Unit 2, Lesson 5
Connecting Factors and Zeros
Algebra 2
5.1
Warm-up
Instructional Routines
Notice and WonderMaterials
To Gather
Math Community ChartActivity Narrative
The purpose of this Warm-up is for students to make connections between zeros of functions and features of their graphs, which will be useful when students identify values of that make factored equations equal 0 in a later activity. While students may notice and wonder many things about these equations and graphs, the connections between the two forms are the important discussion points.
This prompt gives students opportunities to see and make use of structure while making connections between graphs and mathematical expressions (MP7). The specific structure they might notice is the relationship between the horizontal intercepts of a graph and the factors of the expression being graphed.
Launch
Arrange students in groups of 2. Display the functions and graphs for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time and then 1 minute to discuss with their partner the things they notice and wonder.
Activity
NoneWhat do you notice? What do you wonder?
Graph of a polynomial function, xy-plane. Horizontal axis, scale -6 to 5 by 1’s. Vertical axis, scale -30 to 20, by 10’s. Function starts as increasing in Quadrant 3 and ends as increasing in Quadrant 1. The function has roots that are increasing through (-5 comma 0), decreasing through (-1 comma 0) and increasing through (3 comma 0). The function has a local maximum at (0 comma 8), local minimum at (3 comma -4) and intercepts the y-axis at (0 comma -15).
Graph of a polynomial function, xy-plane. Horizontal axis, scale -6 to 5 by 1’s. Vertical axis, scale -30 to 20, by 10’s. Function starts as increasing in Quadrant 3 and ends as increasing in Quadrant 1. The function has roots that are increasing through (-5 comma 0), decreasing through (-1 comma 0) and increasing through (2 comma 0). The function has a local maximum at about (-3 point 5 comma 20), local minimum at about (point 7 comma -12 point 5) and intercepts the y-axis at (0 comma -10).
Graph of a polynomial function, xy-plane. Horizontal axis, scale -6 to 5 by 1’s. Vertical axis, scale -30 to 20, by 10’s. Function starts as increasing in Quadrant 3 and ends as increasing in Quadrant 1. The function has roots that are increasing through (-4 comma 0), decreasing through (-1 comma 0) and increasing through (2 comma 0). The function has a local maximum about (-2 point 5 comma 10) and a local minimum at (point 5 comma -10).
Activity Synthesis
Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary. If possible, record the relevant reasoning on or near the image. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and to respectfully ask for clarification, point out contradicting information, or voice any disagreement.
If not brought up during the conversation, ask students why it makes sense that, for example, has a graph that crosses the horizontal axis at -5. Consider having students calculate the output of the function when to emphasize the connection between the factors and the graph.
Math Community
After the Warm-up, display the Math Community Chart. Remind students that norms are agreements that everyone in the class shares responsibility for, so it is important that everyone understands the intent of each norm and can agree with it. Tell students that today’s Cool-down includes a question asking for feedback on the drafted norms. This feedback will help identify which norms the class currently agrees with and which norms need revising or removing.
Student Lesson Summary
When a polynomial is written in factored form, we can identify several facts about it.
For example, the factored form of the polynomial shown in the graph is .
Graph of a polynomial, xy-axis. Horizontal and vertical axis, both scale -5 to 4, by 1’s. Graph is increasing from left from near (-2 comma -5), passes through (-1 comma 0) and (0 comma 3) as a maximum with a smooth curve. Graph is decreasing from (0 comma 3), passes though (2 comma 0) to a minimum at about (2 point 5 comma -0 point 5) with a smooth curve, and then increasing, passes through (3 comma 0) and keeps increasing up through about (4 comma 4) and beyond.
The graph has -intercepts at 3, 2, and -1. Each of these -values makes one of the factors in the expression equal to 0, and so makes the equation true. The numbers 3, 2, and -1 are known as the zeros of the function.
When a polynomial is not written in factored form, identifying the zeros from the expression for the polynomial can be more challenging. We’ll learn how to do that in future lessons.