11.3: Quadratic Formula
This activity is intended to supplement Algebra I, Chapter 10, Lesson 5.
ID: 12383
Time required: 20 minutes
Activity Overview
In this activity, students will make connections between the visual ways to find zeros of a parabola and algebraic ways. The quadratic formula is heavily emphasized in this activity, and is utilized in both a calculator program and in the Lists screen.
Topic: Quadratic Formula
- Students will see the graph of a parabola, and identify its zeros (\begin{align*}x-\end{align*}intercepts).
- Students will solve the quadratic by using the quadratic formula, with the discriminant being calculated in a formula in lists.
- Students will use a program to solve the quadratic completely.
Teacher Preparation and Notes
- Students will need to enter a formula into a list.
- Students will learn to store a formula in a list. The students will then use the same formula for several problems with different values for \begin{align*}a\end{align*}, \begin{align*}b\end{align*}, and \begin{align*}c\end{align*} \begin{align*}(L_1, \ L_2,\end{align*} and \begin{align*}L_3)\end{align*}.
- To download the calculator file, go to http://www.education.ti.com/calculators/downloads/US/Activities/Detail?id=12383 and select QUAD.8xp.
Associated Materials
- Student Worksheet: Quadratic Formula, http://www.ck12.org/flexr/chapter/9620, scroll down to the third activity.
- QUAD (program)
Exercise 1
Students are asked to graph the equation \begin{align*}y = x^2 - 4\end{align*}, and identify the zeros by looking at the graph or the equation. Students may know the zero product property, and thus may be able to solve by that method on their papers.
Students can also use the zero command, \begin{align*}2^{nd}\end{align*} [TRACE], the CALC function. When prompted for a “Left Bound?,” move the cursor to the left of the zero. For “Right Bound?,” move the cursor to the right of the zero. For “Guess?,” move the cursor near the zero. This process will be repeated if there is more than one zero.
The program QUAD has been provided to evaluate the quadratic formula, both the positive and negative solutions. If you would prefer to use this as a teaching opportunity for programming, you can have the students enter the program.
The program can be sent to student calculators using either TI-Connect\begin{align*}^{TM}\end{align*} or TI-Navigator\begin{align*}^{TM}\end{align*}.
You can also instruct students to enter their own program following your cues.
Exercise 2
The students again graph an equation, determine the zeros, and determine the factored form of the equation \begin{align*}y = x^2 + x - 6\end{align*}.
Now, the procedure is familiar. The QUAD program can be used and the students only needs to enter in the correct values for \begin{align*}a\end{align*}, \begin{align*}b\end{align*}, and \begin{align*}c\end{align*}.
Exercise 3
Students draw the graph, determine the zeros, and write the factored form for \begin{align*}y = x^2 - 4x + 4\end{align*}.
Students can use the QUAD program to confirm their answers, if obtained using the zero command.
Exercise 4
Students may ask why this quadratic \begin{align*}(y = x^2 - 2x - 7)\end{align*} is not factorable using integers and the previous examples were. A discussion of the discriminant and what it tells us about the roots (zeros) would be appropriate here. However, students will discover in the next few exercises that the value of the discriminant for this example is not a perfect square, and will thus create an irrational number in the solution.
Students will first obtain the solutions for the functions in Question 9 by graphing and finding the zeros or by using the QUAD program.
Finally, lists are used to calculate the value of the discriminant for the previous two problems, whose solutions were irrational.
Now, the lists and formula are available for students to use to calculate the discriminant for other problems provided on the student worksheet as homework, or it can be used to rework any previous problems.
Homework/Extensions
Students will use the formula provided in Question 10 to calculate the discriminant for several other quadratics.
Then, students are to examine the flow chart and discuss with students how to use it to answer their homework problems. Ask students to sketch a different graph for each scenario. One is already shown for them.