Lesson Video: Cubic Functions and Their Graphs Mathematics

In this video, we will learn how to graph cubic functions, write their rules from their graphs, and identify their features.

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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 π‘₯ is equal to π‘₯ 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 π‘₯-values and work out their corresponding output or 𝑓 of π‘₯ 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 π‘₯ is positive, negative when π‘₯ is negative, and zero when π‘₯ 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 π‘₯ increases to ∞, then 𝑓 of π‘₯ also increases to ∞. And as π‘₯ decreases, then 𝑓 of π‘₯ also decreases to negative ∞. Finally, we can also say that it is an odd function, so 𝑓 of negative π‘₯ is equal to negative 𝑓 of π‘₯ for all values of π‘₯ 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 π‘₯ is equal to π‘₯ cubed and add in two other functions to consider: 𝑔 of π‘₯ is equal to π‘₯ cubed plus two and β„Ž of π‘₯ is equal to π‘₯ cubed minus one. Notice how the output has changed in these two functions. It isn’t just π‘₯ cubed; it’s π‘₯ cubed plus two and π‘₯ 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 π‘₯ equals π‘₯ cubed plus two passes through the point zero, two. Likewise, β„Ž of π‘₯ is equal to π‘₯ 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 π‘₯ plus π‘˜ for a constant π‘˜ in the real numbers is a vertical translation of the graph of the function 𝑓 of π‘₯ is equal to π‘₯ cubed. If π‘˜ is greater than zero, then its graph is the translation of π‘˜ units upwards of the graph of 𝑓 of π‘₯ is equal to π‘₯ cubed.

We can see this in the function 𝑔 of π‘₯. The π‘˜-value here would be plus two, and the graph was translated two units up. And if π‘˜ is less than zero, then its graph is the translation of the absolute value of π‘˜ units downwards of the graph of 𝑓 of π‘₯ is equal to π‘₯ cubed. If we look at the function β„Ž of π‘₯, the value of π‘˜ here is negative; it’s negative one. And so its absolute value is one. And we can see that β„Ž of π‘₯ is a translation of 𝑓 of π‘₯ equals π‘₯ 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 π‘₯ equals π‘₯ cubed, but we can introduce two new functions 𝑔 of π‘₯ and β„Ž of π‘₯. As we’re changing the input this time, it’s no longer just π‘₯ that’s cubed but π‘₯ plus two cubed and π‘₯ minus one cubed.

We can plot all three graphs on the same axes. This time, we have a horizontal translation. 𝑓 of π‘₯ is equal to π‘₯ cubed does indeed go through zero on the π‘₯-axis. However, 𝑔 of π‘₯ equals π‘₯ plus two cubed goes through negative two on the π‘₯-axis. β„Ž of π‘₯ is equal to π‘₯ minus one cubed goes through positive one on the π‘₯-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 π‘₯ is a translation of 𝑓 of π‘₯ equals π‘₯ 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 π‘₯ to π‘₯ minus β„Ž, then there’s a translation of β„Ž units to the right. For example, in the function β„Ž of π‘₯ is equal to π‘₯ minus one cubed, the value of β„Ž would be equal to one and the translation did occur one unit to the right. When the value of β„Ž is negative, we’re looking for a translation of the absolute value of β„Ž units left. This is what we have in the function 𝑔 of π‘₯, where the value of β„Ž 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 π‘₯ is equal to π‘₯ cubed, 𝑔 of π‘₯ is equal to three π‘₯ cubed, and β„Ž of π‘₯ is equal to a half π‘₯ 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 π‘₯ is equal to π‘₯ cubed has been vertically dilated. 𝑔 of π‘₯ has been vertically dilated by a scale factor of three, and β„Ž of π‘₯ has been vertically dilated by a scale factor of one-half. Thus, we can say that for any positive value of π‘Ž when 𝑓 of π‘₯ maps to π‘Ž times 𝑓 of π‘₯, there is a vertical dilation of scale factor π‘Ž.

And now let’s have a look at how multiplying the input changes the function.

This time, we can compare one other function β„Ž of π‘₯ equals two π‘₯ 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 π‘₯. 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 β„Ž of π‘₯ 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 π‘₯. And so for the input value to be two, that means that π‘₯ must be equal to one. And so the coordinate one, eight lies on the function β„Ž of π‘₯. In general then, we can say that for any positive 𝑏 in the real numbers when the input π‘₯ changes to 𝑏 of π‘₯, there’s a horizontal dilation of 𝑓 of π‘₯ equals π‘₯ 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 π‘₯ is equal to π‘₯ cubed and 𝑔 of π‘₯ is equal to negative π‘₯ cubed. What we have here is a reflection in the horizontal axis. Every output of 𝑔 of π‘₯ is the negative of its value in 𝑓 of π‘₯ is equal to π‘₯ 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 β„Ž of π‘₯ is equal to negative π‘₯ cubed. We can simplify this function by writing it without the parentheses as β„Ž of π‘₯ is equal to negative π‘₯ cubed. This creates an identical function to that of 𝑔 of π‘₯. 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 π‘₯ of a function to negative 𝑓 of π‘₯, that produces a reflection in the horizontal axis. When we change the input to negative π‘₯, that’s a reflection of 𝑓 of π‘₯ in the vertical axis. The fact that the cubic function 𝑓 of π‘₯ is equal to π‘₯ 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 π‘Ž, β„Ž, and π‘˜ are in the set of real numbers with π‘Ž not equal to zero, then the graph of 𝑓 of π‘₯ equals π‘Ž times π‘₯ minus β„Ž cubed plus π‘˜ is a transformation of 𝑦 equals π‘₯ cubed.

We can summarize how the different values of π‘Ž, β„Ž, and π‘˜ 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 π‘Ž-value; secondly, any horizontal translation, that’s the β„Ž-value; and finally, any vertical translation, which is the π‘˜-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 π‘₯ minus two cubed minus one. Option (B) 𝑦 equals π‘₯ plus two cubed minus one. Option (C) 𝑦 equals π‘₯ plus two cubed plus one. Or option (D) 𝑦 equals π‘₯ minus two cubed plus one.

We might begin by noticing that this function looks very similar to the standard cubic function 𝑓 of π‘₯ is equal to π‘₯ cubed, sometimes known as 𝑦 equals π‘₯ cubed. We can sketch 𝑦 equals π‘₯ cubed alongside the given function. The graph of 𝑦 equals π‘₯ 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 π‘₯ 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 π‘Ž times π‘₯ minus β„Ž cubed plus π‘˜ is a transformation of 𝑦 equals π‘₯ cubed for π‘Ž, β„Ž, and π‘˜ in the real numbers and π‘Ž not equal to zero. In this form, the value of π‘Ž indicates the dilation scale factor and a reflection if π‘Ž is less than zero; there’s a horizontal translation of β„Ž units right and a vertical translation of π‘˜ 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 π‘Ž 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 β„Ž units to the right, then this means that our value of β„Ž must be negative. And so β„Ž 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 π‘˜ must be negative. So it’s negative one.

Now all we need to do is fill in the values of π‘Ž, β„Ž, and π‘˜ into this cubic function form. When we simplify, we get the equation 𝑦 equals π‘₯ 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 π‘₯ is equal to negative π‘₯ minus two cubed?

In this question, we’re given a cubic function. So let’s compare this with the standard cubic function 𝑓 of π‘₯ is equal to π‘₯ cubed. We can draw a quick sketch of this function. Let’s recall that a cubic function in the form 𝑓 of π‘₯ is equal to π‘Ž times π‘₯ minus β„Ž cubed plus π‘˜ is the transformation of 𝑓 of π‘₯ equals π‘₯ cubed for π‘Ž, β„Ž, and π‘˜ in the real numbers and π‘Ž not equal to zero.

Here, π‘Ž represents a dilation or reflection, β„Ž gives the number of units the graph has been translated in the horizontal direction, and π‘˜ 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 π‘₯ is equal to negative π‘₯ minus two cubed, then that means that π‘Ž is equal to negative one. This indicates that there is no dilation, or rather a dilation of scale factor one. However, since π‘Ž is negative, this means there is a reflection of the graph in the π‘₯-axis. If we perform just the reflection, then the graph would look like this in pink.

Next, in the given function, β„Ž 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 π‘₯-axis at two, zero.

We can then give the answer that the graph which shows 𝑓 of π‘₯ is equal to negative π‘₯ 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 π‘₯ is equal to π‘₯ 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 π‘₯ is equal to π‘Ž times π‘₯ minus β„Ž cubed plus π‘˜ will allow us to identify all the different transformations of 𝑦 is equal to π‘₯ 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 π‘Ž-value; secondly, any horizontal translations, the β„Ž-value; and finally, any vertical translations, the π‘˜-value.

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