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An Introduction to Data and Functions | |

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Describing Single-Variable Data | |

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Visualizing Single-Variable Data | |

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

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An Introduction to Algebra Aerobics | |

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An Introduction to Explore and Extend | |

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Describing Relationships between Two Variables | |

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Visualizing Two-Variable Data | |

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Constructing a "60-Second Summary" | |

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Using Equations to Describe Change | |

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An Introduction to Functions | |

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What is a Function? | |

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

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Input and Output: Independent and Dependent Variables | |

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When is a Relationship Not a Function? | |

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The Language of Functions | |

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Function Notation | |

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Domain and Range | |

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Visualizing Functions | |

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Is There a Maximum or Minimum Value? | |

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

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Is the Function Increasing or Decreasing? | |

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Is the Graph Concave Up or Concave Down? | |

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Getting the Big Idea | |

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Chapter Summary | |

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Check Your Understanding | |

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Chapter 1 Review: Putting it all Together | |

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Exploration 1.1 Collecting, Representing, and Analyzing Data | |

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Rates of Change and Linear Function | |

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Average Rates of Change | |

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

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Defining the Average Rate of Change | |

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Limitations of the Average Rate of Change | |

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Change in the Average Rate of Change | |

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The Average Rate of Change is a Slope | |

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Calculating Slopes | |

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Putting a Slant on Data | |

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Slanting the Slope: Choosing Different End Points | |

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Slanting the Data with Words and Graphs | |

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Linear Functions: When Rates of Change are Constant | |

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

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Real Examples of a Constant Rate of Change | |

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The General Equation for a Linear Function | |

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Visualizing Linear Functions | |

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The Effect of b | |

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The Effect of m | |

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Finding Graphs and Equations of Linear Functions | |

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Finding the Graph | |

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Finding the Equation | |

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Special Cases | |

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Direct Proportionality | |

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Horizontal and Vertical Lines | |

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Parallel and Perpendicular Lines | |

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Breaking the Line: Piecewise Linear Functions | |

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Piecewise Linear Functions | |

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The absolute value function | |

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Step functions | |

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Constructing Linear Models for Data | |

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Fitting a Line to Data: The Kalama Study | |

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Reinitializing the Independent Variable | |

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Interpolation and Extrapolation: Making Predictions | |

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

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Using U.S. Census Data | |

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Summarizing the Data: Regression Lines | |

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Regression Line: How good a fit? | |

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Interpreting Regression Lines: Correlation vs. Causation | |

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Raising More Questions: Going Deeper | |

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Chapter Summary | |

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Check Your Understanding | |

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Chapter 2 Review: Putting it all Together | |

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Having it Your Way | |

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A Case Study on Education and Earnings | |

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When Lines Meet: Linear Systems | |

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

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When Curves Collide: Non-linear Systems | |

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When Lines Meet: Linear System | |

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Visualizing and Solving Linear Systems | |

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Visualizing Linear Systems | |

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Strategies for Solving Linear Systems | |

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Linear Systems in Economics: Supply and Demand | |

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Reading between the Lines: Linear Inequalities | |

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Above and Below the Line | |

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Reading between the Lines | |

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Manipulating Inequalities | |

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Breakeven Points: Regions of Profit or Loss | |

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Systems with Piecewise Linear Functions: Tax Plans | |

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Graduated vs. Flat Income Tax | |

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

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Chapter Summary | |

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Check Your Understanding | |

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Chapter 3 Review: Putting it all Together | |

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

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A Comparison of Hybrid and Conventional Automobiles | |

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

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The Numbers of Science: Measuring Time and Space | |

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Powers of 10 and the Metric System | |

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Scientific Notation | |

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Positive Integer Exponents | |

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Exponent Rules | |

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Common Errors | |

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Estimating Answers | |

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Zero, Negative and Fractional Exponents | |

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Zero and Negative Exponents | |

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Fractional Exponents | |

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Converting Units | |

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Converting Units within the Metric Systems | |

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Converting between the Metric and English Systems | |

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Using Multiple Conversion Factors | |

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Orders of Magnitude | |

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Comparing Numbers of Widely Differing Sizes | |

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Orders of Magnitude | |

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Graphing Numbers of Widely Differing Sizes: Log Scales | |

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Logarithms Base 10 | |

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Finding the Logarithms of Powers of 10 | |

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Finding the Logarithm of Any Positive Number | |

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Plotting Numbers on a Logarithmic Scale | |

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Chapter Summary | |

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Check Your Understanding | |

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Chapter 4 Review: Putting it all Together | |

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The Scale and the Tale of the Universe | |

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Growth and Decay: An Introduction to Exponential Functions | |

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Exponential Growth | |

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The Growth of E. coli Bacteria | |

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The General Exponential Growth Function | |

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Doubling Time | |

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

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Linear vs. Exponential Functions | |

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General Forms: Linear and Exponential Function | |

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Data Tables: Identifying Linear and Exponential | |

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Finding the Equation for an Exponential Function | |

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Comparing the Average Rates of Change | |

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

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Exponential Decay | |

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The Decay of Iodine-131 | |

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The General Exponential Decay Function | |

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Half-Lives | |

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Visualizing Exponential Functions | |

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The Graphs of Exponential Functions | |

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Horizontal Asymptotes | |

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Exponential Functions: A Constant Percent Change | |

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Exponential Growth: Increasing by a Constant Percent | |

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Exponential Decay: Decreasing by a Constant Percent | |

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Revisiting Linear vs. Exponential Functions | |

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More Interesting Examples of Exponential Growth and Decay | |

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Fitting a Curve | |

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

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The "Rule of 70" | |

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The Malthusian Dilemma | |

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Forming a Fractal Tree | |

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Compound Interest and the Number e | |

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Compounding at Different Intervals | |

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Continuous Compounding Using e | |

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Exponential Functions Base e | |

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Converting ek into a | |

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Semi-log Plots of Exponential Functions | |

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Chapter Summary | |

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Check Your Understanding | |

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Chapter 5 Review: Putting it all Together | |

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Computer Viruses | |

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Logarithmic Links: Logarithmic and Exponential Functions | |

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Using Logarithms to Solve Exponential Equations | |

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Estimating Solutions to Exponential Equations | |

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Rules for Logarithms | |

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Solving Exponential Equations | |

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Solving for Doubling Times and Half-Lives | |

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Using Natural Logarithms to Solve Exponential Equations Base e | |

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The Natural Logarithm | |

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Returning to Doubling Times and Half-Lives | |

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Visualizing and Applying Logarithmic Functions | |

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The Graphs of Logarithmic Functions | |

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Logarithmic Growth | |

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Explore & Extend 6.3: Visualizing Logarithmic Functions | |

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Stretching, Compressing and Reflecting | |

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The Relationship between Logarithmic and Exponential Functions | |

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Logarithmic vs. exponential growth | |

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Logarithmic and exponential functions are inverses of each other | |

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Applications of Logarithmic Functions | |

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Measuring acidity: The pH scale | |

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

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

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Chapter Summary | |

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Check Your Understanding | |

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Chapter 6 Review: Putting it all Together | |

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Properties of Logarithmic Functions | |

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Power Functions | |

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The Tension between Surface Area and Volume | |

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Scaling Up a Cube | |

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Size and Shape | |

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Direct Proportionality: Power Functions with Positive Powers | |

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Direct Proportionality | |

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Properties of Direct Proportionality | |

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Direct Proportionality with more than one Variable | |

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Visualizing Positive Integer Powers | |

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

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Explore & Extend 7.3: Visualizing Power Functions | |

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Odd vs. Even Powers | |

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Symmetry | |

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The Effect of the Coefficient k | |

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Comparing Power and Exponential Functions | |

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

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Inverse Proportionality: Power Functions with Negative Integer Powers | |

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Inverse Proportionality | |

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Properties of Inverse Proportionality | |

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Explore & Extend 7.5: Designing Stringed Instruments | |

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Inverse Square Laws | |

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Visualizing Negative Integer Power Functions | |

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

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Odd vs. Even Powers | |

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Asymptotes | |

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Symmetry | |

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Explore & Extend 7.6: Finding Symmetries | |

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The Effect of the Coefficient k | |

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Using Logarithmic Scales to Find the Best Functional Model | |

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Looking for Lines | |

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

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Translating Power Functions into Equivalent Logarithmic Functions | |

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Analyzing Weight and Height Data | |

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Using a standard plot | |

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Using a semi-log plot | |

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Using a log-log plot | |

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

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Allometry: The Effect of Scale | |

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Chapter Summary | |

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Check Your Understanding | |

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Chapter 7 Review: Putting it all Together | |

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Scaling Objects | |

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Quadratics and the Mathematics of Motion | |

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An Introduction to Quadratic Functions: The Standard Form | |

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The Simplest Quadratic | |

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Designing parabolic devices | |

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The Standard Form of a Quadratic | |

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Properties of Quadratic Functions | |

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Estimating the Vertex and Horizontal Intercepts | |

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Visualizing Quadratics: The Vertex Form | |

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Stretching and Compressing Vertically | |

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Reflections across the Horizontal Axis | |

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Shifting Vertically and Horizontally | |

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Using Transformations to Get the Vertex Form | |

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The Standard Form vs. the Vertex Form | |

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Finding the Vertex from the Standard Form | |

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Converting between Standard and Vertex Forms | |

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Finding the Horizontal Intercepts: The Factored Form | |

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Using Factoring to Find the Horizontal Intercepts | |

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Factoring Quadratics | |

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Using the Quadratic Formula to Find the Horizontal Intercepts | |

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The discriminant | |

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Imaginary and complex numbers | |

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The Factored Form | |

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The Mathematics of Motion | |

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The Scientific Method | |

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Deriving an Equation Relating Distance and Time | |

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Velocity: Change in Distance over Time | |

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Acceleration: Change in Velocity over Time | |

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

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Working with an Initial Upward Velocity | |

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The Average Rate of Change of a Quadratic Function | |

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Chapter Summary | |

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Check Your Understanding | |

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Chapter 8 Review: Putting it all Together | |

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

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New Functions from Old | |

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Transforming a Function | |

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Transformations | |

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Stretching, compressing and shifting | |

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Reflections | |

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Symmetry | |

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Combining Two Functions | |

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The Algebra of Functions | |

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Polynomial Functions | |

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Rational Functions | |

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A Final Example | |

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Chapter Summary | |

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Check Your Understanding | |

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Chapter 9 Review: Putting it all Together | |

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Appendix: Student Data Tables for Exploration 2.1 | |

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Data Dictionary for FAM1000 Data | |

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Solutions | |