Precalculus with Limits Larson Hostetler. `knill/mathmovies/ Assessment Unit 1 Test

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1 Unit 1 Real Numbers and Their Properties 14 days: 45 minutes per day (1 st Nine Weeks) functions using graphs, tables, and symbols Representing & Classifying Real Numbers Ordering Real Numbers Absolute Value Distance Evaluating Algebraic Expressions Properties of Zero Properties & Operations with Fractions `knill/mathmovies/ Unit 1 Test Real numbers Subsets Rational Irrational Real number line Origin Nonnegative Order Inequality Intervals Endpoints Positive infinity Negative infinity Absolute value Variables Algebraic Algebraic expression Constants Terms Variable terms Constant term Coefficient Evaluate Additive inverse Multiplicative inverse Numerator Denominator Factors or divisors Prime number Composite Numerical representations can be used to describe quantitative relationships. How is mathematics used to quantify and compare situations, events and phenomena? Do all types of numbers make sense in all problem situations? Explain.

2 Unit 2 Exponents & Radicals Part I 11 days: 45 minutes per day (1 st Nine Weeks) and graphically functions using graphs, tables, and symbols representation of a function represented graphically and numerically parent functions P.2(B) perform operations including compositions of functions, find inverses, and describe these procedures and results verbally, numerically, symbolically, and graphically P.2(C) investigate identities graphically and verify them symbolically, including logarithmic properties, trigonometric identities, and exponential properites Integer Exponents of degree 2 and 3 Scientific Notation Radicals of root 2 and 3 and Their Properties Simplifying Radicals Rationalizing Denominators and Numerators Exponential form Exponent Base Power Scientific notation Square root Cube root Nth root Principal nth root Radical symbol Index Radicand Perfect squares Perfect cubes Simplest forms Like radicals Conjugate Rational exponent model real-life data Understanding exponents allows us to express numeric values more accurately. Unit 2 Test How are exponents and radicals related? Why is it beneficial to express numbers in exponent form, such as scientific notation?

3 Unit 3 Exponents and Radicals - Part II 7 days: 45 minutes per day (1 st Nine Weeks) represented graphically and numerically - parent functions - P.2(B) perform operations including compositions of functions, find inverses, and describe these procedures and results verbally, numerically, symbolically, P.2(C) investigate identities graphically and verify them symbolically, including logarithmic properties, trigonometric identities, and exponential properties - Radicals of root 3 and higher and Their Properties Simplifying Radicals Rationalizing Denominators and Numerators Rational Exponents Exponential form Exponent Base Power Nth root Principal nth root Index Radicand Simplest forms Like radicals Conjugate Rational exponent Understanding exponents allows us to express numeric values more accurately. Unit 3 Test How are exponents and radicals related? How are rational exponents and radicals related?

4 Unit 4 Polynomials and Factoring Part I 7 days: 45 minutes per day (1 st Nine Weeks) represented graphically and numerically - P.3(A) investigate properties of trigonometric and polynomial functions Polynomials Operations with Polynomials Special Products In mathematics, as in real life, it is always best to make a problem easier before trying to solve it. Polynomial Coefficient Degree Leading coefficient Constant term FOIL Method Unit 4 Test What does a quadratic equation tell me before it is factored, and after? What information do I have? What information do I need? How do I get that information?

5 Unit 5 Polynomials and Factoring Part II 7 days: 45 minutes per day (1 st Nine Weeks) P.3(A) investigate properties of trigonometric and polynomial functions - and solve problems and make predictions Polynomials with Common Factors Factoring Special Polynomial Forms Trinomials with Binomial Factors Factoring by Grouping In mathematics, as in real life, it is always best to make a problem easier before trying to solve it. Unit 5 Test Factoring Prime or irreducible over the integers Completely factored Conjugate pairs Perfect square trinomial Factoring by grouping What does a quadratic equation tell me before it is factored, and after? What information do I have? What information do I need? How do I get that information?

6 Unit 6 Rational Expressions 12 days: 45 minutes per day (2 nd Nine Weeks) represented graphically and numerically - Domain of an Algebraic Expression Simplifying Rational Expression Operations with Rational Expressions Complex Fractions and the Difference Quotient Domain Equivalent Rational expression LCD (least common denominator) Complex fractions Difference quotient parent functions - and solve problems and make predictions - In mathematics, as in real life, it is always best to make a problem easier before trying to solve it. Unit 6 Test Have I seen this before? How does that connection help? Why is it important to factor completely when simplifying a rational expression?

7 Unit 7 Solving Equations 15 days: 45 minutes per day (2 nd Nine Weeks) represented graphically and numerically - parent functions - P.2(B) perform operations including compositions of functions, find inverses, and describe these procedures and results verbally, numerically, symbolically, P.2(C) investigate identities graphically and verify them symbolically, including logarithmic properties, trigonometric identities, and exponential properties - Equations and Solutions of Equations Linear Equations in One Variable Quadratic Equations Polynomial Equations of Higher Degree Equations Involving Radicals Equations with Absolute Values Equation Solve Solutions Identity Conditional equation Equivalent equations Extraneous solution Quadratic equation Second-degree polynomial equation Mathematicians formulate equations or functional relationships to communicate generalizations (general patterns, rules and connections to prior concepts that are at the core of the problem) so that specific problems can be solved more efficiently. Unit 7 Test What information do I have? What information do I need? How do I get that information? What does it mean to have an extraneous solution?

8 Unit 8 Linear Inequalities in One Variable 11 days: 45 minutes per day (2 nd Nine Weeks) P.1(C) describe symmetry of graphs of even and odd functions - represented graphically and numerically - parent functions - Introduction Properties of Inequalities Solving a Linear Inequality in One Variable Inequalities Involving Absolute Values Applications Solve an inequality Solutions Satisfy Properties of Inequalities Equivalent Double inequality P.2(B) perform operations including compositions of functions, find inverses, and describe these procedures and results verbally, numerically, symbolically, P.2(C) investigate identities graphically and verify them symbolically, including logarithmic properties, trigonometric identities, and exponential properties - and solve problems and make predictions - Mathematicians formulate equations or functional relationships to communicate generalizations (general patterns, rules and connections to prior concepts that are at the core of the problem) so that specific problems can be solved more efficiently. Unit 8 Test What are similarities and differences between different types of double inequalities? How can mathematics be used to provide models that help us interpret data and make predictions

9 Unit 9 Functions and their Graphs Part I 12 days: 45 minutes per day (3 rd Nine Weeks) P.1(C) describe symmetry of graphs of even and odd functions - represented graphically and numerically - parent functions - Rectangular Coordinate Graphs of Equations Linear Equations in Two Variables Unit 9 Test Rectangular coordinate system or the Cartesian plane x-axis and y-axis origin quadrants ordered pair x-coordinate and y-coordinate Distance Formula Midpoint Formula Equation in two variables Solution or solution point Graph of an equation Point-plotting method Intercepts Symmetry Circle Standard form of the equation of a circle Linear equation in two variables Slope Slope-intercept form Point-slope form Two-point form Parallel and Perpendicular Ratio Rate or rate of change Linear extrapolation Linear interpolation Mathematicians formulate equations or functional relationships to communicate generalizations (general patterns, rules and connections to prior concepts that are at the core of the problem) so that specific problems can be solved more efficiently. What information do I have? What information do I need? How do I get that information? How can I translate a graphical representation into an algebraic and vice versa?

10 Unit 10 Functions and their Graphs Part II 10 days: 45 minutes per day (3 rd Nine Weeks) P.1(C) describe symmetry of graphs of even and odd functions - parent functions - P.2(B) perform operations including compositions of functions, find inverses, and describe these procedures and results verbally, numerically, symbolically, P.3(A) investigate properties of trigonometric and polynomial functions - and solve problems and make predictions - Functions Analyzing Graphs of Functions A Library of Parent Functions Transformations of Functions Unit 10 Test Relation Function Domain and Range Independent variable Dependent variable Function notation Piecewise-defined function Implied domain Graph of a function Vertical Line Test Zeroes of a function Increasing Decreasing Constant Relative minimum Relative maximum Average rate of change Secant line Even and Odd Linear function Constant functions Identity function Squaring function Cubic, Square Root, and Reciprocal functions Step functions Greatest integer function Vertical and horizontal shifts Reflection Rigid transformations Nonrigid transformations Vertical stretch and vertical shrink Horizontal stretch and horizontal shrink Arithmetic combination

11 Graphs of functions help us visualize a real-life situation in a mathematical format. What properties or characteristics do all parent functions have in common? How does changing an equation affect the graph and vice versa? Unit 11 Functions and their Graphs Part III 10 days: 45 minutes per day (3 rd Nine Weeks) parent functions - P.2(B) perform operations including compositions of functions, find inverses, and describe these procedures and results verbally, numerically, symbolically, Combination of Functions: Composite Functions Inverse Functions Mathematical modeling and Variation Composition Inverse function One-to-one functions Sum of square differences Least squares regression line Correlation coefficient Vary directly or directly proportional Constant of variation or constant of proportionality Varies inversely Inversely proportional Combined variation Jointly Varies jointly or Jointly proportional P.3(C) use regression to determine the appropriateness of a linear function to model real-life data and solve problems and make predictions - Graphs of functions help us visualize a real-life situation in a mathematical format. Unit 11 Test What is a one-to-one function? What does the constant of variation (k) represent in direct variation and inverse variation?

12 Unit 12 Quadratic Functions and Models 9 days: 45 minutes per day (4 th Nine Weeks) P.3(A) investigate properties of trigonometric and polynomial functions - The Graph of a Quadratic Function The Standard Form of a Quadratic Function Applications Polynomial function of x with degree n Quadratic function Parabola Axis of symmetry or axis Vertex Standard form and solve problems and make predictions - Unit 12 Test Mathematicians communicate through words, numbers, graphs, and symbols, moving fluently from one representation to another as the situation requires. What does the equation of a quadratic tell us about its graph and vice versa? What common mistakes do people make when working with this type of problem? What is the misunderstanding that causes this mistake?

13 Unit 13 Polynomial Functions of Higher Degree 8 days: 45 minutes per day (4 th Nine Weeks) represented graphically and numerically - P.3(A) investigate properties of trigonometric and polynomial functions - Graphs of Polynomial Functions The Leading Coefficient Test Zeroes of Polynomial Functions The Intermediate Value Theorem Continuous Power functions Leading Coefficient Test Repeated zero Multiplicity Test intervals Intermediate Value Theorem and solve problems and make predictions - Mathematicians communicate through words, numbers, graphs, and symbols, moving fluently from one representation to another as the situation requires. Unit 13 Test What makes a function continuous? What does the intermediate value theorem tell us?

14 Unit 14 Rational Functions 12 days: 45 minutes per day (4 th Nine Weeks) represented graphically and numerically - and solve problems and make predictions - Introduction Horizontal and Vertical Asymptotes Analyzing Graphs of Rational Functions Slant Asymptotes Applications Unit 14 Test Rational function Vertical asymptote Horizontal asymptote Hyperbolas Slant (oblique) asymptote Mathematicians formulate equations or functional relationships to communicate generalizations so that specific problems can be solved more efficiently. When factors cancel in a rational function, how is that represented graphically? How do you determine vertical and horizontal asymptotes?

15 Unit 15 Exponential and Logarithmic Functions 5 days: 45 minutes per day (4 th Nine Weeks) represented graphically and numerically - P.2(C) investigate identities graphically and verify them symbolically, including logarithmic properties, trigonometric identities, and exponential properites - and solve problems and make predictions - Exponential Functions and their Graphs Logarithmic Functions and their Graphs Properties of Logarithms Exponential and Logarithmic Equations Exponential and Logarithmic Models Unit 15 Test Transcendental functions Exponential function One-to One Property Natural base Natural exponential function Continuous compounding Logarithmic function with base a Common logarithmic function Natural logarithmic function Change-of-base formula Exponential growth model Exponential decay model Gaussian model Logistic growth model Logarithmic models Normally distributed Bell-shaped curve Average value Logistic curve Sigmoidal curve Functions and relations (as well as their inverses) help us formulate potential solutions for real-life problems involving two variables. What information do I have? What information do I need? How do I get that information? How can you prove that exponential and logarithmic functions are inverses? How does that help us?

16 Unit 16 Trigonometry 10 days: 45 minutes per day (4 th Nine Weeks) P.1(C) describe symmetry of graphs of even and odd functions - parent functions - P.2(C) investigate identities graphically and verify them symbolically, including logarithmic properties, trigonometric identities, and exponential properties - P.3(A) investigate properties of trigonometric and polynomial functions - and solve problems and make predictions - P.3(E) solve problems from physical situations using trigonometry, including the use of Law of Sines, Law of Cosines, and area formulas and incorporate radian measure where needed. Radian and Degree Measure Trigonometric Functions: The Unit Circle Right Triangle Trigonometry Trigonometric Functions of Any Angle Graphs of Sine and Cosine Functions Inverse Trigonometric Functions Applications and Models Unit 16 Test Trigonometry Angle Initial side and Terminal side Vertex Standard position Positive angles and negative angles Coterminal Measure of an angle Central angle Radian Complementary and supplementary Degrees Linear speed Angular speed Sector Sine, Cosecant, Cosine, Secant, Tangent, Cotangent Periodic Hypotenuse Opposite side Adjacent side Solving right triangles Angle of elevation Angle of depression Reference angles Since curve One cycle Key points Amplitude Phase sift Damping factor Inverse sine function Inverse tangent function Simple harmonic motion

17 The properties of trigonometry help us determine measurements that cannot be physically calculated. Explain what the acronym SOHCAHTOA means and how is it used in mathematics and measurement? What are the characteristics of a periodic graph?