An Introduction to Data and Functions | |

Describing Single-Variable Data | |

Visualizing Single-Variable Data | |

Numerical Descriptors: What is "Average" Anyway? | |

An Introduction to Algebra Aerobics | |

An Introduction to Explore and Extend | |

Describing Relationships between Two Variables | |

Visualizing Two-Variable Data | |

Constructing a "60-Second Summary" | |

Using Equations to Describe Change | |

An Introduction to Functions | |

What is a Function? | |

Representing Functions: Words, Tables, Graphs and Equations | |

Input and Output: Independent and Dependent Variables | |

When is a Relationship Not a Function? | |

The Language of Functions | |

Function Notation | |

Domain and Range | |

Visualizing Functions | |

Is There a Maximum or Minimum Value? | |

When is the Output of the Function Positive, Negative or Zero? | |

Is the Function Increasing or Decreasing? | |

Is the Graph Concave Up or Concave Down? | |

Getting the Big Idea | |

Chapter Summary | |

Check Your Understanding | |

Chapter 1 Review: Putting it all Together | |

Exploration 1.1 Collecting, Representing, and Analyzing Data | |

Rates of Change and Linear Function | |

Average Rates of Change | |

Describing Change in the U.S. Population over Time | |

Defining the Average Rate of Change | |

Limitations of the Average Rate of Change | |

Change in the Average Rate of Change | |

The Average Rate of Change is a Slope | |

Calculating Slopes | |

Putting a Slant on Data | |

Slanting the Slope: Choosing Different End Points | |

Slanting the Data with Words and Graphs | |

Linear Functions: When Rates of Change are Constant | |

What if the U.S. Population Had Grown at a Constant Rate? | |

Real Examples of a Constant Rate of Change | |

The General Equation for a Linear Function | |

Visualizing Linear Functions | |

The Effect of b | |

The Effect of m | |

Finding Graphs and Equations of Linear Functions | |

Finding the Graph | |

Finding the Equation | |

Special Cases | |

Direct Proportionality | |

Horizontal and Vertical Lines | |

Parallel and Perpendicular Lines | |

Breaking the Line: Piecewise Linear Functions | |

Piecewise Linear Functions | |

The absolute value function | |

Step functions | |

Constructing Linear Models for Data | |

Fitting a Line to Data: The Kalama Study | |

Reinitializing the Independent Variable | |

Interpolation and Extrapolation: Making Predictions | |

Looking for Links between Education and Earnings: Using Regression Lines | |

Using U.S. Census Data | |

Summarizing the Data: Regression Lines | |

Regression Line: How good a fit? | |

Interpreting Regression Lines: Correlation vs. Causation | |

Raising More Questions: Going Deeper | |

Chapter Summary | |

Check Your Understanding | |

Chapter 2 Review: Putting it all Together | |

Having it Your Way | |

A Case Study on Education and Earnings | |

When Lines Meet: Linear Systems | |

Interpreting Intersection Points: Linear and Non-linear Systems | |

When Curves Collide: Non-linear Systems | |

When Lines Meet: Linear System | |

Visualizing and Solving Linear Systems | |

Visualizing Linear Systems | |

Strategies for Solving Linear Systems | |

Linear Systems in Economics: Supply and Demand | |

Reading between the Lines: Linear Inequalities | |

Above and Below the Line | |

Reading between the Lines | |

Manipulating Inequalities | |

Breakeven Points: Regions of Profit or Loss | |

Systems with Piecewise Linear Functions: Tax Plans | |

Graduated vs. Flat Income Tax | |

Comparing the Two Tax Models: Flat vs. Graduated Plans | |

Chapter Summary | |

Check Your Understanding | |

Chapter 3 Review: Putting it all Together | |

Flat vs. Graduated Income Tax: Who Benefits? | |

A Comparison of Hybrid and Conventional Automobiles | |

The Laws of Exponents and Logarithms: Measuring the Universe | |

The Numbers of Science: Measuring Time and Space | |

Powers of 10 and the Metric System | |

Scientific Notation | |

Positive Integer Exponents | |

Exponent Rules | |

Common Errors | |

Estimating Answers | |

Zero, Negative and Fractional Exponents | |

Zero and Negative Exponents | |

Fractional Exponents | |

Converting Units | |

Converting Units within the Metric Systems | |

Converting between the Metric and English Systems | |

Using Multiple Conversion Factors | |

Orders of Magnitude | |

Comparing Numbers of Widely Differing Sizes | |

Orders of Magnitude | |

Graphing Numbers of Widely Differing Sizes: Log Scales | |

Logarithms Base 10 | |

Finding the Logarithms of Powers of 10 | |

Finding the Logarithm of Any Positive Number | |

Plotting Numbers on a Logarithmic Scale | |

Chapter Summary | |

Check Your Understanding | |

Chapter 4 Review: Putting it all Together | |

The Scale and the Tale of the Universe | |

Growth and Decay: An Introduction to Exponential Functions | |

Exponential Growth | |

The Growth of E. coli Bacteria | |

The General Exponential Growth Function | |

Doubling Time | |

Looking at Real Growth Data for E. coli Bacteria | |

Linear vs. Exponential Functions | |

General Forms: Linear and Exponential Function | |

Data Tables: Identifying Linear and Exponential | |

Finding the Equation for an Exponential Function | |

Comparing the Average Rates of Change | |

In the Long Run, Exponential Growth Will Always Outpace Linear Growth | |

Exponential Decay | |

The Decay of Iodine-131 | |

The General Exponential Decay Function | |

Half-Lives | |

Visualizing Exponential Functions | |

The Graphs of Exponential Functions | |

Horizontal Asymptotes | |

Exponential Functions: A Constant Percent Change | |

Exponential Growth: Increasing by a Constant Percent | |

Exponential Decay: Decreasing by a Constant Percent | |

Revisiting Linear vs. Exponential Functions | |

More Interesting Examples of Exponential Growth and Decay | |

Fitting a Curve | |

Doubling Time and Half-Life: Translating between Forms | |

The "Rule of 70" | |

The Malthusian Dilemma | |

Forming a Fractal Tree | |

Compound Interest and the Number e | |

Compounding at Different Intervals | |

Continuous Compounding Using e | |

Exponential Functions Base e | |

Converting ek into a | |

Semi-log Plots of Exponential Functions | |

Chapter Summary | |

Check Your Understanding | |

Chapter 5 Review: Putting it all Together | |

Computer Viruses | |

Logarithmic Links: Logarithmic and Exponential Functions | |

Using Logarithms to Solve Exponential Equations | |

Estimating Solutions to Exponential Equations | |

Rules for Logarithms | |

Solving Exponential Equations | |

Solving for Doubling Times and Half-Lives | |

Using Natural Logarithms to Solve Exponential Equations Base e | |

The Natural Logarithm | |

Returning to Doubling Times and Half-Lives | |

Visualizing and Applying Logarithmic Functions | |

The Graphs of Logarithmic Functions | |

Logarithmic Growth | |

Explore & Extend 6.3: Visualizing Logarithmic Functions | |

Stretching, Compressing and Reflecting | |

The Relationship between Logarithmic and Exponential Functions | |

Logarithmic vs. exponential growth | |

Logarithmic and exponential functions are inverses of each other | |

Applications of Logarithmic Functions | |

Measuring acidity: The pH scale | |

Using Semi-log Plots to Construct Exponential Models for Data | |

Why Do Semi-Log Plots of Exponential Functions Produce Straight Lines? | |

Chapter Summary | |

Check Your Understanding | |

Chapter 6 Review: Putting it all Together | |

Properties of Logarithmic Functions | |

Power Functions | |

The Tension between Surface Area and Volume | |

Scaling Up a Cube | |

Size and Shape | |

Direct Proportionality: Power Functions with Positive Powers | |

Direct Proportionality | |

Properties of Direct Proportionality | |

Direct Proportionality with more than one Variable | |

Visualizing Positive Integer Powers | |

The Graphs of f(x)=x 2 and g(x)=x 3 | |

Explore & Extend 7.3: Visualizing Power Functions | |

Odd vs. Even Powers | |

Symmetry | |

The Effect of the Coefficient k | |

Comparing Power and Exponential Functions | |

Which Eventually Grows Faster, a Power Function or an Exponential Function? | |

Inverse Proportionality: Power Functions with Negative Integer Powers | |

Inverse Proportionality | |

Properties of Inverse Proportionality | |

Explore & Extend 7.5: Designing Stringed Instruments | |

Inverse Square Laws | |

Visualizing Negative Integer Power Functions | |

The Graphs of f(x)=x^-1 and g(x)=x^-2 | |

Odd vs. Even Powers | |

Asymptotes | |

Symmetry | |

Explore & Extend 7.6: Finding Symmetries | |

The Effect of the Coefficient k | |

Using Logarithmic Scales to Find the Best Functional Model | |

Looking for Lines | |

Why is a Log-Log Plot of a Power Function a Straight Line? | |

Translating Power Functions into Equivalent Logarithmic Functions | |

Analyzing Weight and Height Data | |

Using a standard plot | |

Using a semi-log plot | |

Using a log-log plot | |

Explore & Extend 7.7: Constructing Functions from Log-Log Plots | |

Allometry: The Effect of Scale | |

Chapter Summary | |

Check Your Understanding | |

Chapter 7 Review: Putting it all Together | |

Scaling Objects | |

Quadratics and the Mathematics of Motion | |

An Introduction to Quadratic Functions: The Standard Form | |

The Simplest Quadratic | |

Designing parabolic devices | |

The Standard Form of a Quadratic | |

Properties of Quadratic Functions | |

Estimating the Vertex and Horizontal Intercepts | |

Visualizing Quadratics: The Vertex Form | |

Stretching and Compressing Vertically | |

Reflections across the Horizontal Axis | |

Shifting Vertically and Horizontally | |

Using Transformations to Get the Vertex Form | |

The Standard Form vs. the Vertex Form | |

Finding the Vertex from the Standard Form | |

Converting between Standard and Vertex Forms | |

Finding the Horizontal Intercepts: The Factored Form | |

Using Factoring to Find the Horizontal Intercepts | |

Factoring Quadratics | |

Using the Quadratic Formula to Find the Horizontal Intercepts | |

The discriminant | |

Imaginary and complex numbers | |

The Factored Form | |

The Mathematics of Motion | |

The Scientific Method | |

Deriving an Equation Relating Distance and Time | |

Velocity: Change in Distance over Time | |

Acceleration: Change in Velocity over Time | |

Deriving an Equation for the Height of an Object in Free Fall | |

Working with an Initial Upward Velocity | |

The Average Rate of Change of a Quadratic Function | |

Chapter Summary | |

Check Your Understanding | |

Chapter 8 Review: Putting it all Together | |

How Fast are You? Using a Ruler to Make a Reaction Timer | |

New Functions from Old | |

Transforming a Function | |

Transformations | |

Stretching, compressing and shifting | |

Reflections | |

Symmetry | |

Combining Two Functions | |

The Algebra of Functions | |

Polynomial Functions | |

Rational Functions | |

A Final Example | |

Chapter Summary | |

Check Your Understanding | |

Chapter 9 Review: Putting it all Together | |

Appendix: Student Data Tables for Exploration 2.1 | |

Data Dictionary for FAM1000 Data | |

Solutions | |

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