WAYNESBORO AREA SCHOOL DISTRICT CURRICULUM PRE-CALCULUS (June 2014) COURSE NAME: Pre-Calculus UNIT: Chapter 1 NO. OF DAYS: KEY LEARNING (S): UNIT ESSENTIAL QUESTIONS: What methods are used to solve equations and inequalities? How can symmetry, the domain and range, and knowing whether a function is even or odd be used when graphing? STANDARD CH 1: PC.1.1.1 PC.1.1.2 PC.1.2.1 CONCEPTS Eligible Content & Skills Solving Equations and Inequalities in One Variable (1.1) Solve equations in one variable algebraically. Include: Factoring out GCF Factor by grouping Square root method ESSENTIAL QUESTIONS A = Acquisition ET = Extended Thinking What methods are used for finding solutions to equations and inequalities? RESOURCES/ MATERIALS/SUGGESTED ACTIVITIES TIER 2 Inequality Factor Radical Square Root Solution TIER 3 Interval Notation Greatest Common Factor Factor By Grouping Quadratic Formula Square Root Method Use interval notation. Solve inequalities on one variable algebraically.
Symmetry; Graphing Key Equations; Circles (1.4) Test an equation for symmetry with respect to the x-axis, y-axis, and origin. Know how to graph key equations. How is symmetry used when graphing equations? What information is needed to write the standard form equation of circle? Symmetry Center Radius Distance Formula Midpoint Formula Standard Form Equation of a Circle Completing the Square Find the intercepts. Write the standard form equation of a circle. Find the center and radius of a circle from an equation in general form. Introduction to Functions (1.6) Determine whether a relation represents a function. Find the value of a function. Find the domain of a function. How do you determine whether a relation is a function? How do you determine the domain of a function given an equat Relation Function Domain Range Function Notation Difference Quotient Implicit Form Explicit Form Horizontal Line Test Find the sum, difference, product, and quotient of two functions.
The Graph of Functions (1.7) Obtain information from or about the graph of a function. How do you determine the domain and range of a function given a graph? x-intercept y-intercept Vertical Line Test Find the domain and range of a function given the function and its graph. Properties of Functions (1.8) Determine even and odd functions. From a graph From the equation Use a graph to determine where a function is increasing, decreasing, or constant. How do you determine whether a function is even or odd given its equation or graph? Constant Average Rate of Change Odd Function Even Function Increasing Function Decreasing Function Constant Function Local Maximum Local Minimum Use a graph to locate local maxima and minima. Find the average rate of change of a function.
COURSE NAME: Pre-Calculus UNIT: Chapter 2 NO. OF DAYS: KEY LEARNING (S): UNIT ESSENTIAL QUESTIONS: What information is needed to write an equation of a line? How do you graph a line given an equation? STANDARD CH 2: PC.1.1.2 Review 2.2 Algebraic Concepts CONCEPTS Eligible Content & Skills Equations of Lines (2.2) Graph a linear function. Use average rate of change to identify linear functions. Determine whether a linear function is increasing, decreasing, or constant. ESSENTIAL QUESTIONS A = Acquisition ET = Extended Thinking What information is needed to write an equation of a line? How do you graph a line given an equation? RESOURCES/ MATERIALS/SUGGESTED ACTIVITIES TIER 2 Linear Slope Parallel Lines Perpendicular Lines TIER 3 Linear Function Equations: Vertical Line Horizontal Line Point-Slope Form Slope-Intercept Form General Form Identify the slope and y-intercept of a line from its equation. Define/Use: Parallel lines Perpendicular lines Find the equation of: Parallel lines Perpendicular lines Secant line Vertical line Horizontal line A line given two points A line given the slope and one point
COURSE NAME: Pre-Calculus UNIT: Chapter 3 NO. OF DAYS: KEY LEARNING (S): UNIT ESSENTIAL QUESTIONS: How are the properties of quadratic functions used when graphing? How are quadratic functions used to m and solve real-life applications? STANDARD CH 3: PC.1.1.2 PC.1.1.3 PC.1.2.3 CONCEPTS Eligible Content & Skills Quadratic Functions (3.1) Solve quadratic equations by: Factoring Completing the Square Quadratic Formula Solve applied problems. ESSENTIAL QUESTIONS A = Acquisition ET = Extended Thinking What methods are used for finding solutions to quadratic equati RESOURCES/ MATERIALS/SUGGESTED ACTIVITIES TIER 2 TIER 3 Quadratic Function Standard Form Quadratic Formula Completing the Square The Graph of a Quadratic Function (3.2) Graph quadratic functions. Identify/Use: Graph opens up or down Vertex Axis of symmetry Maximum or minimum value How are the properties of quadratic functions used when graphing? Vertex Parabola Vertex Form Equation of a Parabola Axis of Symmetry Use the maximum or minimum value of a quadratic function to solve applied problems.
Quadratic Model: Building Quadratic Functions (3.4) Solve applied problems by building quadratic functions. How are the properties of quadratic functions used to model and solve real-life problems?
COURSE NAME: Pre-Calculus UNIT: Chapter 4 NO. OF DAYS: KEY LEARNING (S): UNIT ESSENTIAL QUESTIONS: How are transformations used when graphing a function? How are functions used to model and solve real-life applications? STANDARD CH 4: PC.1.1.6 PC.1.1.7 PC.1.2.3 CONCEPTS Eligible Content & Skills Radical Equations; Absolute Value Equations; Absolute Value Inequalities (4.1) Solve: Radical equations Absolute value equations Absolute value inequalities ESSENTIAL QUESTIONS A = Acquisition ET = Extended Thinking What is the procedure for solving radical equations? What is the procedure for solving absolute value equations and absolute value inequa How are absolute value equations and inequalities used to model and solve real-life problems? RESOURCES/ MATERIALS/SUGGESTED ACTIVITIES TIER 2 TIER 3 Radical Equation Extraneous Roots (Solutions) Absolute Value Equations Absolute Value Inequalities Library of Functions; Piecewise- Defined Functions (4.2) Identify and graph functions listed in the Library of Functions. Evaluate and graph piecewise functions. Evaluate greatest-integer functions. What is the procedure for evaluating a piecewise function? How do you graph a piecewise func How are piece-wise functions used to model and solve real-life problems? Piecewise-Defined Functions Greatest-Integer Functions Library of Functions: Linear Identity Constant Reciprocal Square Square Root Cubic Absolute Value
Graphing Techniques: Transformations (4.3) Graph functions using transformations. Quadratic Cubic Rational Absolute value Square root Cube root Graph functions using: Horizontal and vertical shifts Reflections Compressions and stretches How is a function identified by its equation and by its graph? How do you apply function transformations to a parent graph? Transformation Transformations: Vertical Shift Horizontal Shift Compression Stretch Reflection
COURSE NAME: Pre-Calculus UNIT: Chapter 5 NO. OF DAYS: KEY LEARNING (S): UNIT ESSENTIAL QUESTIONS: What methods and properties are used to graph polynomial and rational functions? How can functions be u to model and solve real-life applications? STANDARD CH 5: PC.1.1.2 PC.1.1.3 PC.1.1.4 PC.1.1.5 PC.1.1.6 CONCEPTS Eligible Content & Skills Polynomial Functions and Models (5.2) Graph transformations of power functions. Find the power function of best fit to data. Polynomial functions - Identify/Analyze/Use: Degree Zeros Multiplicities Intercepts Arrow behavior of the graph ESSENTIAL QUESTIONS A = Acquisition ET = Extended Thinking How do the degree, end behavior, zeros, and multiplicities of a polynomial function relate to its graph? RESOURCES/ MATERIALS/SUGGESTED ACTIVITIES TIER 2 TIER 3 Power Function Polynomial Function Multiplicity Repeated Zeros End Behavior Turning Points Leading Coefficient Test Graph polynomial functions.
The Real Zeros of a Polynomial Function (5.6) Use the Remainder and Factor Theorems. Use the Rational Root Theorem to list the potential rational zeros of a polynomial function. Find the real zeros of a polynomial function. Solve polynomial functions. Graph polynomial functions. What are some strategies in using the Rational Root Theorem when graphing polynomial functions? Polynomial Long Division Synthetic Division Real Zeros (Roots) Rational Zeros Test (Rational Root Theorem) Coefficient Leading Term Polynomial and Rational Inequalities (5.5) Solve polynomial inequalities. Solve rational inequalities. How do you determine the solution set for polynomial and rational inequalities? Polynomial Inequality Rational Inequality
Rational Functions I (5.3) Find the domain of a rational function. Determine the vertical, horizontal, and oblique asymptotes of rational functions. What techniques are used to analyze and sketch rational functions? Rational Function Vertical Asymptote Horizontal Asymptote Oblique Asymptote Rational Functions II: Analyzing Graphs (5.4) Analyze the graph of a rational function with the aid of a graphing calculator. What techniques are used to analyze and sketch rational functions? Rational Function Vertical Asymptote Horizontal Asymptote Oblique Asymptote Odd or Even Function
COURSE NAME: Pre-Calculus UNIT: Chapter 6 NO. OF DAYS: KEY LEARNING (S): UNIT ESSENTIAL QUESTIONS: What properties are used to solve exponential and logarithmic equations? How are exponential and logarithmic functions used to model and solve real-life applications? STANDARD CH 6: PC.1.1.6 PC.1.1.7 PC.1.2.2 CONCEPTS Eligible Content & Skills Composite Functions (6.1) Form the composite function and find its domain. ESSENTIAL QUESTIONS A = Acquisition ET = Extended Thinking Which operations can be performed on a pair of functions to obtain a third function? RESOURCES/ MATERIALS/SUGGESTED ACTIVITIES TIER 2 Rule TIER 3 Composite Function Inverse Functions (6.2) Find the inverse of a function, Obtain the graph of the inverse function from the graph of the function. What are the steps for finding the inverse of a function? How do you graph the inverse of a function, if it exists? Inverse Interchange Replace Inverse Function One-To-One Function Vertical Line Test Horizontal Line Test Solve for y in terms of x Exponential Functions (6.3) Evaluate exponential functions by writing as powers of same base. What are the characteristics of exponential functions? How are exponential functions evaluated? Base Power Exponent Exponential Function Natural Number e Graph basic exponential functions. Define the number e. Solve exponential equations.
Logarithmic Functions (6.4) Change exponential expressions to logarithmic expressions. Change logarithmic expressions to exponential expressions. Evaluate logarithmic functions by rewriting as exponential functions and then as powers of same base. Solve logarithmic equations. Graph basic logarithmic functions. What are the characteristics of logarithmic functions? What is the relationship between an exponential function and a logarithmic function? How are logarithmic functions evaluated? How are logarithmic equations solved? Logarithmic Function Natural Log Common Log Properties of Logarithms (6.5) Work with the properties of logarithms. Write a logarithmic expression as a sum or difference. Write a logarithmic expression as a single log or rational number. Evaluate logarithmic functions whose base is neither 10 or e using a change of base formula. How are properties of logarithms used when solving logarithmic equations? Change of Base Formula
Logarithmic and Exponential Equations (6.6) Solve logarithmic equations using the properties of logs. How are properties used to solve exponential and logarithmic equations? Solve exponential equations. Compound Interest (6.7) Determine/Calculate: Future value of a lump sum of money Present value of a lump sum of money Time required to double a lump sum of money How are formulas used to model and solve real-life problems? Interest Principle Rate of Interest Simple Interest Compound Interest Compounded Exponential Growth and Decay (6.8) Find equations of populations that obey the Law of Uninhibited Growth. Find equations of populations that obey the Law of Decay. How are exponential growth and decay functions and logarithmic functions used to model and solve real-life problems? Exponential Growth Exponential Decay
COURSE NAME: Pre-Calculus UNIT: Chapter 7 NO. OF DAYS: KEY LEARNING (S): UNIT ESSENTIAL QUESTIONS: How are the exact values of trigonometric functions determined? How is the unit circle used to graph trigonometric functions? STANDARD CH 7: PC.1.1.6 PC.1.2.3 PC.2.1.1 PC.2.1.2 PC.2.1.3 PC.2.1.4 PC.2.1.5 PC.2.2.1 CONCEPTS Eligible Content & Skills Angles and Their Measures (7.1) Convert between measures: Degrees, minutes, seconds and decimal degrees Radian measure and degree measure Draw standard position angles. ESSENTIAL QUESTIONS A = Acquisition ET = Extended Thinking How are positive and negative angles sketched? How are radian and degree measures of an angle related? RESOURCES/ MATERIALS/SUGGESTED ACTIVITIES TIER 2 Counterclockwise Clockwise Degree Measure Decimal Degrees Degrees-Minutes- Seconds (DMS) TIER 3 Initial Side Terminal Side Positive Angle Negative Angle Standard Position Radian Measure
Trigonometric Functions: Unit Circle Approach, Properties of the Trigonometric Functions (7.2/7.3) Find the exact values of the trigonometric functions: Using a point on the unit circle Of quadrantal angles Of the reference angles in radians and degrees Utilizing fundamental identities Of an angle given on the functions and the quadrant of the angle Use a calculator to approximate the value of a trigonometric function. Determine the signs of the trigonometric functions in each quadrant. How are the exact values of trigonom functions determined given any angle in degrees or radians? Unit Circle Coterminal Angles Reference Angles Reference Triangles Trigonometric Ratios
Graphs of the Sine and Cosine Functions (7.4/7.6) Define/Identify/Use: Cosine curve Sine curve Amplitude Period Phase shift Vertical shift Describe how the graphs of sine and cosine to the unit circle. Graph sinusoidal functions. How are the graphs of sine and cosine related to the values of the unit circle? What effect do changes in amplitude, period, phase shift, and vertical shift have on the graphs of sine and cosine? Amplitude Period Phase Shift Vertical Shift Unit Circle Radian Measure Trigonometric Ratios Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions (7.5) Recognize the graphs of tangent, cotangent, secant, and cosecant. Describe how the graphs of tangent, cotangent, secant, and cosecant relate to the unit circle. How do the graphs of tangent, cotangent, cosecant, and secant relate to the graphs of their reciprocal functions and the unit circle?
COURSE NAME: Pre-Calculus UNIT: Chapter 8 NO. OF DAYS: KEY LEARNING (S): UNIT ESSENTIAL QUESTIONS: How can trigonometric identities be used to evaluate trigonometric functions, simplify trigonometric expressions, and rewrite trigonometric expressions? STANDARD CH 8 PC.1.1.6 PC.2.1.6 PC.2.1.7 PC.2.2.3 CONCEPTS Eligible Content & Skills Inverse Trigonometric Functions (8.1) Find the exact value of the inverse of sine, cosine, and tangent functions. ESSENTIAL QUESTIONS A = Acquisition ET = Extended Thinking How are inverse trigonometric funct evaluated? RESOURCES/ MATERIALS/SUGGESTED ACTIVITIES TIER 2 TIER 3 Restricted Domain Inverse Trigonometric Functions Find the approximate value of the inverse sine, cosine, and tangent functions. Trigonometric Identities (8.3) Use algebra to simplify trigonometric expressions. Establish identities. What strategies are used to verify trigonometric identities? Trigonometric Identities Even-Odd Functions Pythagorean Identities Sum and Difference Formulas (8.4) Use sum and difference formulas to find exact values. How are the sum and difference formulas used to evaluate trigonometric functions? Sum Formula Difference Formula
Double-Angle and Half-Angle Formulas (8.5) Use double-angle formulas to find exact values. How are double-angle and half-angle formulas used to evaluate trigonometric functions? Double-Angle Formula Half-Angle Formula Trigonometric Equations 1 (8.7) Solve equations involving a single trigonometric function. How are properties and trigonometric identities used to solve trigonometric equations? Trigonometric Equation Trigonometric Equations 2 (8.8) Solve trigonometric equations using identities and algebraic properties. How are properties and trigonometric identities used to solve trigonometric equations?
COURSE NAME: Pre-Calculus UNIT: Chapter 9 NO. OF DAYS: KEY LEARNING (S): UNIT ESSENTIAL QUESTIONS: How do you find the unknown sides and angles of triangles? How is trigonometry used to model and solve real-life applications? STANDARD CH 9: PC.2.1.8 CONCEPTS Eligible Content & Skills Right Triangle Trigonometry (9.1) Find the value of the trigonometric functions of acute angles. ESSENTIAL QUESTIONS A = Acquisition ET = Extended Thinking How are trigonometric functions used to solve right triangles? How are trigonometric functions used to model and solve reallife applications? RESOURCES/ MATERIALS/SUGGESTED ACTIVITIES TIER 2 TIER 3 Trigonometric Ratios Pythagorean Theorem Solve right triangles. Solve applied problems using right triangle trigonometry. Review theorems from Geometry concerning triangles. The Law of Sines (9.2) Solve SAA, ASA, or SSA triangles. Solve applied problems using the Law of Sines. When is the Law of Sines used to solve a triangle? What is the ambiguous case in the Law of Sines? How do you determine which situation is present in a given triangle? Oblique Triangle Law of Sines Ambiguous Case How is the Law of Sines used to model and solve real-life applications?
The Law of Cosines (9.3) Solve SAS and SSS triangles. Solve applied problems using the Law of Cosines. When is the Law of Cosines used to solve a triangle? How is the Law of Cosines used to model and solve real-life applications? Law of Cosines Area of a Triangle (9.4) Find the area of SAS and SSS triangles. How is Heron s formula and the SAS theorem used to find the area of an oblique triangle? Heron s Formula
COURSE NAME: Pre-Calculus UNIT: Chapter 10 NO. OF DAYS: KEY LEARNING (S): UNIT ESSENTIAL QUESTIONS: How are trigonometric functions used to convert points in a plane in both rectangular coordinates and polar coordinates? STANDARD CH 10: PC.2.2.2 PC.3.2.1 CONCEPTS Eligible Content & Skills Polar Coordinates (10.1) Plot points using polar coordinates. Convert between polar coordinate and rectangular coordinate measures. ESSENTIAL QUESTIONS A = Acquisition ET = Extended Thinking How are trigonometric functions used to convert points in a plane in both rectangular and polar forms? RESOURCES/ MATERIALS/SUGGESTED ACTIVITIES TIER 2 TIER 3 Rectangular Coordinates Polar Coordinates Pole Polar Axis Directed Distance Directed Angle Polar Equations and Graphs (10.2) Graph polar equations using a graphing calculator. What characteristics may be present in the graphs of polar equations? Polar Equation Polar Graph Determine characteristics of a polar equation and its graph.