Quadratic Functions and Roots

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SWBAT visually determine the roots of a quadratic function.

Big Idea

Students will learn the meaning of roots (zeros) by numerically and graphically inspecting quadratic functions.


15 minutes

Students will work in partners on today's quadratics_and_roots opener.  This warm-up serves as a review of important vocabulary from the previous lesson.  The emphasis throughout the first half of this unit will be on solving quadratic equations by finding the roots/zeros. At the same time, I want students to be familiar with the vertex and axis of symmetry of a parabola, since they can make meaningful connections between these features and the roots of a function.  I will ask students to draw and label the axis of symmetry on each graph.

I will not give much instruction before letting students work through the various examples.  I encourage students to discuss vocabulary and how to identify features from the graphs (MP3). Once students have worked through each example, pull the class back together to discuss key points.

I will ask students to process each of these questions:

  1. What were the roots for question #4?
  2. So is it possible to have no y-intercept? (this is a repeat question from the previous lesson but is worth asking again).
  3. Which examples have only one root (zero)?

If I think it will help, I will employ a strategy like Think-Pair-Share to help students develop their fluency with vocabulary and interpreting graphs of quadratic functions. 


20 minutes

Students will work with their partner on the quadratics_and_roots investigation.  I will allow students to use technology such as a graphing calculator or www.desmos.com.  My students find that Desmos is a little more user friendly for finding roots because they can tap on the roots to see the coordinates (see desmos_screenshot).

During this investigation my goal is for students to connect the fact that when a polynomial function is evaluated at the x-coordinate of one of its zeros, the result is zero (hence the name!). I encourage each pair to make note of any similarities that they notice form one example to the next. I plan to focus on problem number 3 and problem number 5 when I interact with students as they are working.  Problem 3 has a single zero. Problem 5 has three zeros.  


Teaching Note: I sometimes find it is necessary to review how to evaluate x^2 when x is a negative integer (for example -4).  In some of my classes, students will evaluate (-4)^2 and get -16.  This will obviously introduce bad data into their investigation. It is important to keep in mind that technology can contribute to this error. If a students types -4^2 into a calculator, they will get the incorrect answer of -16. This is one of the famed "Lies my calculator told me!"


5 minutes

With about 5-6 minutes left, I will bring students back together. I plan to ask them to take out a half sheet of paper that can be used by both members of each partnership.  Then, students should explain in words what they have noticed during this investigation (MP8).  

Here are a couple of more specific prompts: 

  • What was the connection between the zeros of a function graphically and the value for evaluating the function at each zero?
  • Why do you think this relationship exists? 

This closure will help me to assess how successful students were at making meaning from their work during today's investigation. If the lesson went as planned, this closing reflection will serve as a natural segway into solving quadratic equations tomorrow. When students look at solving quadratic equations algebraically, I want them to be thinking that they are trying to find the values of x when the function will be equal to zero.  Several techniques for accomplishing this goal will be discussed during the next few lessons in this unit.