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Precalculus Enhanced with Graphing Utilities (2012)
1: Graphs
1.1: The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing Equations

Distance Formula
Explore the distance formula as an application of the Pythagorean theorem. Learn to see any two points as the endpoints of the hypotenuse of a right triangle. Drag those points and examine changes to the triangle and the distance calculation. 5 Minute Preview
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Points, Lines, and Equations
Compare the graph of a linear function to its rule and to a table of its values. Change the function by dragging two points on the line. Examine how the rule and table change. 5 Minute Preview
1.2: Intercepts; Symmetry; Graphing Key Equations
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Slope-Intercept Form of a Line
Compare the slope-intercept form of a linear equation to its graph. Vary the coefficients and explore how the graph changes in response. 5 Minute Preview
1.3: Solving Equations Using a Graphing Utility
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Solving Equations by Graphing Each Side
Solve an equation by graphing each side and finding the intersection of the lines. Vary the coefficients in the equation and explore how the graph changes in response. 5 Minute Preview
1.4: Lines
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Point-Slope Form of a Line
Compare the point-slope form of a linear equation to its graph. Vary the coefficients and explore how the graph changes in response. 5 Minute Preview

Slope-Intercept Form of a Line
Compare the slope-intercept form of a linear equation to its graph. Vary the coefficients and explore how the graph changes in response. 5 Minute Preview

Standard Form of a Line
Compare the standard form of a linear equation to its graph. Vary the coefficients and explore how the graph changes in response. 5 Minute Preview
1.5: Circles
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Circles
Compare the graph of a circle with its equation. Vary the terms in the equation and explore how the circle is translated and scaled in response. 5 Minute Preview
2: Functions and Their Graphs
2.1: Functions
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Function Machines 1 (Functions and Tables)
Drop a number into a function machine, and see what number comes out! You can use one of the six pre-set function machines, or program your own function rule into one of the blank machines. Stack up to three function machines together. Input and output can be recorded in a table and on a graph. 5 Minute Preview

Introduction to Functions
Determine if a relation is a function using the mapping diagram, ordered pairs, or the graph of the relation. Drag arrows from the domain to the range, type in ordered pairs, or drag points to the graph to add inputs and outputs to the relation. 5 Minute Preview
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Linear Functions
Determine if a relation is a function from the mapping diagram, ordered pairs, or graph. Use the graph to determine if it is linear. 5 Minute Preview
2.2: The Graph of a Function
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Function Machines 2 (Functions, Tables, and Graphs)
Drop a number into a function machine, and see what number comes out! You can use one of the six pre-set function machines, or program your own function rule into one of the blank machines. Stack up to three function machines together. Input and output can be recorded in a table and on a graph. 5 Minute Preview
2.5: Graphing Techniques; Transformations
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Translating and Scaling Functions
Vary the coefficients in the equation of a function and examine how the graph of the function is translated or scaled. Select different functions to translate and scale, and determine what they have in common. 5 Minute Preview
2.6: Mathematical Models: Building Functions
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Dye Elimination
When a foreign substance such as a drug is ingested, it often remains in the bloodstream for a long time. This Gizmo models the elimination of substances from the bloodstream using water and dye. Add dye to a container of water, and then add beakers of pure water while removing beakers of dyed water. The amount of dye remaining is recorded after each cycle. The volumes of all containers can be adjusted, as well as the amount of dye used. This provides a good example of exponential decay. 5 Minute Preview

Half-life
Investigate the decay of a radioactive substance. The half-life and the number of radioactive atoms can be adjusted, and theoretical or random decay can be observed. Data can be interpreted visually using a dynamic graph, a bar chart, and a table. Determine the half-lives of two sample isotopes as well as samples with randomly generated half-lives. 5 Minute Preview

Compound Interest
Explore compound interest in-depth, from compounded annually to compounded continuously. In addition, compare the END POINTS graph, with dots that fit an exponential curve, to the ALL TIME graph, which has a more step-like appearance. 5 Minute Preview
3: Linear and Quadratic Functions
3.1: Linear Functions and Their Properties
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Linear Functions
Determine if a relation is a function from the mapping diagram, ordered pairs, or graph. Use the graph to determine if it is linear. 5 Minute Preview
3.2: Linear Models: Building Linear Functions from Data
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Correlation
Explore the relationship between the correlation coefficient of a data set and its graph. Fit a line to the data and compare the least-squares fit line. 5 Minute Preview
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Least-Squares Best Fit Lines
Fit a line to the data in a scatter plot using your own judgment. Then compare the least squares line of best fit. 5 Minute Preview
3.3: Quadratic Functions and Their Properties
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Quadratics in Polynomial Form
Compare the graph of a quadratic to its equation in polynomial form. Vary the coefficients of the equation and explore how the graph changes in response. 5 Minute Preview
4: Polynomial and Rational Functions
4.1: Polynomials Functions and Models
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Graphs of Polynomial Functions
Study the graphs of polynomials up to the fourth degree. Vary the coefficients of the equation and investigate how the graph changes in response. Explore things like intercepts, end behavior, and even near-zero behavior. 5 Minute Preview
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Zap It! Game
Adjust the values in a quadratic function, in vertex form or in polynomial form, to "zap" as many data points as possible. 5 Minute Preview
4.2: The Real Zeros of a Polynomial Function
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Dividing Polynomials Using Synthetic Division
Divide a polynomial by dragging the correct numbers into the correct positions for synthetic division. Compare the interpreted polynomial division to the synthetic division. 5 Minute Preview
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Polynomials and Linear Factors
Create a polynomial as a product of linear factors. Vary the values in the linear factors to see how their connection to the roots of the function. 5 Minute Preview
4.4: Properties of Rational Functions
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General Form of a Rational Function
Compare the equation of a rational function to its graph. Multiply or divide the numerator and denominator by linear factors and explore how the graph changes in response. 5 Minute Preview
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Rational Functions
Compare the graph of a rational function to its equation. Vary the terms of the equation and explore how the graph is translated and stretched as a result. Examine the domain on a number line and compare it to the graph of the equation. 5 Minute Preview
4.5: The Graph of a Rational Function
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General Form of a Rational Function
Compare the equation of a rational function to its graph. Multiply or divide the numerator and denominator by linear factors and explore how the graph changes in response. 5 Minute Preview
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Rational Functions
Compare the graph of a rational function to its equation. Vary the terms of the equation and explore how the graph is translated and stretched as a result. Examine the domain on a number line and compare it to the graph of the equation. 5 Minute Preview
5: Exponential and Logarithmic Functions
5.3: Exponential Functions
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Exponential Functions
Explore the graph of an exponential function. Vary the coefficient and base of the function and investigate the changes to the graph of the function. 5 Minute Preview
5.4: Logarithmic Functions
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Logarithmic Functions
Compare the equation of a logarithmic function to its graph. Change the base of the logarithmic function and examine how the graph changes in response. Use the line y = x to compare the associated exponential function. 5 Minute Preview
5.5: Proprieties of Logarithms
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Logarithmic Functions: Translating and Scaling
Vary the values in the equation of a logarithmic function and examine how the graph is translated or scaled. Connect these transformations with the domain of the function, and the asymptote in the graph. 5 Minute Preview
5.8: Exponential Growth and Decay Models; Newton's Law; Logistic Growth and Decay Models
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Dye Elimination
When a foreign substance such as a drug is ingested, it often remains in the bloodstream for a long time. This Gizmo models the elimination of substances from the bloodstream using water and dye. Add dye to a container of water, and then add beakers of pure water while removing beakers of dyed water. The amount of dye remaining is recorded after each cycle. The volumes of all containers can be adjusted, as well as the amount of dye used. This provides a good example of exponential decay. 5 Minute Preview

Exponential Growth and Decay
Explore the graph of the exponential growth or decay function. Vary the initial amount and the rate of growth or decay and investigate the changes to the graph. 5 Minute Preview

Half-life
Investigate the decay of a radioactive substance. The half-life and the number of radioactive atoms can be adjusted, and theoretical or random decay can be observed. Data can be interpreted visually using a dynamic graph, a bar chart, and a table. Determine the half-lives of two sample isotopes as well as samples with randomly generated half-lives. 5 Minute Preview
5.9: Building Exponential, Logarithmic, and Logistic Models from Data

Dye Elimination
When a foreign substance such as a drug is ingested, it often remains in the bloodstream for a long time. This Gizmo models the elimination of substances from the bloodstream using water and dye. Add dye to a container of water, and then add beakers of pure water while removing beakers of dyed water. The amount of dye remaining is recorded after each cycle. The volumes of all containers can be adjusted, as well as the amount of dye used. This provides a good example of exponential decay. 5 Minute Preview

Half-life
Investigate the decay of a radioactive substance. The half-life and the number of radioactive atoms can be adjusted, and theoretical or random decay can be observed. Data can be interpreted visually using a dynamic graph, a bar chart, and a table. Determine the half-lives of two sample isotopes as well as samples with randomly generated half-lives. 5 Minute Preview

Compound Interest
Explore compound interest in-depth, from compounded annually to compounded continuously. In addition, compare the END POINTS graph, with dots that fit an exponential curve, to the ALL TIME graph, which has a more step-like appearance. 5 Minute Preview
6: Trigonometric Functions
6.2: Trigonometric Functions: Unit Circle Approach
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Sine Function
Compare the graph of the sine function with the graph of the angle on the unit circle. Drag a point along the sine curve and see the corresponding angle on the unit circle. 5 Minute Preview
6.4: Graphs of the Sine and Cosine Functions
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Cosine Function
Compare the graph of the cosine function with the graph of the angle on the unit circle. Drag a point along the cosine curve and see the corresponding angle on the unit circle. 5 Minute Preview
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Sine Function
Compare the graph of the sine function with the graph of the angle on the unit circle. Drag a point along the sine curve and see the corresponding angle on the unit circle. 5 Minute Preview
6.5: Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions
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Tangent Function
Compare the graph of the tangent function with the graph of the angle on the unit circle. Drag a point along the tangent curve and see the corresponding angle on the unit circle. 5 Minute Preview
6.6: Phase Shift, Sinusoidal Curve Fitting
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Translating and Scaling Sine and Cosine Functions
Experiment with the graph of a sine or cosine function. Explore how changing the values in the equation can translate or scale the graph of the function. 5 Minute Preview
7: Analytic Trigonometry
7.4: Trigonometric Identities
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Simplifying Trigonometric Expressions
Choose the correct steps to simplify a trigonometric function. Use step-by-step feedback to diagnose incorrect steps. 5 Minute Preview
7.5: Sum and Difference Formulas
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Sum and Difference Identities for Sine and Cosine
Choose the correct steps to evaluate a trigonometric expression using sum and difference identities. Use step-by-step feedback to diagnose incorrect steps. 5 Minute Preview
9: Polar Coordinates; Vectors
9.1: Polar Coordinates
9.3: The Complex Plane; De Moivre's Theorem
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Points in the Complex Plane
Identify the imaginary and real coordinates of a point in the complex plane. Drag the point in the plane and investigate how the coordinates change in response. 5 Minute Preview
9.4: Vectors
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Vectors
Manipulate the magnitudes and directions of two vectors to generate a sum and learn vector addition. The x and y components can be displayed, along with the dot product of the two vectors. 5 Minute Preview
10: Analytic Geometry
10.2: The Parabola
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Parabolas
Explore parabolas in a conic section context. Find the relationship among the vertex, focus, and directrix of a parabola, and how that relates to its equation. 5 Minute Preview
10.3: The Ellipse
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Ellipses
Compare the equation of an ellipse to its graph. Vary the terms of the equation of the ellipse and examine how the graph changes in response. Drag the vertices and foci, explore their Pythagorean relationship, and discover the string property. 5 Minute Preview
10.4: The Hyperbola
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Hyperbolas
Compare the equation of a hyperbola to its graph. Vary the terms of the equation of the hyperbola. Examine how the graph of the hyperbola and its asymptotes changes in response. 5 Minute Preview
11: Systems of Equations and Inequalities
11.1: Systems of Linear Equations: Substitution and Elimination
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Solving Linear Systems (Matrices and Special Solutions)
Explore systems of linear equations, and how many solutions a system can have. Express systems in matrix form. See how the determinant of the coefficient matrix reveals how many solutions a system of equations has. Also, use a draggable green point to see what it means for an (x, y) point to be a solution of an equation, or of a system of equations. 5 Minute Preview
11.7: Systems of Inequalities
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Systems of Linear Inequalities (Slope-intercept form)
Compare a system of linear inequalities to its graph. Vary the coefficients and inequality symbols in the system and explore how the boundary lines, shaded regions, and the intersection of the shaded regions change in response. 5 Minute Preview
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Systems of Linear Inequalities (Slope-intercept form)
Compare a system of linear inequalities to its graph. Vary the coefficients and inequality symbols in the system and explore how the boundary lines, shaded regions, and the intersection of the shaded regions change in response. 5 Minute Preview
11.8: Linear Programming
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Linear Programming
Use the graph of the feasible region to find the maximum or minimum value of the objective function. Vary the coefficients of the objective function and vary the constraints. Explore how the graph of the feasible region changes in response. 5 Minute Preview
12: Sequences; Induction; the Binomial Theorem
12.2: Arithmetic Sequences
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Arithmetic Sequences
Find the value of individual terms in arithmetic sequences using graphs of the sequences and direct computation. Vary the common difference and examine how the sequences change in response. 5 Minute Preview
12.3: Geometric Sequences; Geometric Series
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Geometric Sequences
Explore geometric sequences by varying the initial term and the common ratio and examining the graph. Compute specific terms in the sequence using the explicit and recursive formulas. 5 Minute Preview
13: Counting and Probability
13.2: Permutations and Combinations
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Permutations and Combinations
Experiment with permutations and combinations of a number of letters represented by letter tiles selected at random from a box. Count the permutations and combinations using a dynamic tree diagram, a dynamic list of permutations, and a dynamic computation by the counting principle. 5 Minute Preview
13.3: Probability
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Independent and Dependent Events
Compare the theoretical and experimental probabilities of drawing colored marbles from a bag. Record results of successive draws to find the experimental probability. Perform the drawings with replacement of the marbles to study independent events, or without replacement to explore dependent events. 5 Minute Preview
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Probability Simulations
Experiment with spinners and compare the experimental probability of particular outcomes to the theoretical probability. Select the number of spinners, the number of sections on a spinner, and a favorable outcome of a spin. Then tally the number of favorable outcomes. 5 Minute Preview
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Theoretical and Experimental Probability
Experiment with spinners and compare the experimental probability of a particular outcome to the theoretical probability. Select the number of spinners, the number of sections on a spinner, and a favorable outcome of a spin. Then tally the number of favorable outcomes. 5 Minute Preview
14: A Preview of Calculus: The Limit, Derivative, and Integral of a Function
14.5: The Area Problem; The Integral
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Riemann Sum
Approximate the area under a curve in an interval using rectangles. Compare the results of left-hand summation to the results of right-hand summation. Vary the interval and the number of rectangles and explore how the graph of the rectangles and curve change in response. 5 Minute Preview
How Free Gizmos Work
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Start teaching with 20-40 Free Gizmos. See the full list.
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Access lesson materials for Free Gizmos including teacher guides, lesson plans, and more.
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All other Gizmos are limited to a 5 Minute Preview and can only be used for 5 minutes a day.
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Free Gizmos change each semester. The new collection will be available January 1 and July 1.
About STEM Cases
Students assume the role of a scientist trying to solve a real world problem. They use scientific practices to collect and analyze data, and form and test a hypothesis as they solve the problems.
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Each STEM Case uses realtime reporting to show live student results.
Introduction to the Heatmap
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STEM Cases take between 30-90 minutes for students to complete, depending on the case.
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Student progress is automatically saved so that STEM Cases can be completed over multiple sessions.
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Multiple grade-appropriate versions, or levels, exist for each STEM Case.
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Each STEM Case level has an associated Handbook. These are interactive guides that focus on the science concepts underlying the case.
Find Your Solution
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