PRE-CALCULUS General Specific Math Skills

Transcription

1 PRE-CALCULUS Welcome to Pre-Calculus! Pre-Calculus will be challenging but rewarding!! This full year course requires that everyone work hard and study for the entirety of the class. You will need a large binder or notebook and a graphing calculator for the course. Student expectations: General keep an organized notebook take good notes complete homework every night be active learners ask questions and participate in class seek help outside of class if needed work with others work with and without a calculator Specific Math Skills 1) Algebra a. Can manipulate with ease fractions, decimals and variables in a variety of settings including in equations, expressions, and rational functions b. Comfortable with all forms of factoring including quadratics, sum and difference of cubes, quartics and factoring by grouping c. Add, subtract, multiply and divide radical expressions including rationalizing denominators d. Solve equations and inequalities, including but not limited to i. Polynomial ii. Logarithmic iii. Rational iv. Radical v. Exponential vi. Absolute Value vii. Systems e. Manipulate equations to solve for a single variable f. Know and use the laws of exponents to simplify expressions g. Solve systems of equations h. Identify functions from graphs and tables i. Be able to write a polynomial from a given set of roots j. Know the Pythagorean Theorem k. Demonstrate mastery of finding the inverse of a function 2) Graphing a. Be familiar with the graphs of linear, absolute value, quadratic, cubic, quartic, logarithmic and exponential(growth and decay) functions b. Linear Graphs i. Be able to find the slope and y-intercept of a line from both a graph and an equation ii. Be able to determine whether two lines are perpendicular c. Recognize end behaviors of graphs d. Be familiar with transformations of graphs

2 3) Trigonometry a. Work with the basic six trig functions including manipulating them to simplify expressions and solve equations by finding all solutions b. Know the trig identities i. Reciprocal ii. Quotient iii. Pythagorean c. Know common values of the six basic trig functions using at least one of the following methods: i. Memorizing the trig table of values (link for practice) ii. Memorizing the unit circle iii. Finding the values from special right triangles d. Know the graphs of the six basic trig functions including their domain and range e. Convert easily between degrees and radians f. Be able to verify equality in trig equations You should be able to use, describe, and give examples of the following vocabulary: Reference Angles Coterminal Angles Asymptotes Zeros (of a polynomial function) Roots (of a function) *NOTE: Show all of your work. Your teacher will give a quiz on this material at the beginning of the year. You should Google the topic if you are unsure how to complete the examples. Useful resources are also linked at the end of this document. Some answers are given for particular problems for you to check your work. Review Problems I. Graphing Sketch the graph of each function & state the domain and range of each. Then, choose the name of the function from the word bank below. Not every word is used, some words are used more than once. Cubic Linear Radical Exponential (Growth) Logarithmic (log) Rational Exponential (Decay) Quadratic Trigonometric (Trig)

3 A) y = x B) y = x2

4 II. Factoring Factor each of the following completely or state that it is prime.

5 III. Laws of Exponents IV. Manipulating Equations For problems 1 4, solve the Linear Equation for the indicated variable. 1) 2(g h) + 3 = 2h; solve for h 2) 3m 2n 2 = 4; solve for n 3) ab 2 + c = k; solve for b 4) 3rt 2mn = 4ab; solve for r

6 Question 5. Question 6. The preceding formula gives the monthly payment m needed to pay off a loan of P dollars at r percent annual interest over N months. Give P in terms of m, r, and N? Answer:

7 V. Simplifying Expressions and Solving Equations Simplify. 1) 2) 3) 4) Simplify.

8 Solve. 1) ln (x-2) = 4 2) -3ln(x+1) = 2 3) ln(x-2) + ln(3)= 5 4) 100 = 200e 0.06x Solve each system. Then explain how the solutions would appear on a coordinate grid. 5) 6) Solve each equation. Then state which values make the functions undefined. 7) 8)

9 9) 10) 11) 12) Solve each of the following equations by the method of your choice and check your solutions below. 13. x = x 2 + x = y 2 + 6y 8 = y 2 25 = 0

10 Answers: A polynomial function with rational coefficients has the following zeros. Find all additional zeros.

11 Answers to check your work A polynomial function with rational coefficients has the following zeros. Find all additional zeros.

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13 Estimate the zeros of each function graphed below.

14 Answers to check your work

15 VI. Linear Functions Graph each of the following equations, then identify the slope and y-intercept. 1) y = 3x + 4 2) 8x + 2y = 32 3) y =7 4) x = 9 5) Write an equation that creates a line parallel to the function y = ¼ x 20. 6) Write an equation that creates a line perpendicular to the function y = ¼ x 20. Write the formula for each transformation. VII. Transformations A) y = x! a translation right four units and up 3 units B) y = x a reflection about the x-axis and a vertical stretch to 3 times its height C) 2 x a reflection about the y-axis and then translate right 3 units D) y = x a translation left 3 units and a horizontal shrink to ¼ its usual size

16 VIII. Identifying Functions Determine whether or not each graph below represents a function. Identifying Functions from Tables

17 Reference Angles and Coterminal Angles VIII. Trigonometry

18 Trig Facts: You will need to know these as you know your multiplication tables. The values for sin, cos, and tan in Quadrant I should be memorized cold. The other Quadrants values should not take you more than a few seconds to find when quizzed. Fill in the blank: Every trig function takes an angle as input and returns a ratio as output. So, every inverse trig function takes a ratio as input and returns as output.

19 Fill out the following 16-point unit circle by finding the following: 1) The measures for each angle in radians and in degrees. 2) The coordinate pair for each angle. Converting between radians and degrees. 1. Convert the following angles given in degrees, to radians. Do not use a calculator and give your answers as multiples of π. a) 90 b) 72 c) Convert each of the following angles given in radians, to degrees. Give your answers correct to two decimal places. a) 3 radians b) 2.4 radians c) 1 radian

20 3. Convert each of the following angles given in radians, to degrees. Do not use a calculator. a)!!" b)!! Answers 1. a)! radians, b)!!!! radians, c) radians!!! 2. a) , b) , c) a) 12, b) 36 Finding Trigonometric (Trig) Values Find cos θ and tan θ. Find sin θ and tan θ. Find sin θ and tan θ. Find sin θ and tan θ. Online Instructional Resources (Short 5-10 minute videos for classes through Algebra 2) (Good for basic math, algebra, and geometry) (Click Lessons on the homepage) (Great math lessons and practice problems with solutions) (Or search West Texas A&M Virtual Math Lab on yahoo) (Videos on just about any topic) (Short videos on just about any topic. View the ratings and/or user comments first.) (collection of short videos/tutorials for all classes; many topics found under the Algebra link) (simple explanations and examples for sophisticated topics)