Precalculus

Topic

Lesson Topic

Subtopics

Objectives

1

Geometry and Measurement

1. Geometric Figures

1. Construct and deconstruct standard solid geometry figures.
2. Calculate perimeter, area, surface area and volume for standard geometric figures.

2

Basic Algebraic Operations

1. Real Numbers and Polynomials
2. Rational Expressions
3. Rational Exponents and Radicals

1. Identify and use properties of real numbers.
2. Simplify algebraic expressions.
3. Identify and classify polynomial expressions.
4. Perform operations on polynomials.
5. Factor polynomials.
6. Write a rational expression in simplest  form.
7. Compute rational expressions.
8. Simplify radical expressions.
9. Multiply and divide radical expressions.
10. Factor quadratic trinomials when a leading coefficient equals 1 and when leading coefficient is different  from 1

3

Linear Equations and Inequalities in One Variable

1. Linear Equations and Applications
2. Linear Inequalities and Applications
3. Absolute Value in Equations and Inequalities

1. Solve linear equations by using all properties of equality and the rules.
2. Solve word problems using linear equations.
3. Use the notation of inequalities.
4. Solve and graph linear inequalities.
5. Solve an application using inequalities.
6. Solve absolute value equations and inequalities.

4

Polynomial and Other Equations

1. Solving Polynomial Equations
2. Equations Involving Radicals and Rational Exponents
3. Complex Numbers

1. Solve quadratic equations using the quadratic formula.
2. Solve word problems involving quadratic Radicals and Rational equations.
3. Solve polynomial equations using the zero factor property.
4. Solve applications using these equation types.
5. Identify and simplify complex numbers.
6. Add and subtract complex numbers.
7. Multiply and divide complex numbers.
8. Solve rational and radical equations.
9. Solve quadratic equations using factoring, the square root property, and completing the square.
10. Solve quadratic equations by completing the square.

5

Functions and Graphs

1. Rectangular Coordinates and the Graph of a Line
2. Relations, Functions, and Graphs
3. Linear Functions

1. Use a table of values to graph linear equations
2. Determine when lines are parallel or perpendicular.
3. Use linear graphs in an applied context.
4. Identify functions and state their domain and range.
5. Use function notation.
6. Write a linear equation in function form.
7. Use function form to identify the slope.
8. Use slope-intercept form to graph linear functions.
9. Write a linear equation in point-intercept form.
10. Use these forms to solve applications.

6

Operations on Functions and Analyzing Graphs

1. The Algebra and Composition of Functions
2. One-to-One and Inverse Functions 
3. Transformations and Symmetry

1. Compose two functions and find the domain.
2. Identify one-to-one functions.
3. Find inverse functions using an algebraic  method.
4. Graph a function and its inverse.
5. Use symmetry as an aid to graphing.
6. Perform stretches and compressions on a basic graph.
7. Perform vertical and horizontal shifts and reflections of a basic graph.

7

Graphing Polynomial and Rational Functions

1. Graphing Polynomial Functions
2. Asymptotes and Rational Functions
3. Graphing Rational Functions

1. Graph quadratic functions by completing the square and using transformations.
2. Graph a general quadratic function using the vertex formula.
3. Solve applications involving quadratic  functions.
4. Graph polynomial functions.
5. Identify horizontal and vertical asymptotes.
6. Use asymptotes to determine the equation of a rational function from its graph.
7. Graph general rational functions.
8. Solve applications involving rational functions.
9. Find the domain and intercepts of a rational function.

8

Exponential and Logarithmic Equations

1. Exponential Functions
2. Logarithms and Logarithmic Functions
3. The Exponential Function and Natural Logarithm

1. Evaluate an exponential function.
2. Graph exponential functions.
3. Solve certain exponential equations.
4. Write exponential equations in logarithmic form.
5. Graph logarithmic functions and find their domains.
6. Apply the properties of logarithms.
7. Evaluate and graph the natural logarithm and exponential functions.
8. Solve applications of logarithmic and exponential functions.

9

Exponential and Logarithmic Equations

1. Exponential Equations
2. Logarithmic Equations
3. Applications of Exponential and Logarithmic Equations

1. Write logarithmic and exponential  equations in simplified form.
2. Solve exponential equations.
3. Solve logarithmic equations.
4. Solve applications involving exponential and logarithmic equations.
5. Use exponential equations to find the interest compounded n times per year.
6. Use exponential equations to find the interest compounded continuously.

10

An Introduction to Trigonometric Functions

1. Special Angels and the Unit Circle
2. Graphs of Basic Trigonometric Functions
3. Applications of Basic Trigonometric Functions

1. Correctly use vocabulary associated with a study of angles and triangles.
2. Convert between degrees and radians for nonstandard angles.
3. Define the six trigonometric functions in terms of a point on the unit circle or in terms of a real number.
4. Identify and discuss important characteristics of tangent and cotangent.
5. Solve applications of trigonometric functions.
6. Find values of the six trigonometric functions from their ratio definition.
7. Graph the basic trigonometric functions.

11

Trigonometric Identities

1. Transformations and the Applications of Trigonometric Graphs
2. Family of Trigonometric Identities
3. The Inverse Trigonometric Functions and Their Applications

1. Use fundamental identities to express a given trigonometric function in terms of the other five.
2. Solve applications using these identities.
3. Find the inverse trigonometric functions and evaluate related expressions.
4. Apply the definition and notation of inverse trigonometric functions to simplify expressions.
5. Graph sine and cosine functions with various amplitudes and periods.
6. Write the equation for a given graph.

12

Applications of Trigonometry

1. The Law of Sines
2. The Law of Cosines
3. More Applications of Trigonometry
4. Vectors
5. Polar Coordinates
6. Parametric Equations

1. Solve ASA and AAS triangles.
2. Use the Law of Sines to solve  applications.
3. Apply the Law of Cosines when two sides  and an included angle are known (SAS).
4. Apply the Law of Cosines when three sides are known (SSS).
5. Solve applications using the Law of Cosines.
6. Solve more applications involving trigonometric functions.
7. Solve the SSA case, including the ambiguous case.
8. Perform basic operations with vectors graphically and component-wise.
9. Solve applied problems involving vectors.
10. Convert polar coordinates and equations into rectangular coordinates and vice versa.
11. Sketch the graph of a polar equation.
12. Evaluate powers and roots of complex numbers using the polar form of complex numbers.
13. Given a set of parametric equations, convert them into a Cartesian equation or vice versa.
14. Sketch the graph of a set of parametric equations.

13

Systems of Equations and Inequalities

1. Solving Systems Graphically, by Substitution, and Using Elimination
2. Solving Linear Systems Using Matrix Equations
3. Applications of Linear Systems
4. Non-linear Equations and Linear Inequalities

1. Solve linear systems by graphing, by substitution, and by elimination.
2. Use system of equations to mathematically model and solve applications.
3. Form the augmented matrix of a system of equations.
4. Solve a system of equations using row operations.
5. Recognize inconsistent and dependent systems.
6. Use system of equations to mathematically model and solve applications.
7. Solve systems of equations involving quadratic and other non-linear functions.
8. Graph systems of linear inequalities.

14

Conic Sections

1. The Parabola
2. The Ellipse and the Circle
3. The Hyperbola

1. Define and identify a parabola.
2. Graph a parabola.
3. Solve applications of parabolas.
4. Define and identify an ellipse and a circle.
5. Graph an ellipse and a circle.
6. Solve applications of ellipses and circles.
7. Define and identify a hyperbola.
8. Graph a hyperbola.
9. Solve applications of hyperbolas.

15

Sequences and Series

1. Sequences and Series
2. Arithmetic Sequences
3. Geometric Sequences

1. Write the terms of a sequence given the general term.
2. Determine the general term of a sequence.
3. Find the partial sum of a series.
4. Use summation notation to write and evaluate series.
5. Solve applications involving arithmetic sequences.
6. Find the sum of a geometric series.
7. Solve application problems involving geometric sequences and series.

16

Course Review

1. Course Review

1. None