# Intermediate Algebra : Functions and Authentic Applications

**by**Lehmann, Jay

### 9780321620958

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## Summary

Unique in its approach, the **Lehmann Algebra Series** uses curve fitting to model compelling, authentic situations, while answering the perennial question “But what is this good for?” **Lehmann** begins with interesting data sets, and then uses the data to find models and derive equations that fit the scenario. This interactive approach to the data helps readers connect concepts and motivates them to learn. The curve-fitting approach encourages readers to understand functions graphically, numerically, and symbolically. Because of the multi-faceted understanding that they gain, readers are able to verbally describe the concepts related to functions.

Introduction to Modeling, Operations and Expressions, Using the Slope to Graph Linear Equations, Simplifying Expressions and Solving Equations, Linear Functions and Linear Inequalities in One Variable, Systems of Linear Equations and Systems of Linear Inequalities, Polynomial Functions and Properties of Exponents, Factoring Polynomials and Solving Polynomial Equations, Quadratic Functions, Exponential Functions, Logarithmic Functions, Rational Functions, Radical Functions, Sequences and Series, Additional Topics For all readers interested in algebra.

## Author Biography

**In the words of the author:**

Before writing my algebra series, it was painfully apparent that my students couldn't relate to the applications in the course. I was plagued with the question, "What is this good for?" To try to bridge that gap, I wrote some labs, which facilitated my students in collecting data, finding models via curve fitting, and using the models to make estimates and predictions. My students really loved working with the current, compelling, and authentic data and experiencing how mathematics truly is useful.

My students' response was so strong that I decided to write an algebra series. Little did I know that to realize this goal, I would need to embark on a 15-year challenging journey, but the rewards of hearing such excitement from students and faculty across the country has made it all worthwhile! I'm proud to have played even a small role in raising peoples' respect and enthusiasm for mathematics.

I have tried to honor my inspiration: by working with authentic data, students can experience the power of mathematics. A random-sample study at my college suggests that I am achieving this goal. The study concludes that students who used my series were more likely to feel that mathematics would be useful in their lives (*P*-value 0.0061) as well as their careers (*P*-value 0.024).

In addition to curve fitting, my approach includes other types of meaningful modeling, directed-discovery explorations, conceptual questions, and of course, a large bank of skill problems. The curve-fitting applications serve as a portal for students to see the usefulness of mathematics so that they become fully engaged in the class. Once involved, they are more receptive to all aspects of the course.

## Table of Contents

**1. Linear Equations and Linear Functions**

1.1 Using Qualitative Graphs to Describe Situations

1.2 Graphing Linear Equations

1.3 Slope of a Line

1.4 Meaning of Slope for Equations, Graphs, and Tables

1.5 Finding Linear Equations

1.6 Functions

*Chapter Summary *

*Key Points of Chapter 1*

*Chapter 1 Review Exercises *

*Chapter 1 Test *

**2. Modeling with Linear Functions **

2.1 Using Lines to Model Data

2.2 Finding Equations of Linear Models

2.3 Function Notation and Making Predictions

2.4 Slope Is a Rate of Change

*Chapter Summary *

*Key Points of Chapter 2*

*Chapter 2 Review Exercises*

*Chapter 2 Test*

**3. Systems of Linear Equations**

3.1 Using Graphs and Tables to Solve Systems

3.2 Using Substitution and Elimination to Solve Systems

3.3 Using Systems to Model Data

3.4 Value, Interest, and Mixture Problems

3.5 Using Linear Inequalities in One Variable to Make Predictions

*Chapter Summary *

*Key Points of Chapter 3*

*Chapter 3 Review Exercises *

*Chapter 3 Test *

*Cumulative Review of Chapters 1—3 *

**4. Exponential Functions**

4.1 Properties of Exponents

4.2 Rational Exponents

4.3 Graphing Exponential Functions

4.4 Finding Equations of Exponential Functions

4.5 Using Exponential Functions to Model Data

*Chapter Summary *

*Key Points of Chapter 4*

*Chapter 4 Review Exercises *

*Chapter 4 Test *

**5. Logarithmic Functions**

5.1 Inverse Functions

5.2 Logarithmic Functions

5.3 Properties of Logarithms

5.4 Using the Power Property with Exponential Models to Make Predictions

5.5 More Properties of Logarithms

5.6 Natural Logarithms

*Chapter Summary *

*Key Points of Chapter 5 *

*Chapter 5 Review Exercises *

*Chapter 5 Test *

*Cumulative Review of Chapters 1—5*

**6. Polynomial Functions**

6.1 Adding and Subtracting Polynomial Expressions and Functions

6.2 Multiplying Polynomial Expressions and Functions

6.3 Factoring Trinomials of the Form *x*2 + *bx *+ *c*; Factoring out the GCF

6.4 Factoring Polynomials

6.5 Factoring Special Binomials; A Factoring Strategy

6.6 Using Factoring to Solve Polynomial Equations

*Chapter Summary *

*Key Points of Chapter 6 *

*Chapter 6 Review Exercises *

*Chapter 6 Test *

**7. Quadratic Functions**

7.1 Graphing Quadratic Functions in Vertex Form

7.2 Graphing Quadratic Functions in Standard Form

7.3 Using the Square Root Property to Solve Quadratic Equations

7.4 Solving Quadratic Equations by Completing the Square

7.5 Using the Quadratic Formula to Solve Quadratic Equations

7.6 Solving Systems of Linear Equations in Three Variables; Finding Quadratic Functions

7.7 Finding Quadratic Models

7.8 Modeling with Quadratic Functions

*Chapter Summary *

*Key Points of Chapter 7 *

*Chapter 7 Review Exercises *

*Chapter 7 Test *

*Cumulative Review of Chapters 1—7 *

**8. Rational Functions **

8.1 Finding the Domains of Rational Functions and Simplifying

Rational Expressions

8.2 Multiplying and Dividing Rational Expressions

8.3 Adding and Subtracting Rational Expressions

8.4 Simplifying Complex Rational Expressions

8.5 Solving Rational Equations

8.6 Modeling with Rational Functions

8.7 Variation

*Chapter Summary *

*Key Points of Chapter 8 *

*Chapter 8 Review Exercises *

*Chapter 8 Test *

**9. Radical Functions **

9.1 Simplifying Radical Expressions

9.2 Adding, Subtracting, and Multiplying Radical Expressions

9.3 Rationalizing Denominators and Simplifying Quotients of Radical Expressions

9.4 Graphing and Combining Square Root Functions

9.5 Solving Radical Equations

9.6 Modeling with Square Root Functions

*Chapter Summary *

*Key Points of Chapter 9 *

*Chapter 9 Review Exercises *

*Chapter 9 Test *

**10. Sequences and Series **

10.1 Arithmetic Sequences

10.2 Geometric Sequences

10.3 Arithmetic Series

10.4 Geometric Series

*Chapter Summary *

*Key Points of Chapter 10 *

*Chapter 10 Review Exercises *

*Chapter 10 Test *

*Cumulative Review of Chapters 1—10 *

**11. Additional Topics **

11.1 Absolute Value: Equations and Inequalities

*Key Points of Section 11.1 *

11.2 Linear Inequalities in Two Variables; Systems of Linear Inequalities

*Key Points of Section 11.2 *

11.3 Performing Operations with Complex Numbers

*Key Points of Section 11.3 *

11.4 Pythagorean Theorem, Distance Formula, and Circles

*Key Points of Section 11.4 *

11.5 Ellipses and Hyperbolas

*Key Points of Section 11.5 *

11.6 Solving Nonlinear Systems of Equations

*Key Points of Section 11.6 *

**A. Reviewing Prerequisite Material **

A.1 Plotting Points

A.2 Identifying Types of Numbers

A.3 Absolute Value

A.4 Performing Operations with Real Numbers

A.5 Exponents

A.6 Order of Operations

A.7 Constants, Variables, Expressions, and Equations

A.8 Distributive Law

A.9 Combining Like Terms

A.10 Solving Linear Equations in One Variable

A.11 Solving Equations in Two or More Variables

A.12 Equivalent Expressions and Equivalent Equations

**B Using a TI-83 or TI-84 Graphing Calculator**

B.1 Turning a Graphing Calculator On or Off

B.2 Making the Screen Lighter or Darker

B.3 Entering an Equation

B.4 Graphing an Equation

B.5 Tracing a Curve without a Scattergram

B.6 Zooming

B.7 Setting the Window Format

B.8 Plotting Points in a Scattergram

B.9 Tracing a Scattergram

B.10 Graphing Equations with a Scattergram

B.11 Tracing a Curve with a Scattergram

B.12 Turning a Plotter On or Off

B.13 Creating a Table

B.14 Creating a Table for Two Equations

B.15 Using “Ask” in a Table

B.16 Finding the Regression Curve for Some Data

B.17 Plotting Points in Two Scattergrams

B.18 Finding the Intersection Point(s) of Two Curves

B.19 Finding the Minimum Point(s) or Maximum Point(s) of a Curve

B.20 Storing a Value

B.21 Finding Any *x*-Intercepts of a Curve

B.22 Turning an Equation On or Off

B.23 Finding Coordinates of Points

B.24 Graphing Equations with Axes “Turned Off”

B.25 Entering an Equation by Using *Yn *References

B.26 Responding to Error Messages