Graphing Polynomial Functions
Part 1
Graphing polynomial functions will not introduce a new concept! To make these graphs we will combine several previously understood concepts and procedures. The thing is, it is a new application.
Because you’re using old things in new ways, this can seem overwhelming at first. But once you see it, boom, it will be clear!
Work through the notes below. Then, watch the video. After that, work on the assignment. Remember, learning is difficult. You might have to go back and re-read and re-watch the text and videos. But, that’s okay. Well worth the effort!
In this section you will learn how to graph a polynomial function that is factorable. In an upcoming section you will learn another factoring method (factoring difference or sum of cubes), but for this section if you can factor out a monomial, basic quadratic equations, and factor by grouping, you’re good to go.
To graph a polynomial you must be solid with all of the material we’ve learned up until now. If you find you’re confused by something like what turning points are, you need to review that concept before trying to move on with this! However, sometimes applying something like number of turning points helps you to understand! The point is this: learning is hard and often take redoubling of your efforts. Also, learning doesn’t occur unless you’ve worked through confusion. That’s hard to do!
Let’s look at a big picture approach to graphing a polynomial function. Ready?
- Use the degree (and leading coefficient) to inform you about:
- Turning points
- x – intercepts
- End Behavior
- Find the y – intercept
- Find the x – intercepts
- Plot all intercepts.
- To determine what the graph does between consecutive x – intercepts, plug in a value within that interval. If the output is positive, the graph goes up. Do this for all intervals between x – intercepts
- Sketch the graph
Let’s start by seeing an example where factoring is not an issue.
Graph f(x) = (x + 4)(x + 1)(1 – x)
Step 1: Determine the degree.
The degree of this function is 3.
We can see this by carrying out the multiplication of the xs.
The degree will tell you the following:
- Maximum number of x – intercepts
- Here we have a maximum of 3 x – intercepts
- Maximum number of turning points
- We have a maximum number of 2 turning points
- Possible end behavior
- Down left and up right, or up left and down right
Step 2: Determine the sign of the leading coefficient.
Here the leading coefficient is negative.
The sign of the leading coefficient tells you about the end behavior.
Since we have an odd degree and a negative leading coefficient, the end behavior will be
“up left down right”
Step 3: Find the y – intercepts.
Plug in zero for x.
f(0) = (0 + 4)(0 + 1)(1 – 0)
(4)(1)(1)
f(0) = 4
Step 4: Find the x – intercepts.
Set the function equal to zero. Factor. Set each factor equal to zero in a new set of equations. Solve each equation to find the x – coordinates of the x – intercepts.
Our x – intercepts are (-4, 0), (-1, 0) and (1, 0). Be careful with sign errors and simple mistakes when find the intercepts!
Observation: Consider the two x – intercepts (-1, 0) and (1, 0). Do you know if the graph goes up or down between those intercepts? You do know that it does one or the other. It cannot cross the x – axis again. We know this because the degree is 3, and we found all of the intercepts.
The y – intercept is in this interval from -1 < x < 1, and it is positive, (0, 4). That means the graph goes up between. This is exactly how you carry out the last step. You pick a test number that exists between consecutive x – intercepts. Then, you plug it into the function and see what the sign is. If the output is negative, the function goes down. A turning point will exist, along with a local minimum value.
Step 5: Determine if the function is above or below the x – axis between consecutive x – intercepts.
Now, how this math is done on paper is very different from how it is shown in text. However, you can set up a table just like this to conduct your exploration.
We are determining if the outputs within the interval between x – intercepts are positive or negative. That tells us where the graph goes.
When trying these practice problems, use a graphing software to check your answers, once you’re done. Feedback will be very important for you on this. The application here is new, but the concepts at play are not. But, because you’re using several old things together in a new way, this can be confusing at first. But, once you see it, it’ll be simple for you.
