GRAPHING CUBIC FUNCTIONS
Objective
THE DOMAIN OF A FUNCTION
Now that we understand how to represent an interval on a Number line, let’s talk about Domain.
So, for the official definition, the Domain of a Function, is all of the possible values of x.
Now, for some graphs, there is a limit on what x can be, but those are usually stated before hand.
For most other’s, there is no limit, because the function is unbounded.
For example:
If we look only at the x-axis of this graph, we can see that this graph has no limit on what x can be.
Therefore, we would say that:
Domain:
Inequality:
Set Notation:
Interval Notation:
However, if we are given a graph such as:
We would then see that:
THE RANGE OF A FUNCTION
Now that we understand how to identify the domain of a function, let’s talk about Range.
So, for the official definition, the Range of a Function, is all of the possible values of y.
Now, for some graphs, there is a limit on what y can be, but those are usually stated before hand.
However, it is important to note, they are more common than having a limited domain.
For most other’s, like with domain, there is no limit, because the function is unbounded.
For example, looking at our past example:
If we look only at the y-axis of this graph, we can see that y must be greater than or equal to zero.
Therefore, we would say that:
Range:
Inequality:
Set Notation:
Interval Notation:
However, if we are given a graph such as:
We would then see that:
SO, SINCE WE NOW REMEMBER DOMAIN AND RANGE, LET’S TRY IT OUT:
Identify the domain and range of the following:
So, first thing we do is graph the equation:
So now that we can see the graph, we can examine the x-coordinate behavior (or the domain).
As we can see, it doesn’t look like the graph has any boundaries.
So, for the domain, it’s safe to say:
Inequality:
Set Notation:
Interval Notation:
Now, let’s look at the y-coordinate behavior (or the range)
As we can see, it doesn’t look like the graph has any boundaries.
So, for the range, it’s safe to say:
Domain:
Range:
LOCAL MAXIMUM AND LOCAL MINIMUM OF A FUNCTION
A local maximum is a maximum that can be seen given a specific interval.
Like the graph we saw before, some graphs have what we call valleys (or low points) and peaks (or high points).
These highs and lows are considered the local maximums and local minimums of a function.
The peaks are the maximums, and the valleys are the minimum.
So why do we call them local?
Because they are specific to the interval given.
The actual maximum of a function is usually referred to as the global maximum, and is the highest point of the function at all intervals. �Or, in other words, the highest point of the entire function.
So let’s look at a quick example to tell the difference.
THE LOCAL MAXIMUM
Let’s use the same graph (because I’m lazy and don’t want to go making another one).
Again, it’s important to point out that the local maximum is not the highest point of the entire graph, it’s just the highest point of that interval.
THE LOCAL MINIMUM
Now let’s talk about the local minimum.
The local minimum is the same as the local maximum, except that it’s the lowest point of the graph.
So, take the same graph (again, lazy!).
The zeroes of a function
Finally, the zeroes of a function are where the graph crosses the x-axis.
This only occurs when you set y = 0.
They are also fairly easy to spot, so let’s look at an example:
As we can see from this graph, at (0,0) and at (3,0) it crosses/touches the x-axis.
So the zeroes of this graph would be: x = 0, x = 3.
And yes, that’s how you would write that.
Example:
As we can see, it seems like the graph is decreasing from -4 to -3, but then drastically increases from -3 on, so we would say this is locally increasing.
The local maximum is going to be at y = 9 since that’s the highest point of the graph, and the local minimum would be at y = 0, the lowest point of the graph.
The zero of the function (given the interval) would be at x = -3
End behavior
Last thing we’ll talk about concerning graphs (at least for right now) is their end behavior.
Basically, the end behavior is what the graph will do constantly in either the positive direction, or the negative direction.
The way we determine what the end behavior of a graph is, is by determining the direction we want to face, then determine what the graph does.
This is more complicated to explain than show, so here’s an example:
Example:
Determine the end behavior from the following function:
As we can see, as x increases, we can see that this graph seems
to fall to negative infinity.
So, to answer this question, we would say:
Likewise, if we want to look at as x decreases, we can say that
the graph seems to rise to positive infinity.
So, we would also say:
So what do the graphs of different polynomials look like?
Well, they are different depending on what power the leading coefficient is.
The leading coefficient is the number written in front of the variable with the largest exponent.
So, for example:
Each polynomial with a different leading coefficient will have a different graph
However, they all follow patterns depending on the coefficient.
Here are some examples:
DOMAIN, RANGE, AND END BEHAVIOR OF DIFFERENT POLYNOMIALS
So, to start, let’s look at different graphs to see if we can find some similarities:
So, as we can see, even degree and odd degree polynomials follow a pattern with their domain and range, as well as their end behavior.
DOMAIN, RANGE, AND END BEHAVIOR OF NEGATIVE POLYNOMIALS
So, to start, let’s look at different graphs to see if we can find some similarities:
So, as we can see, even degree and odd degree polynomials follow a pattern with their domain and range, as well as their end behavior even if their leading coefficient is negative.
TURNING POINTS OF POLYNOMIAL FUNCTIONS
So, the turning points of polynomials is where the polynomial changes from going in the positive direction, to the negative direction, or vice versa.
For example, with the graph:
We can see the graph turns at around x = -2
We can see the graph turns again at around x = 0
So the turning points are:�x = -2, x = 0
FINDING THE X - INTERCEPTS
So, to find the x-intercepts of a polynomial, you set the polynomial equal to 0, and solve.
Whatever x equals, those are the x-intercepts of the function.
For example:
Example
Find the x-intercepts from the following:
f(x) = x(x - 7)(x + 7)
So, again, to find the x-intercepts, set the function equal to 0, then solve.
So:
0 = x(x - 7)(x + 7)
Now, before we lose it, let’s think about this.
We have three parts of this function being multiplied together.
But, if only 1 of those parts ends up being 0, then the whole function is 0.
So, we can set each part to 0, and get the answers we want.
So:
x
x - 7
x + 7
= 0
= 0
= 0
+7 +7
-7 -7
x = 7
x = -7
So then x = 0, 7, -7
Graphing functions
So we know the basics of a graph right?
But what if you’re asked to skew the graph some?
For example:
But what if we want to skew this graph to the left three slots?
What about to the right 3 slots?
What about up 3 slots?
Down 3 slots?
SO HOW DO WE DO IT?
MOVING THE GRAPH UP
Let’s look at a quick x-y chart to see what this means.
X | Y |
0 | 0 |
1 | 1 |
2 | 4 |
3 | 9 |
| |
| |
| |
| |
| |
X Y
0
3
1
2
3
4
7
12
So what’s the graph look like?
MOVING THE GRAPH DOWN
X | Y |
0 | 0 |
1 | 1 |
2 | 4 |
3 | 9 |
X | Y |
| |
| |
| |
| |
0
-3
1
2
3
-2
1
6
SO WHAT OTHER WAYS CAN WE MOVE THE GRAPH?
So moving the graph up and down was pretty simple, but what about moving the graph left or right?
This gets a little more complicated, so let’s experiment.
First off, we know that y goes up and down, so adding or subtracting something from y is pretty easy.
However, x is the variable that determines left to right, and we can’t just add something to the equation because that’s how we move up and down.
So, any ideas?
Well, let’s just try a few things and see what happens
MOVING THE GRAPH HORIZONTALLY
MOVING THE GRAPH LEFT
X | Y |
| |
| |
| |
| |
-3
0
-2
-1
0
1
4
9
That was different.
So then, what’s the graph look like?
So, why did we pick -3 as our first point and not 0?
MOVING THE GRAPH TO THE RIGHT
X | Y |
| |
| |
| |
| |
3
0
2
1
0
1
4
9
It makes sense, so what does the graph look like?
So as we can see, it did shift to the right 3 slots.
But that’s weird, why is it that subtracting from x moves the graph to the right,
but adding moves it to the left?
It almost seems like it’s an inverse operation.
Shrinking a parabola
X | Y |
| |
| |
| |
| |
0
0
1
2
3
2
8
18
So our graph changes from:
To:
STRETCHING A PARABOLA
X | Y |
| |
| |
| |
| |
0
0
2
4
6
2
8
18
Again, we went from this:
To this:
So how do we graph cubic functions?
Well, honestly, they’re harder to graph than quadratics, but there are some tricks to make it easier to graph.
Of course, one of the easiest ways to graph it is to look at the original cubic function, and then try to find some ways to manipulate it so we don’t have to work so hard.
So, to start off, we need to original function.
So, to start off, let’s start like we would with any graph.
We pick a point.
X | Y |
| |
| |
| |
| |
| |
0
So, when x = 0, then:
0
One of the easiest points to work with is 0, so:
Now, let’s try 1
So, when x = 1, then:
1
1
Now, let’s try -1
-1
So, when x = -1, then:
-1
Now, let’s try 2
So, when x = 2, then:
2
8
Now, let’s try -2
-2
So, when x = -2, then:
-8
Alright, great. So, what now?
Well, now there are a few things we can do with this graph.
We can: stretch this graph
We can move the graph left or right
And finally we can move the graph up or down.
So, let’s look at some examples of how to do all of these things.
Stretching the graph, the first way:
So, stretching the graph is very similar to how we stretch a quadratic.
So, to start, let’s look at a quick example:
Now, let’s look at the graph:
So, obviously this graph looks like it’s been stretched up some.
But how do we know by how much?
Well, it’s because, each point on the x axis is being pushed twice as far as the point on the y-axis.
I know this may sound confusing, but looking at the original graph we can see it:
As you can see, we can tell that this graph is skinnier, or in other words, it’s stretched.
So what if we’re just given the graph, how can we tell what the stretch is?
Well, we look at the two closest points to the center of the graph.
However far away they are from the center, that’s the stretch.
For example:
2
1
So the stretch is 2/1, or 2.
But:
1
1
So the stretch is 1/1, or 1.
MOVING THE GRAPH TO THE LEFT OR RIGHT
So, to move the graph left or right, we do the exact same as if it’s a quadratic.
For example:
The graph of this looks like:
As we can see, the graph is the exact same, but moved 2 units to the right.
Now let’s look at another example:
As we can see, the graph is moved to the left 2 units now.
So, just like with quadratics, the graph will move the opposite way that the number is.
So, if the number is negative, the graph moves to the right.
If the number is positive, the graph moves to the left.
MOVING THE GRAPH TO UP OR DOWN
Lastly, to move the graph up or down, we do the exact same as if it’s a quadratic.
For example:
The graph of this looks like:
As we can see, the graph is the exact same, but moved 1 unit up.
Now let’s look at another example:
As we can see, the graph is moved down 1 unit.
So, just like with quadratics, the graph will move in whatever direction the number’s sign is.
So, if the number is negative, the graph moves down.
If the number is positive, the graph moves up.
SO THAT’S HOW CUBIC FUNCTIONS WORK
So now let’s look at some examples of all of these different types of transformations mixed together.
Example 1
Find the equation from the following graph:
So, as we can see, the graph seems like it was moved to the right 3 units.
Which means our equation starts with:
We can also see, the graph looks like it went down 2 units.
Adding to our equation, we can see that:
Finally, we can see that from the first point:
To the middle point:
The graph seems to go up 2, and over 1:
2
1
So, our equation looks like:
Example 2
Find the equation from the following graph:
So, as we can see, the graph seems like it was moved to the left 2 units.
Which means our equation starts with:
We can also see, the graph looks like it went down 1 unit.
Adding to our equation, we can see that:
Finally, we can see that from the first point:
To the middle point:
The graph seems to go up 4, and over 1:
So, our equation looks like:
4
1