PreCalculus Summer Assignment (2018/2019)

Transcription

1 PreCalculus Summer Assignment (2018/2019) We are thrilled to have you join the Pre-Calculus family next year, and we want you to get a jump-start over the summer! You have learned so much valuable information in Math 1-3, that we do not want you to lose over the next three months! Pre-Calculus is more rigorous in nature, and it is vital that you understand the content in this summer assignment prior to the school year. This assignment is meant to take up a maximum of minutes of your time each week. If you are struggling with a topic, try watching some videos on Khan Academy as a refresher or take advantage of lots of other great resources on the Internet. We encourage you to work with other classmates and to form study groups, but remember you are ultimately responsible for understanding the material. Try not to procrastinate your work until the last week of summer. This assignment will be due the first week of class. If you turn it in complete on the first day of school you will get bonus points. To receive full credit, please make sure you show ALL work! Please see the suggested pacing guide below to help you stay on track! Week # Dates Assignment Done 1 6/4-6/10 Fraction Basics & Exponent Laws 2 6/11-6/17 Function Notation & Definition 3 6/18-6/24 Graphing Linear Functions 4 6/25-7/1 Expanding & Factoring Quadratics 5 7/2-7/8 Solving Quadratic Equations 6 7/9-7/15 Graphing Quadratic Equations 7 7/ Classifying Functions & Degree of Polynomials 8 7/23-7/29 Adding/Subtracting/Multiplying Polynomials 9 7/30-8/5 Dividing Polynomials 10 8/6-8/12 Logarithms 11 8/13-8/19 Trigonometry 13 8/20-8/26 Transformations 14 8/27-8/29 Conclusion Page

2 Fraction Basics & Exponent Laws Fraction Rules Addition Subtraction Multiplication Dividing Laws of Exponents Type Law Example Product x m x n = x m+n x 3 x 4 = (x x x)(x x x x) = x 7 Quotient x m = xm n xn x 7 x x x x x x x = = x 4 x3 x x x Power (x m ) n = x m n (x 2 ) 3 = (x x)(x x)(x x) = x 6 Inverse x m = 1 x m x 4 = 1 x 4 Zero Power x 0 = = 1 Rational Exponent x m n n = x m x = x 2 8 2/3 3 = = 64 = 4

3 Fraction Basics & Exponent Laws Solve each of the following. All answers should be completely reduced. Do not use a calculator, and show all work Simplify each of the following expressions completely using the law of exponents. Try to rewrite all of your answers without negative exponents = 10. (x) 2 (x) = = 12. (5 2 ) 2 = = 14. (4 5 )3 = 15. x8 x 4 = y4 16. = 17. y 7 (32 s 3 ) 6 = 18. (4 0 w 2 ) 5 = 19. (2m 3 n 1 )(8m 4 n 2 ) = a 3 b 9 12a 2 b 5 =

4 Function Notation & Definition Definition For every x-value there is only one y-value. (For every input there is only one output) Functions Vertical Line Test If you can draw a vertical line anywhere on a graph so that it hits the graph in more than one spot, then the graph is NOT a function. Example Non-Example X Y X Y /2 Function Notation NAME OF THE FUNCTION THE FUNCTION f(x) = 2x 2 3x + 2 INPUT OUTPUT Example: Evaluate f( 2) = 2( 2) 2 3( 2) + 2 = 2(4) = = 0

5 Function Notation & Definition Determine whether or not the following represent functions. Explain why or why not If you are at the Rapid City Airport, is the temperature recorded a function of time? Is your grade level in school a function 7. of your GPA? Use the functions given below to evaluate the following. Show all work. f(x) = 2x + 5 g(x) = 3(2 x ) h(x) = 3x 2 2x f ( 1 ) = 9. g(3) = 10. h( 1) = f( 10) = 12. g( 3) = 13. h(5) = 14. Suppose that at All Sport Shoes, the manager estimates the monthly operating cost for the store (in dollars) as a function of the number of pairs of shoes that the store purchases from its suppliers. The rule for that function is C(x) = 17, x. a. Calculate C(100) and explain the meaning. b. What value of x satisfies C(x) = 24,500? What does that value tell about the store s monthly business costs?

6 Graphing Linear Functions Standard Form Ax + By = C (Rewrite into slope-intercept form) Example: 4x + 2y = 6 +4x + 4x 2y 2 = 4x y = 2x + 3 Move the x-term over Divide by 2 to get y alone Simplify Linear Equations Slope-Intercept Form y = mx + b Definition: m = rise run = y x = y 2 y 1 x 2 x 1 Slope of a Line Example: Find the slope of the that crosses through the points ( 8,3)and ( 4,6). m = 4 ( 8) 6 3 = 4 3 Parallel Lines Must have the SAME slope! Perpendicular Lines Must have OPPOSITE RECIPROCAL slopes! y = 2 3 x + 3 and y = 2 3 x 2 y = 1 x + 5 and y = 2x 4 2

7 Graphing Linear Functions 1. Find the slope of the line that passes through each pair of points given below. Then decide how the line relates to y = 3 x 1. (Parallel, Perpendicular, or Neither) 4 a. (7, 5) and (10, 1) b. ( 12,8) and ( 15,12) c. ( 6,8) and ( 2,5) m = m = m = Related? Related? Related? 2. Identify the slope and y-intercept for each of the linear functions, then use them to sketch a graphs. a. f(x) = 5 4 x 1 b. f(x) = 3x + 4 c. f(x) = 1 2 x m = y int = m = y int = m = y int = Rewrite each of the following linear equations to express y as a function of x (slope-intercept form). Then determine the slope and y-intercept for each equation. a. 2x + y = 6 b. 8x 5y = 20 c. 4x 3y = 15 y = y = y = m = y int = m = y int = m = y int =

8 Expanding & Factoring Quadratic Expressions Expanding Quadratic Equations Algebra Tiles Box Method (x + 3)(x + 2) = x 2 + 5x + 6 (2x + 1)(x 3) = 2x 2 5x 3 Distributive Property (FOIL) Standard Form: ax 2 + bx + c Factored Form: (x ± m)(x ± n) Factoring Quadratic Equations Diamond Box Method (You do not have to use this method if you can use the FOIL method backwards) Greatest Common Factor(GCF) ax 2 + ax = ax(x + 1) 15x 2 6x = 3x(5x + 2)

9 Expanding & Factoring Quadratic Expressions Rewrite each of the quadratic expressions in expanded standard form. 1. (x 10)(x + 10) 2. (3x + 5)(x + 2) 3. (x 7) 2 4. (x 6)(x + 8) 5. 2x(3x 8) 6. (2x 4)(3x + 7) 7. 5x(x + 8) 8. (3x 1)(3x + 1) 9. (4x 1)(4x 1) Rewrite each of these quadratic expressions in an equivalent factored form. Some may not be factorable. 10. x 2 10x x x 2 5x 13. x 2 12x x x x 2 + 6x x x 2 16x 18. x 2 x 12

10 Solving Quadratic Equations Factoring (a, b, and c term) 3x 2 + 4x 4 = 0 (3x 2)(x + 2) = 0 3x 2 = 0 and x + 2 = 0 x = 2 3 and x = 2 Taking Square Roots (a and c term) 4x 2 64 = x 2 4 = 64 4 x 2 = 16 x 2 = 16 x = ±4 Solving Methods Factoring(GCF) (a and b term) 5x x = 0 5x(x 3) = 0 5x = 0 and x 3 = 0 x = 0 and x = 3 Quadratic Formula (ax 2 x = b ± b2 4ac + bx + c = 0) 2a 2x 2 + 3x 20 = 0 x = x = (3) ± (3)2 4(2)( 20) 2(2) 3 ± 9 ( 160) x = 4 3 ± ± 13 x = = x = and x = and x = Imaginary/Complex Numbers i = 1 and i 2 = 1 Complex Number: a ± bi

11 Solving Quadratic Equations Solve each of the following quadratic equations using an appropriate method. 1. x 2 + 6x + 5 = x + x 2 = 0 3. x x + 20 = x + x 2 = = 7 + 4x 2 6. x 2 + 3x + 4 = x 2 + 3x + 1 = x 2 5x = x 2 3x = x 2 12x + 18 = x = x 2 6x + 25 = 0

12 Graphing Quadratic Equations Points are mirrored over this line Occurs when f(x) = 0 Occurs at f(0) Occurs halfway between the x-intercepts Direction a > 0 Open up a < 0 Open down X-intercepts Occurs when f(x) = 0 Y-intercepts Occurs at f(0) f(x) = 2x 2 + 4x + 16 Parabola will open down since a = 2 f(x) = 2x 2 + 2x 12 0 = 2(x 2 2x 8) 0 = 2(x 4)(x + 2) x = 3 and x = 2 f(0) = 2(0) 2 + 4(0) + 16 f(0) = 16 Max/Min Point(vertex) Occurs at axis of symmetry (hallway between x- intercepts) f(1) = 2(1) 2 + 4(1) + 16 f(1) = f(1) = 18

13 Graphing Quadratic Equations Sketch graphs of the following functions. Label these key points with their coordinates on the graphs: 1. f(x) = x 2 + 2x 3 x-intercepts: y-intercept: Vertex: 2. f(x) = 2x 2 + 2x + 12 x-intercepts: y-intercept: Vertex: 3. f(x) = 0.5(x 6) 2 x-intercepts: y-intercept: Vertex:

14 Classifying Polynomials & Degree of Polynomials Examples of Different Functions Exponential Function f(x) = a b x Inverse Function f(x) = k x r Linear Function f(x) = mx + b Quadratic Function f(x) = ax 2 + bx + c Cubic Function f(x) = ax 3 + bx 2 + cx + d Quartic Function f(x) = ax 4 + bx 3 + cx 2 + dx + e

15 Classifying Polynomials & Degree of Polynomials Classify the following graphs and equations as a linear function(l), exponential function(e), inverse function(i), quadratic function(q), cubic function(3), quartic function(4), or quintic function(5). 1. f(x) = x g(x) = 1 2 (3)x 3. h(x) = 2x x 3 4. f(x) = 2(x 4) 2 (x + 2) 2 5. g(x) = 2 3 x 6. h(x) = (x 3) f(x) = 5 x 3 8. g(x) = 20 x h(x) = 5 + 2x 5 f(x) = 1 x

16 Adding, Subtracting, and Multiplying Polynomials Adding Polynomials Subtracting Polynomials Multiplying Polynomials

17 Adding, Subtracting, and Multiplying Polynomials Given the polynomials below, perform the given operation for each problem. Make sure all answers are simplified completely. f(x) = 2x 2 + 3x + 6 h(x) = 2x 4 g(x) = 3x 3 + x 2 4x 5 w(x) = 5x 2 4x 3 d(x) = 3x 4 3x 3 + 2x 2 4x f(x) + d(x) 2. g(x) + h(x) 3. d(x) + w(x) 4. h(x) + w(x) 5. d(x) f(x) 6. h(x) g(x) 7. h(x) f(x) 8. h(x) w(x) 9. h(x) g(x) 10. x d(x)

18 Dividing Polynomials

19 Dividing Polynomials Divide the polynomials below using long division. 1. (x 3 + 7x 2 + 7x 6) (x + 2) 2. (x 3 + 2x 2 25x 50) (x + 5) Divide the polynomials below using synthetic division. 3. (x 4 + 2x 3 + x 2 + 5x + 6) (x + 2) 4. (x 4 4x 3 7x x 24) (x + 3)

20 Logarithms Logarithm What it does A logarithm finds an exponent for a base of 10 when it equals a specific value. Definition log 10 y = x if and only if 10 x = y Example 10 x = 517 log 517 = 2.17 x = 2.17 since = 517 Use logarithms to solve exponential equations Example 2(10) 2x 4 = 96 Rewrite so the base of 10 is by itself 2(10) 2x = 100 (10) 2x = 50 Use the logarithm to rewrite both sides of the equation with a base 10 log x = Set the exponents equal and solve 2x = x.8495

21 Logarithms Solve the following equations using logarithms x = x+2 = x+2 = (10) x = (10) x+4 = (10) 2x = x+2 = (10) 3x+2 = (10) x = The population of the U.S. in 2010 was about 309 million and growing exponentially at a rate of about 0.8% per year. Use the equation below to predict about how long will it be until the population of the United States reaches 400 million? P(t) = 309( t )

22 Trigonometry

23 Trigonometry 1. Find the indicated measures in each triangle. Show all of your work below and place your answers in the correct spaces. a. b. c. m A = DE = m K = BC = FD = m M = 2. Demetri leans a ladder, 30 feet in length, against a wall. The base of the ladder is 10 feet from the wall. a. How high up the wall does the ladder reach? Draw a picture and show your work. b. What angle does the ladder make with the ground? Show your work. 3. Find x for the following triangles. You will need to use the Law of Sines or Law of Cosines. a. b. c.

24 Function Transformations Vertical Translations A shift may be referred to as a translation. If c is added to the function, where the function becomes y = f(x) + c, then the graph of f(x) will vertically shift upward by c units. If c is subtracted from the function, where the function becomes y = f(x) c then the graph of f(x) will vertically shift downward by c units. In general, a vertical translation means that every point (x, y) on the graph of f(x) is transformed to (x, y + c) or (x, y c) on the graphs of y = f(x) + c or y = f(x) c respectively. Horizontal Translations If c is added to the variable of the function, where the function becomes y = f(x + c), then the graph of f(x)will horizontally shift to the left c units. If c is subtracted from the variable of the function, where the function becomes y = f(x c), then the graph of f(x) will horizontally shift to the right c units. In general, a horizontal translation means that every point (x, y) on the graph of is transformed to (x c, y) or (x + c, y) on the graphs of y = f(x + c) or y = f(x c) respectively. If the function or the variable of the function is multiplied by -1, the graph of the function will undergo a reflection. When the function is multiplied by -1 where y = f(x) becomes y = f(x), the graph of y = f(x) is reflected across the xaxis. Reflections On the other hand, if the variable is multiplied by -1, where y = f(x) becomes y = f( x), the graph of y = f(x) is reflected across the y-axis.

25 Function Transformations Vertical Stretching and Shrinking If c is multiplied to the function then the graph of the function will undergo a vertical stretching or compression. So when the function becomes y = cf(x) and 0 < c < 1, a vertical shrinking of the graph of y = f(x) will occur. Graphically, a vertical shrinking pulls the graph of y = f(x) toward the x-axis. When c > 1 in the functiony = cf(x), a vertical stretching of the graph of y = f(x) will occur. A vertical stretching pushes the graph of y = f(x)away from the x-axis. In general, a vertical stretching or shrinking means that every point (x, y) on the graph of f(x) is transformed to (x, cy) on the graph of y = cf(x). If c is multiplied to the variable of the function then the graph of the function will undergo a horizontal stretching or compression. So when the function becomes y = f(cx)and 0 < c < 1, a horizontal stretching of the graph of y = f(x) will occur. Graphically, a vertical stretching pulls the graph of y = f(x) away from the y-axis. When c > 1 in the function y = f(x), a horizontal shrinking of the graph of will occur. A horizontal shrinking pushes the graph of y = f(x) toward the y-axis. In general, a horizontal stretching or shrinking means that every point (x, y) on the graph of f(x) is transformed to (x/c, y) on the graph of y = f(cx). Horizontal Stretching and Shrinking

26 Function Transformations Match each transformation of f(x) listed below with its graph from the bottom of the page. The original graph of f(x) is shown at the right. 1. f(x) 2 2. f(2x) 3. f(x + 2) 4. 2f( x) 5. f(x) 2 6. f(x 2) 7. f ( x 2 ) 8. f( x) f(x) 2 A. B. C. D. E. F. G. H. I. The graphs on this worksheet were produced with InquiCalc 2.0, available at InquiSoft. Reproduction for educational use permitted provided that this footer text is retained.

27 Function Transformations 10. Write g(x)(thinner) in terms of f(x)(thicker). a. b. g(x) = g(x) = c. d. g(x) = g(x) = 11. Use the given parent function,f(x), to sketch the transformed function g(x). a. f(x) = x g(x) = x b. f(x) = x g(x) = 2 x

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29 Conclusion Name: Learning About You What name do you prefer to be called? Are you new to Stevens? If so, where did you attend school before? What is your favorite thing to do? What is your favorite class? What activities are you involved with? What do you want to do when you leave high school? About Math Do you feel like you are good at math? Do you enjoy math? Was there ever part of math that you really liked? What was it? Was there ever part of math class that you really disliked or found difficult? What was it? Who was your math teacher last year? About Learning How do you learn best? Reading, listening, watching, practicing, etc? Do you prefer to work alone or with a group?

30 About this Class What do you want to accomplish in this course? What would you most like to learn in this course? What concerns you about this course? What grade are you planning to achieve in this course? What do you plan to do to be successful in this course? How can I help you be successful in this course? What else would you like me to know about you?