Algebra 2 Unit 8 Interactive Notebooks: Sketching Polynomial Graphs

This year I’ve committed to posting each unit of both my Algebra 1 and Algebra 2 INBs.

My district is moving to a standards based curriculum, and has identified priority standards for every course. These are the standards we are required to address and assess our students over, so they pretty much form our units.

You can find my Algebra 1 (year long class) INB posts here:

Unit 1 | Unit 2 | Unit 3 | Unit 4 | Unit 5 | Unit 6 | Unit 7

And my Algebra 2 INB posts here:

Unit 1 | Unit 2 | Unit 3 | Unit 4 | Unit 5 | Unit 6 | Unit 7

And finally, my posts from a 2nd go around I’m teaching of Algebra 1 here:

Unit 1 | Unit 2 | Unit 3 | Unit 4

 

Our 8th priority standard in Algebra 2 is A.APR.3:

Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

 

This is a no calculator allowed unit, which makes sense because the goal is to create a ROUGH graph based on the factored form of the function. My students actually found the freedom from calculators nice – as one of them put it, “It makes me feel like I’m not going to make it more complicated than it needs to be, because I know I shouldn’t need a calculator to do it.”

 

Here’s how I broke down this unit into skills (you can see from the index page picture that I changed the wording of one of our goals after I made it):

Skill 1: I can identify zeros of a polynomial when it is given in factored form

When we had done our solving polynomials unit earlier in the year, my students expressed a lot of confusion regarding the words root, zero, solution, and x-intercept. I decided to address this for this unit – I actually only got really clear about which term should be used in which circumstance when I saw a graphic someone had posted on twitter last year, so I’m unsurprised that students were confused! I took extra care throughout this unit to use the correct terms with the correct situations and I think my students feel a lot more comfortable using each of them now.

After discussing those terms, we practiced finding the zeros of several factored functions, and then matching them to a graph with x-intercepts that made sense. My students found this easy, although we had to take a brief detour into the world of fractions since they couldn’t use a calculator to find a decimal equivalent. There were a lot of number lines and counting by the denominator that you don’t see in these notes, but I think it was a really good revisit of fraction concepts!

Skill 2: I can identify the total number of solutions, maximum number of extrema, and end behavior based on the degree of a polynomial function

This first page on information you can tell from a polynomial function could use some reformatting. I really liked the root types and the possibilities for what degree 3 solutions could look like on a graph, but I want to switch their places on the page. My students also kept forgetting the difference between finding the degree of a factored polynomial function and one in standard form, so I would add that at the top of the page.

The graph shape and end behavior page is the one that got used the most during this unit. Pretty much all of the other information got memorized pretty quickly, but this one was the page they kept referencing. We used Desmos to explore what would happen for each of the scenarios, with them choosing an even degree and a positive leading coefficient, then having another student choose an even degree and positive leading coefficient, etc., which was really helpful for them to see that this end behavior would hold true for ANY even degree and ANY positive leading coefficient. We also got to see some interesting graphs, like y=57x^100…

Skill 3: I can sketch a rough graph of a polynomial using its factored equation

We’re putting it all together! I liked the organization of the information to identify here. The only struggle my students had was that the zeros don’t always end up in the order that the x-intercepts appear on the graph, so for the example on the front, they kept putting the double root on (4,0) instead of (-1,0) because that was the order it was listed in the equation. I might add a place for them to rearrange the x-intercepts and types in order to try to prevent this.

Skill 4: I can write a polynomial from given constraints

This is going backwards from what we had just been doing, so it went pretty well intuitively for the students. I always frame this section as them being the teachers and trying to come up with problems. For their assignment, I actually had them write functions, then find the roots and initial value and write them on an index card, and then they traded to see if they could get back to the original function! We mostly looked at having each factor having degree 1, but the last example I showed them how you could come up with alternatives by using other exponents on factors.

 

You can find the files for these notes here, in PDF and Publisher form.