Connecting Quadratic Functions and Quadratic Equations
Lesson 8 of 16
Objective: SWBAT describe graphical and algebraic methods for determining the x-intercepts of a quadratic equation.
To begin class, students complete Warm-up: Converting Quadratic Forms to practice converting among quadratic forms and begin to link the process of solving quadratic equations to the work we have done so far with quadratic functions [MP2]. Students are given 15 minutes to complete this Warm-Up while I project an online timer on the board. They can work with table partners to get the work done and solutions are projected on the board when the timer goes off.
The work we do in translating among the various algebraic forms of quadratics and the graph addresses several Common Core Content Standards as well as MP7. I discuss the content alignment of this work in the video, Supporting Standards with Quadratics.
Recommendations for learning progressions within the Common Core Standards for Math suggest that students are often confused about the difference between equations and functions and are therefore confused about the difference between solving, simplifying and factoring. Therefore, I draw attention to this distinction as we transition from working with quadratic functions in general to solving for values of x that make the function equal to some value.
I initiate this discussion by sending Quick Poll: Conceptual Questions About Quadratic Functions to students via the NSpire Navigator system. The questions in these polls are designed to reveal whether students are clear on the following concepts [MP2]:
- The algebraic form of a quadratic function has two variables
- Setting the y-value of a quadratic function equal to some number results in a quadratic equation that can be solved in several ways.
- There can be 0, 1, or 2 solutions to a quadratic equation in one variable, but there are infinitely many solutions to a quadratic equation in two variables. The collection of solutions forms a parabola if they are plotted in the coordinate plane.
After revealing the poll results for each question, I leave time for a discussion of each concept and why students answered the way they did. Practicing discussing these concepts out loud helps students refine their thinking, so I include several opportunities to "turn to your partner and explain" during this discussion [MP3].
After reinforcing the idea that replacing "y" with some number leads to a quadratic equation that can be solved, we consider the best methods to use when trying to determine the x-intercepts of a quadratic equation.
- If the function is in intercept form, we can set y=0 then use the zero product property
- if the function is in vertex form, we can set y=0 and take square roots of both sides of the equation
- If the function is in standard form, we can set y=0 and use the Quadratic Formula
I walk students through several examples of solving for the x-intercepts using the methods presented above. I frame each question in the context of functions so that students will understand the connection between quadratic functions and quadratic equations [MP6].
Although my initial examples will have rational x-intercepts, I will move to examples with irrational x-intercepts and also include an example of a quadratic function that has no x-intercepts. Students have not yet been introduced to imaginary numbers, so for now we will avoid this topic. Some of the examples I use in this discussion are included in the resource Examples for Finding x-Intercepts.
Practice and Extend
To practice the skills learned in the day's lesson, students will cycle through Stations - Finding x-Intercepts of Quadratic Functions. These stations are tables for four that have multiple copies of exercises printed on half sheets of colored paper (one color for each station). I use stations for practice when there are a few distinct skills that are best learned by practicing each individually before seeing them mixed together in a problem set.
I know the students well by this time of the year and can easily assign them to groups in which they will be most likely to stay on task and help each other persevere [MP1]. I choose this type of "productivity" grouping for this activity and give each group 3 or 4 students so that there are 8 groups in all. Each group will cycle through 4 unique stations (there are two of each). The stations will be organized as follows.
- Station 1: find the x-intercept(s) of the following quadratic functions presented in intercept form. Sketch a quick graph of the function after you have determined the intercepts.
- Station 2: find the x-intercept(s) of the following quadratic functions presented in standard form. Sketch a quick graph of the function after you have determined the intercepts.
- Station 3: find the x-intercept(s) of the following quadratic functions presented in vertex form. Sketch a quick graph of the function after you have determined the intercepts.
- Station 4: sketch the graph of the following quadratic equations without using the graphing calculator (quadratic functions in each of the 3 forms will be presented). Label the x-intercepts, where they exist.
I give students a quick overview of the problem types found at each station, reminding them that they should pick up a paper for their folder at each station they visit. Each group will start at one of the 4 stations, spend 8 minutes at each station, and then rotate to the next station. I use the overhead timer to make everyone aware of how much time is left. After the timer rings, each group advances to the next station.
As students work, I use the 3-Cup System to encourage perseverance provide help where it is needed [MP1]. The goal of this activity is for students to practice solving quadratic equations and make connections between the concept of a quadratic functions and a quadratic equations [MP2].
Closure and Assignment
With the few remaining minutes, I ask if any questions came up during the station activity. If there is sufficient time, I will do one or two of the problems on the board. If not, I tell students to look at the answer key on Edmodo before the next class period to see if that answers their question. The first part of the next lesson can also be used to answer questions on the station activity.
The homework assignment is to complete Finding x-Intercepts of Quadratic Functions, a set of 12 excercises in which students solve quadratic equations in order to find the x-intercepts of quadratic functions presented in standard, intercept and vertex forms.