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Mathematics TEKS Refinement 2006 9-12

Tarleton State University

Quadratic and Square Root Functions Activity: TEKS: Predicting Extraneous Roots (2A.9) Quadratic and square root functions. The student formulates equations and inequalities based on square root functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is expected to: (D) determine solutions of square root equations using graphs, tables, and algebraic methods. The intent of this exercise is to increase the students' understanding of extraneous roots. They will use the tabular and graphical forms of the functions to help them understand when, where, and how extraneous roots occur while manipulating the algebraic forms of the functions in an equation. Equations: A Question about Functions Worksheet Graphing calculator Colored markers 3- 4 students 1 class period

Overview:

Materials:

Grouping: Time: Lesson: 1. 2.

Procedures Put students in groups of 4. Tell the groups to solve the equation below symbolically. 4- x = x+2

Notes

The solution to this problem is an input of 0.

3.

Put on the overhead Equations: A Question About Functions. Have the entire class discuss and analyze the algebraic steps used in solving the equation. Give students the Worksheet. Have them completely fill in the tables and the graphs.

Make sure that every student understands the solutions to this problem and the algebraic steps that were used to answer the question.

4.

It is very important that they use the same spacing on each graph. If one grid mark represents 2 units on the graph for stage 1, then one grid mark

Algebra II Page 1

Quadratic and Square Root Functions Predicting Extraneous Roots

Mathematics TEKS Refinement 2006 9-12

Tarleton State University

Procedures

Notes should represent 2 units for stages 2 and 3 also. Allow them time to discover where and why extraneous roots appear in the tables and the graphs. The solutions of the equation will appear in each table and graph for each stage.

5.

Have the students discuss, in their groups, anything they see in the tables and graphs. Have the students circle the solutions of the equation on the tables and graphs with a blue marker. Have the students circle the extraneous solution on the tables and graphs with a red marker. The extraneous roots of the equation will appear at stage 2. Ask the students : "Why does this happen?" Have the students look at the tables and graphs in stage 1 and come up with a method for predicting where the extraneous roots will occur.

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9.

Before both sides of the equation are squared, input values whose output values differ only by a sign will become extraneous roots. For example, if we have f ( x) = g ( x) with f (2) = 3 and g (2) = -3 , then squaring both sides of f ( x) = g ( x) will create an extraneous root at 2 because now f (2) = g (2) = 9 .

Homework:

Ask them to solve other equations involving radicals and test their method. On-going assessment during the activity is necessary for the teacher to gauge whether or not students understand what they are doing. The questions outlined above will generate open discussion of the key concepts. The teacher should pay close attention to the responses and guide the discussion carefully.

Assessment:

Quadratic and Square Root Functions Predicting Extraneous Roots

Algebra II Page 2

Mathematics TEKS Refinement 2006 9-12

Tarleton State University

Equations: A Question About Functions

Stage 1:

4- x = x+2

f1(x) = g1(x)

The first algebraic step is to square both sides of the equation. Stage 2:

4 - x = x 2 + 4x + 4

f2(x) = g2(x)

The next algebraic step is to bring everything to one side of the equation. Stage 3:

x 2 + 5x = 0

f3(x) = g3(x)

The next step would be to factor and use the property of zero to find the answers to this question, i.e. the roots of the equation. They are the inputs 0 and -5. Now checking both of these inputs in the original equation we see that 0 is a true root but -5 is an extraneous root.

Quadratic and Square Root Functions Predicting Extraneous Roots

Algebra II Page 3

Mathematics TEKS Refinement 2006 9-12

Tarleton State University

Worksheet

Quadratic and Square Root Functions Predicting Extraneous Roots

Algebra II Page 4

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