𝑔 of 𝑥 equals 𝑥 cubed plus two
passes through the point zero, two. Likewise, ℎ of 𝑥 is equal to 𝑥
cubed minus one passes through the coordinate zero, negative one. What we have here is a vertical
translation. In general, the graph of a function
𝑓 of 𝑥 plus 𝑘 for a constant 𝑘 in the real numbers is a vertical translation of
the graph of the function 𝑓 of 𝑥 is equal to 𝑥 cubed. If 𝑘 is greater than zero, then
its graph is the translation of 𝑘 units upwards of the graph of 𝑓 of 𝑥 is equal
to 𝑥 cubed.
We can see this in the function 𝑔
of 𝑥. The 𝑘-value here would be plus
two, and the graph was translated two units up. And if 𝑘 is less than zero, then
its graph is the translation of the absolute value of 𝑘 units downwards of the
graph of 𝑓 of 𝑥 is equal to 𝑥 cubed. If we look at the function ℎ of 𝑥,
the value of 𝑘 here is negative; it’s negative one. And so its absolute value is
one. And we can see that ℎ of 𝑥 is a
translation of 𝑓 of 𝑥 equals 𝑥 cubed by one unit down.
Next, we’ll have a look at how
adding or subtracting values from the input changes the function. We’ll keep the standard cubic
function 𝑓 of 𝑥 equals 𝑥 cubed, but we can introduce two new functions 𝑔 of 𝑥
and ℎ of 𝑥. As we’re changing the input this
time, it’s no longer just 𝑥 that’s cubed but 𝑥 plus two cubed and 𝑥 minus one
cubed.
We can plot all three graphs on the
same axes. This time, we have a horizontal
translation. 𝑓 of 𝑥 is equal to 𝑥 cubed does
indeed go through zero on the 𝑥-axis. However, 𝑔 of 𝑥 equals 𝑥 plus
two cubed goes through negative two on the 𝑥-axis. ℎ of 𝑥 is equal to 𝑥 minus one
cubed goes through positive one on the 𝑥-axis. This can be quite a difficult
transformation to recall. For example, when we add two to a
function, it’s tempting to think that this goes in the positive direction or, in
this case, two to the right. However, 𝑔 of 𝑥 is a translation
of 𝑓 of 𝑥 equals 𝑥 cubed by two units to the left.
Likewise, when we subtract one, we
think that the translation should go to the left. But in fact it goes to the
right. In order to help us remember this,
we can say that if we change the input 𝑥 to 𝑥 minus ℎ, then there’s a translation
of ℎ units to the right. For example, in the function ℎ of
𝑥 is equal to 𝑥 minus one cubed, the value of ℎ would be equal to one and the
translation did occur one unit to the right. When the value of ℎ is negative,
we’re looking for a translation of the absolute value of ℎ units left. This is what we have in the
function 𝑔 of 𝑥, where the value of ℎ here would be negative two, which is a
translation two units left.
Now let’s have a look at how
multiplication changes the function. This time, we have 𝑓 of 𝑥 is
equal to 𝑥 cubed, 𝑔 of 𝑥 is equal to three 𝑥 cubed, and ℎ of 𝑥 is equal to a
half 𝑥 cubed. Here are the three functions
plotted together. So what do we notice this time? Well, it’s not a translation, since
all three functions go through the same coordinate zero, zero. What we can say is that the
function 𝑓 of 𝑥 is equal to 𝑥 cubed has been vertically dilated. 𝑔 of 𝑥 has been vertically
dilated by a scale factor of three, and ℎ of 𝑥 has been vertically dilated by a
scale factor of one-half. Thus, we can say that for any
positive value of 𝑎 when 𝑓 of 𝑥 maps to 𝑎 times 𝑓 of 𝑥, there is a vertical
dilation of scale factor 𝑎.
And now let’s have a look at how
multiplying the input changes the function.
This time, we can compare one other
function ℎ of 𝑥 equals two 𝑥 cubed alongside the standard cubic function. In order to fully understand what’s
happening here, let’s consider some of the coordinates in each of the functions. Let’s take the coordinate two,
eight in the function 𝑓 of 𝑥. We can even consider this in terms
of a function machine. An input of two would give an
output of eight. We can then look at the function ℎ
of 𝑥 and consider what value must be input in order to give the same output of
eight.
Well, we know that when we input a
value of two, we get eight. But this time, we’re not simply
inputting two; we’re inputting two 𝑥. And so for the input value to be
two, that means that 𝑥 must be equal to one. And so the coordinate one, eight
lies on the function ℎ of 𝑥. In general then, we can say that
for any positive 𝑏 in the real numbers when the input 𝑥 changes to 𝑏 of 𝑥,
there’s a horizontal dilation of 𝑓 of 𝑥 equals 𝑥 cubed by a factor of one over
𝑏.
Finally, let’s look at how negation
changes the standard cubic function. Let’s consider the graphs of the
functions 𝑓 of 𝑥 is equal to 𝑥 cubed and 𝑔 of 𝑥 is equal to negative 𝑥
cubed. What we have here is a reflection
in the horizontal axis. Every output of 𝑔 of 𝑥 is the
negative of its value in 𝑓 of 𝑥 is equal to 𝑥 cubed. For example, the coordinate two,
eight in the original function is two, negative eight in the transformed
function.
You might wonder why this is a
reflection in the horizontal axis and not in the vertical axis. Well, let’s consider what happens
when we change the input. So we’ll take this new function ℎ
of 𝑥 is equal to negative 𝑥 cubed. We can simplify this function by
writing it without the parentheses as ℎ of 𝑥 is equal to negative 𝑥 cubed. This creates an identical function
to that of 𝑔 of 𝑥. This is a feature of the fact that
the cubic graph is an odd function.
In general, we can say that if we
change the output 𝑓 of 𝑥 of a function to negative 𝑓 of 𝑥, that produces a
reflection in the horizontal axis. When we change the input to
negative 𝑥, that’s a reflection of 𝑓 of 𝑥 in the vertical axis. The fact that the cubic function 𝑓
of 𝑥 is equal to 𝑥 cubed is odd means that negating either the input or the output
produces the same graphical result.
So far, we have seen how changing
different things about the input or output changes the function. But of course we can change a cubic
function in more than one way at a time. We can combine all of the
transformations into one cubic function form. We can say that if 𝑎, ℎ, and 𝑘
are in the set of real numbers with 𝑎 not equal to zero, then the graph of 𝑓 of 𝑥
equals 𝑎 times 𝑥 minus ℎ cubed plus 𝑘 is a transformation of 𝑦 equals 𝑥
cubed.
We can summarize how the different
values of 𝑎, ℎ, and 𝑘 change the shape of the function. You may wish to pause the screen
and make a note of these. One final note, the order in which
we perform these transformations is important, even if sometimes we get the same
graph regardless. Firstly, we carry out a vertical
dilation, that’s the 𝑎-value; secondly, any horizontal translation, that’s the
ℎ-value; and finally, any vertical translation, which is the 𝑘-value. We’ll now look at some examples,
beginning with one where we identify the correct equation of a given graph.
Which equation matches the
graph? Option (A) 𝑦 equals 𝑥 minus two
cubed minus one. Option (B) 𝑦 equals 𝑥 plus two
cubed minus one. Option (C) 𝑦 equals 𝑥 plus two
cubed plus one. Or option (D) 𝑦 equals 𝑥 minus
two cubed plus one.
We might begin by noticing that
this function looks very similar to the standard cubic function 𝑓 of 𝑥 is equal to
𝑥 cubed, sometimes known as 𝑦 equals 𝑥 cubed. We can sketch 𝑦 equals 𝑥 cubed
alongside the given function. The graph of 𝑦 equals 𝑥 cubed has
an inflection point at zero, zero. The inflection point of the given
function is at negative two, negative one. We could therefore say that the
function 𝑦 equals 𝑥 cubed must have been translated two units left and one unit
down. Both of these functions have the
same steepness, and they have not been reflected, so there are no further
transformations.
We can recall that a cubic function
in the form 𝑦 is equal to 𝑎 times 𝑥 minus ℎ cubed plus 𝑘 is a transformation of
𝑦 equals 𝑥 cubed for 𝑎, ℎ, and 𝑘 in the real numbers and 𝑎 not equal to
zero. In this form, the value of 𝑎
indicates the dilation scale factor and a reflection if 𝑎 is less than zero;
there’s a horizontal translation of ℎ units right and a vertical translation of 𝑘
units up. We perform these transformations
with the vertical stretch first, horizontal translation second, and vertical
translation third.
In this question, the graph has not
been reflected or dilated, so 𝑎 is equal to one. Next, we identified that this graph
has a translation of two units left. Because this cubic function form
has a horizontal translation of ℎ units to the right, then this means that our value
of ℎ must be negative. And so ℎ is equal to negative
two. Finally, we identified that there
must be a vertical translation of one unit down. Since this form gives us a vertical
translation in terms of units upwards, then our value of 𝑘 must be negative. So it’s negative one.
Now all we need to do is fill in
the values of 𝑎, ℎ, and 𝑘 into this cubic function form. When we simplify, we get the
equation 𝑦 equals 𝑥 plus two cubed minus one. This is the equation given in
answer option (B).
We’ll now look at an example where
we identify the correct shape of a graph of a cubic function.
Which of the following is the graph
of 𝑓 of 𝑥 is equal to negative 𝑥 minus two cubed?
In this question, we’re given a
cubic function. So let’s compare this with the
standard cubic function 𝑓 of 𝑥 is equal to 𝑥 cubed. We can draw a quick sketch of this
function. Let’s recall that a cubic function
in the form 𝑓 of 𝑥 is equal to 𝑎 times 𝑥 minus ℎ cubed plus 𝑘 is the
transformation of 𝑓 of 𝑥 equals 𝑥 cubed for 𝑎, ℎ, and 𝑘 in the real numbers and
𝑎 not equal to zero.
Here, 𝑎 represents a dilation or
reflection, ℎ gives the number of units the graph has been translated in the
horizontal direction, and 𝑘 is the number of units the graph is translated in the
vertical direction. We perform these transformations
with the vertical dilation first, horizontal translation second, and vertical
translation third.
So let’s consider the function we
were given. Since 𝑓 of 𝑥 is equal to negative
𝑥 minus two cubed, then that means that 𝑎 is equal to negative one. This indicates that there is no
dilation, or rather a dilation of scale factor one. However, since 𝑎 is negative, this
means there is a reflection of the graph in the 𝑥-axis. If we perform just the reflection,
then the graph would look like this in pink.
Next, in the given function, ℎ is
equal to two. So this means that there is a
translation of two units to the right. This moves the inflection point
from zero, zero to two, zero. Therefore, the function would look
something like this. It’s always a good idea to mark on
any important information onto a sketch. And of course, we know that this
graph crosses the 𝑥-axis at two, zero.
We can then give the answer that
the graph which shows 𝑓 of 𝑥 is equal to negative 𝑥 minus two cubed is that given
in option (E).
We can now summarize the key points
of this video. We began with the standard cubic
function 𝑓 of 𝑥 is equal to 𝑥 cubed. We then identified a number of
important properties. Next, we looked at how changing the
inputs and outputs of the function affect how it appears. Next, we saw how the function form
𝑓 of 𝑥 is equal to 𝑎 times 𝑥 minus ℎ cubed plus 𝑘 will allow us to identify all
the different transformations of 𝑦 is equal to 𝑥 cubed. Finally, we saw how the order in
which we carry out the transformations of this cubic function is important. First, it’s a vertical dilation,
the 𝑎-value; secondly, any horizontal translations, the ℎ-value; and finally, any
vertical translations, the 𝑘-value.
