Students come into my Precalculus class knowing a lot about quadratic and cubic functions, but their knowledge of fourth-degree polynomial functions or higher is sometimes limited. In this lesson you can leverage the students' existing knowledge to build bridges to graphs of any polynomial function.
To begin, remind students that we just worked with quadratic functions in the previous lesson, but now our goal is to be able to produce a rough sketch of the graph of any polynomial function. Ask a student to remind you what a polynomial function is.
To begin, I give students this worksheet and explain that they will given equations and they are to sketch the graphs. You can start up this PowerPoint to explain the rules in the second slide. You really want to stress that students should not be calculating random points - they should use the points they can easily find and then use other knowledge about the function from the equation. The important aspects that I want my students to be aware of are end behavior, if the function opens up or down, x- and y-intercepts, etc. Obviously we want students to be able to reason about the graphs using their own sense, so I would not allow them to use graphing calculators at all.
As explained in the PowerPoint, give an equation and have students sketch it on their own, then discuss as a group, and then choose someone to share with the entire class. As you go over their sketch, ask them to justify what part of the equation informed their decision. For example, if you ask why y = -x^3 is decreasing from left to right, they should be able to say that the negative sign reflects the parent function over the x-axis. I would use Desmos to verify if a sketch is correct once we have the whole class discussion.
The graphs have been sequenced in a very deliberate way. My hope is that there is just enough variation and scaffolding in the sequence of equations to make it challenging, yet a very logical progression. This video (Scaffolding within the Lesson) goes into more detail about the scaffolding of the lesson.
After the graphs are all complete, ask students to summarize some of the trends they noticed while looking at all of the polynomial graphs. It will be a good idea to make an exhaustive record of their ideas so you can refer back to them during the unit.
You want to draw the important aspects out of their summaries. Here is a good list that you should reach by the end of this activity:
1. Even degree functions have end behavior that point in the same direction.
2. Odd degree functions have end behavior that point in opposite directions.
3. Even degree functions that have a positive leading coefficient open upwards.
4. Even degree functions that have a negative leading coefficient open downwards.
5. Odd degree functions that have a positive leading coefficient rises from left to right.
6. Odd degree functions that have a negative leading coefficient falls from left to right.
7. Zeros that are repeated an even number of times "bounce back" off of the x-axis.
8. Zeros that are repeated an odd number of times "pass through" the x-axis.
Thinking about why many of these trends occur will happen during their assignment. It is certainly okay if it comes out during the activity.
Students will likely notice many patterns and observations while working through the graphs in the activity. This assignment (Assignment - Sketching Graphs of Polynomial Functions) will reinforce many of the rules that we observed, but also get them thinking about why some of the rules are true. Students can work on this assignment for the remainder of class and finish it up for homework.