Precalculus Prerequisites Ridgewood High School

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1 Precalculus Prerequisites Ridgewood High School Future Precalculus Student: Mathematics builds! To be successful in Precalculus, there are certain skills that you are expected to have already mastered. Ten of these prerequisites are summarized in this packet. Although some of the topics listed here may be reviewed in Precalculus, you are expected to have familiarity with them so that we can quickly move beyond the basics to higher-level discussions. Successful completion of Algebra I, Geometry, and Algebra II are all prerequisites for taking Precalculus. We are excited you have chosen to enroll in this course; you will begin to see many powerful connections between the mathematics courses you have previously taken while also preparing to take Calculus at the collegiate level. Sample prerequisite problems (with solutions) are included in this packet. There will be a formal assessment over this material within the first two weeks of school; the assessment will consist of problems that have a similar format to the problems included in this packet. To best prepare for this assessment and for success in Precalculus, you should begin working on these problems to gauge your familiarity with them. At the very latest, make sure to begin this packet by August 1 to allow yourself sufficient time to review the material before returning to school. If you cannot successfully complete the practice problems for a learning objective or have difficulty with any of these topics, review your notes from prior classes or visit Interactmath.com is a free companion website to our Precalculus textbook and your prior math course textbooks. This website offers excellent step-by-step instructions and helpful hints for solving each type of problem. Each prerequisite learning objective listed in this packet includes a suggestion for where to go on to practice the topic. Additionally, you can also look on the internet for tutorials on specific topics on websites such as or any other websites from your preferred search engine. Be persistent and resourceful until you find a tutorial that is helpful, understandable, and provides good examples with answers for you to follow. Do not accept simply getting an answer; it is important that you understand how to successfully complete these types of review problems. Don t panic if you are rusty on a couple of the topics listed in this packet. The first few days of class will be devoted to some review and filling in gaps. Also, the RHS Math Department teachers are all available to help you. It s important, however, that you get this material at your fingertips right away, because we ll be drawing on these skills frequently. You will be held accountable for mastery of this material throughout the year. We are excited to have your in our Precalculus class next fall! The Ridgewood Mathematics Department 1

2 Precalculus Prerequisites List of Learning Objectives Geometry 1. Use the Pythagorean Theorem to find side lengths of right triangles Interactmath.com: Prentice Hall Geometry 2011, Chapter 8, Section 1 2. Use properties of special right triangles to find side lengths and angle measurements Interactmath.com: Prentice Hall Geometry 2011, Chapter 8, Section 2 3. Solve right triangles using trigonometry Interactmath.com: Prentice Hall Geometry 2011, Chapter 8, Section 3 Algebra 1 4. Simplify radicals and rational denominators Interactmath.com: Prentice Hall Algebra , Chapter 10, Section 2 5. Solve linear equations Interactmath.com: Prentice Hall Algebra , Chapter 2, Sections 3, 4, and 5 6. Factor expressions including greatest common term, difference of squares, and trinomials Interactmath.com: Prentice Hall Algebra , Chapter 8, Sections 2, 3, 4, 5, and 7 7. Manipulate fractions without a calculator Interactmath.com: Clark Applied Basic Mathematics, Chapter 2, Sections 1-7 Algebra 2 8. Solve quadratic equations Interactmath.com: Prentice Hall Algebra , Chapter 4, Sections 4, 5, and 7 9. Analyze functions graphically (including identifying transformations, finding domain/range and writing equations of asymptotes) Interactmath.com: Prentice Hall Algebra , Chapter 9, Section 1, 3 rd objective only Interactmath.com: Prentice Hall Algebra , Chapter 10, Section 5 Interactmath.com: Prentice Hall Algebra , Chapter 4, Section 1 Interactmath.com: Prentice Hall Algebra , Chapter 6, Section 8 Interactmath.com: Prentice Hall Algebra , Chapter 7, Section 1 and 2 Interactmath.com: Prentice Hall Algebra , Chapter 7, Section 3, 3 rd and 4 th objectives only 10. Simplify exponential and logarithmic expressions and solve basic exponential and logarithmic equations Interactmath.com: Prentice Hall Algebra , Chapter 7, Sections 1-7 Interactmath.com: Prentice Hall Algebra , Chapter 7, Section 3, 1 st and 2 nd objectives only Interactmath.com: Prentice Hall Algebra , Chapter 7, Section 5 2

3 1. Go to (Google Chrome and Internet Explorer browsers work best). 2. Click Enter, choose the appropriate 3. Go to the appropriate chapter and section textbook from the drop-down menu, click Submit. to find the skill you want to practice. 4. Click on the first exercise and begin solving. 3

4 1. Use the Pythagorean Theorem to find side lengths of right triangles (Prentice Hall Geometry 2011, Ch 8, Sect 1) The Pythagorean Theorem can be used to find side lengths of right triangles. The formula is often represented as a 2 + b 2 = c 2, however (trust us!) it will be much more helpful to remember this relationship as: Example: leg 2 + leg 2 = hypotenuse 2 It is also useful to recall common Pythagorean triples (sides of right triangles that are all integers). Example: Other Pythagorean triples include: 3/4/5, 5/12/13, 7/24/25, 8/15/17, 9/40/41, 11/60/61, 12/35/37, 13/84/85 Practice: Find the length of each side. Tell whether the side lengths form a Pythagorean triple. a. b. c. d. 4

5 2. Use properties of special right triangles (Prentice Hall Geometry 2011, Ch 8, Sect 2) Examples: Practice: Find the value of each variable. a. b. c. d. 5

6 3. Solve right triangles using trigonometry (Prentice Hall Geometry 2011, Ch 8, Sect 3) We will quickly move away from this basic definition, but for now refresh your S o h C a h T o a skills. Remember sine, cosine, and tangent are simply ratios of sides of right triangles (opposite to the indicated angle, adjacent to the indicated angle, or hypotenuse); the sine of 30 is always equal to ½ because of the power of similar triangles. Examples: Practice: a. Find the sine, cosine, and tangent of A. b. In the sled-riding example, find the height of the hill if the angle of elevation of the hill is 42. c. If the angle of elevation from your position on the ground to the top of a building is 67 and you are standing 30 meters from the foot of the building, approximate the height of the building. 6

7 To solve a right triangle means to find all its missing measurements (angles and sides). When working these problems make sure your graphing calculator is in DEGREE mode. Press the MODE button to the right of the 2 ND button. Make sure DEGREE is highlighted (if you are prone to playing calculator games sometimes your mode will switch to radian). In Precalculus you will finally get to learn all about the radians! This Geometry practice is exceptionally important because it is content you most likely learned over a year ago and we begin Precalculus with trigonometry.you must have these skills down pat to be successful! Examples: Practice: Solve each right triangle. d. e, f. g. 7

8 4. Simplify radicals and rational denominators (Prentice Hall Algebra , Ch 10, Sect 2) Simplifying an answer with a radical means that 1) no perfect squares (except for 1) are under the radicand (the square root sign) and 2) there are no radicals in the denominator. Example: Simplify 147. Practice: Simplify each square root. a. 98 b. 52 c. 300 d. 99 Examples of rationalizing the denominator: Practice: 8 e. 100 f. 4 5 g h i

9 5. Solve linear equations (Prentice Hall Algebra , Ch 2, Sections 3, 4, and 5) For examples and instruction visit the sections listed above. Practice: a. x 4 1 = 5 b. 10(z 2) = c. A parking garage charges $3 plus $1.50 per hour. You have $12 to spend for parking. Write and solve an equation to find the number of hours that you can park. d. The sum of the ages of two sisters is 25. The second sister s age is 5 more than three times the first sister s age. Find the two ages. e. f. g. h. 9

10 6. Factor expressions (Prentice Hall Algebra , Ch 8, Sections 2, 3, 4, 5, and 8) Example: Factoring using greatest common factor. 2x 5 y 10x 2 y 4 z 14x 3 y 2 To find the common factors in an expression, determine the common factor in each term. If all the terms have common factors, divide by the highest power possible of each common variable. In the example, each term has a factor of 2, x 2 and y. 2x 5 y 2x 2 y 10x2 y 4 z 2x 2 y 14x3 y 2 2x 2 y 2x 2 y (x 3 5y 3 z 7xy) Practice: Factor each expression using the greatest common factor. a. 33x 3 y 11x 2 y b. 4x x 2 12x 3 c. 30a 15a ab Example: Factoring difference of two squares. (x 2 25) First you must identify the expression as a difference between two perfect squares (there must be only two terms, the coefficients must be perfect squares, the variables raised to an even power, and one term is positive while the other is negative). This example is a difference of two squares, so find the square root of each term and factor appropriately. (x 5)(x + 5) Practice: Factor each expression using the difference of two squares. d x 2 e. 100c 2 25d 2 f. x 2 y 2 w 2 z 2 g. j 2 k 4 h. 81k 6 l 4 m 2 16 Example: Factoring trinomials. (x 2 x 12) You most likely learned how to factor trinomials during Algebra, if the method you originally learned still works for you then don t change it now. However, if factoring is fuzzy for you then use the online resources listed at the beginning of this packet to find a better alternative. Practice: Factor each trinomial. (x 2 x 12) (x 4)(x + 3) i. b 2 + 8b + 7 j. n 2 10n + 9 k. m 2 + m 90 l. b 2 6b + 8 m. w 2 + 2w 24 10

11 7. Manipulate fractions without a calculator (Clark Applied Basic Mathematics, Ch 2, Sections 1 7) Fractions are an important part of number sense and understanding certain trigonometric relationships. Your brain will not be able to focus on the new, interesting Precalculus material if it is stuck trying to figure out what ½ minus ¾ equals. Reach back to elementary school, retrieve your fraction skills, and put your calculator away! Example: To add or subtract fractions find the least common denominator (in this example, 12). Subtract the numerators Practice: Simplify each expression without using a calculator. Write your answer as a fraction in lowest terms. a b c d e f g h

12 8. Solve quadratic equations (Prentice Hall Algebra , Ch 4, Sections 4, 5, and 7) Example: Find the zeros of the following function f(x) = x 2 6x Practice: Find the zeros of each function by factoring. Check your solution on your graphing calculator. a. f(x) = x 2 + x - 20 b. f(x) = x 2 1 c. f(x) = x 2 + 3x - 10 Example: Solve the equation x = 10 5(x 7)2 = 135 Practice: Solve each equation. d. 4x 2 5 = -1 e. 12 2y 2 = 4 f. p2 4 3 = 33 g. 2(x 3) 2 = -12 h. (r 8) 2 = 50 i. 1 3 (z + 3)2 = 5 12

13 Example: Solve the equation 8x 2 5x = x 2 1 Practice: Solve each equation. j. x 2 9x + 5 = 0 k. 5x 2 + 3x 1 = 0 l. x 2 + 2x + 4 = 0 m. x 2 6x + 9 = 0 n. x 2 + 4x + 4 = 0 o.x x + 25 = 0 13

14 9. Analyze functions graphically (Prentice Hall Algebra , Ch 9, Sect 1, 3 rd objective only and Ch 10, Sect 5 and Prentice Hall Algebra , Ch 4 Sect 1, Ch 6 Sect 8, Ch 7 Sect 1 and 2, and Ch 7 Sect 3, 3 rd and 4 th objective ) Example: Practice: Graph the function. State the domain and range. a. f(x) = 3 2 x+4 b. f(x) = 5 2 x 1 c. f(x) = 3 x d. f(x) = log 3 x + 1 e. f(x) = ln (x 2) f. f(x) = x g. f(x) = x 3 h. f(x) = (x + 3) 2 2 i. f(x) = 2 + 3(x 1) 2 14

15 10. Simplify exponential and logarithmic expressions and solve basic exponential and logarithmic equations (Prentice Hall Algebra , Ch 7, Sections 1-7 and Prentice Hall Algebra , Ch 7, Sect 3, 1 st and 2 nd objective and Sect 5) Example: Simplify log expression without a calculator. Practice: Simplify each log expression without a calculator. a. log 4 64 b. log c. log 7 1 d. log e. log 9 9 f. log1 3 3 Example: Using properties of logs to expand and condense expressions. Practice: Expand and condense log expressions. g. log 9x h. log 2 6x 3 i. ln 2x2 y j. 3 log 2 3 log 2 w k. ln 3 + ln 6 ln 9 l. 6 log 7 w + 3 log 7 v 15

16 Example: Solve exponential equation e x 6 = 9 Practice: Solve each exponential equation m. 5 x = 8 n. e x = 5 o. 2 x + 1 = 5 p x = 26 q. 11 n 8 5 = 54 r. 10 e 2n 10 5 = 73 Example: Solve logarithmic equation 4 log 3 3x = 20 Practice: Solve each logarithmic equation s. log 8 (x 5) = 2 3 t. 3log 5 (x + 2) = 6 u. 4 ln 2x = 5 v. log x + log 8 = 1 w. log 3 (x 2 + 8) log 3 4 = 3 x. log 5 3 log 5 5x = 16

17 Precalculus Prerequisites - Solutions 1. Use the Pythagorean Theorem to find side lengths of right triangles a. 113; not a Pythagorean Triple b.13; Pythagorean Triple c. 135; not a Pythagorean Triple d. 5.04; not a Pythagorean Triple 2. Use properties of special right triangles a. x = 14 3, y = 28 b.x = 7 3, y = 14 3 c. x and y = d. x = 42 2, y = Solve right triangles using trigonometry. a. b. c. d. e. f. g. 4. Simplify radicals and rationalize denominators. a. b. c. d. e. f. g. h. i. 5. Solve linear equations. a. x = 24 b. z = 2.5 c. f(x) = x; x = 6 d. Ages are 5 and 20 e. -5 f. ½ g. -2 h Factor. a. 11x 2 y(3x 1) b. 2x 2 (2x x) c. 5a(6 3a + 2b) d. (7 8x)(7 + 8x) e. (10c 5d)(10c + 5d) f. (xy wz)(xy + wz) g. (j k 2 )(j + k 2 ) h. (9k 3 l 2 m 4)(9k 3 l 2 m + 4) i. (b + 7)(b + 1) j. (n 9)(n 1) k. (m + 10)(m 9) l. (b 4)(b 2) m. (w + 6)(w 4) 17

18 7. Manipulate fractions without a calculator. a. 5 4 b c. 5 6 d. 9 4 e f. 7 4 g. 3 4 h Solve quadratic equations. a. -5, 4 b. -1, 1 c. -5, 2 d. -1, 1 e. -2, 2 f. -12, 12 g. 3 ± 6 h. 8 ± 5 2 i. 3 ± 15 j. 9 ± 61 2 k 3± l. 1 ± 5 m. 3 n. 2 o Graphically (Check problems without graphs shown below on your calculator) a. Domain (, ) b. Domain(, ) c. Domain(, ) Range (, 0) Range (0, ) Range (4, ) d. Domain (0, ) e. Domain(2, ) f. Domain[ 2, ) Range (, ) Range (, ) Range [1, ) g. Domain [3, ) h. Domain(, ) i. Domain(, ) Range [ 4, ) Range [ 2, ) Range [2, ) 10. Exponentials and logs a. 4 b. 3 c. 0 d. -3 e 1 f. -1 g. log 9 + log x h. log log 2 x i.ln 2 + 2ln x ln y 27 j. log 2 k. ln 2 l.log w 7 w 6 v 3 m. x = n. x = o. x = 2 p. x = q. x = r. n = s. x = 9 t. x = 23 u. n = v. x = 5 w. x = 10, 10 x. x =