3. If a coordinate is zero the point must be on an axis. If the x-coordinate is zero, where will the point be?

Transcription

1 Chapter 2: Equations and Inequalities Section 1: The Rectangular Coordinate Systems and Graphs 1. Cartesian Coordinate System. 2. Plot the points ( 3, 5), (4, 3), (3, 4), ( 3, 0) 3. If a coordinate is zero the point must be on an axis. If the x-coordinate is zero, where will the point be? If the y-coordinate is zero, where will the point be? 4. Different ways to graph 2x + 3y = 18. (A) (B) (C) (D) (E) 1

2 5. Drawing graphs: (a) Draw axes and label with variables (especially important for P = 5A + 10) (b) Scale axes appropriately (label tick marks). (c) If drawing two graphs label both, clearly! 6. Graph the equation y = x + 2 by plotting points. 7. Graph 3x 5y = 10 by finding the intercepts. 8. Distance Formula: d = (x 2 x 1 ) 2 + (y 2 y 1 ) Find the distance between the two points: (Be careful if the answer should be rounded or exact!) (a) ( 2, 2) and (3, 6) (b) ( 3, 1) and (2, 3) 2

3 (c) ( 1, 5) and ( 1, 2) (THINK!) 10. Application. Finding distance between two locations: A man drove 10 miles directly east from his home, made a left turn at an intersection, and then traveled 5 miles north to his place of work. If a road was made directly from his home to his place of work, what would its distance be to the nearest tenth of a mile? 11. The coordinates on a map for San Francisco are (53, 17) and those for Sacramento are (123, 78). Note that the coordinates represent miles. Find the distance between the cities to the nearest mile. 12. The point ( 2, 5) is on a circle that has (3, 1) as its center. Find the length of the radius of the circle. 3

4 13. Midpoint of the two points (x 1, y 1 ) and (x 2, y 2 ) is 14. Find midpoint of segment whose end points are ( 4, 2) and (2, 5) 15. Find the center of circle whose diameter connects points (2, 3) and (6, 4). 4

5 Section 2.2: Linear Equations in One Variable 1. A linear equation is an equation of a straight line, written in one variable. The only power of the variable is 1. Linear equations in one variable take on the form ax + b = 0. There are three types of linear equations: identity, conditional or inconsistent. In each of these types there is a solution set which consists of all values that make the equation true. An identity equation is true for all values of the variable. Example: 3x = 2x + x A conditional equation is true for only some values of the variable. Example: 5x + 2 = 3x 6 An inconsistent equation results in a false statement. Example: 5x 15 = 5(x 4) 2. Solve the following equations: (a) 2x + 1 = 9 (b) 2(3x 1) + x = 14 x 1

6 3. Rational equations have a variable in the denominator in at least one of the terms. In this section we will be able to manipulate rational equations and turn them into a linear equation and solve using techniques above. (Note, there are a variety of ways to solve these equations, in class we will only focus on a couple.) 4. Solve the following: (a) 7 2x 5 3x = 22 3 (b) (Solve without factoring) 2 x 3 2 = 7 2x (c) (Solve by factoring) 1 x = x 2

7 5. Solve the following rational equations and state the excluded values: (a) 3 x 6 = 5 x (b) x x 3 = 5 x (c) x x 2 = 5 x

8 (d) 2 x x 1 = 2x x A linear function: can be written as f(x) = mx + b (or y = mx + b) where the following constants are defined: We can also write in point slope formula y y 1 = m(x x 1 ). Or the Standard Form: where A, B, and C are integers. Ax + By = C 4

9 7. Find the slope of the line that passes through the points (a) ( 2, 6) and (1, 4). (b) ( 2, 6) and (1, 6). (c) ( 2, 6) and ( 2, 4). 8. Find the slope and y-intercept of the line with equation 3x 6y 7 = Find the equation of the line, in slope intercept form, given the slope is 4 and it passes through the point (2, 5). 5

10 10. Find the equation of the line passing through the points (3, 4) and (0, 3). Write answer in slope-intercept form and standard form. 11. Vertical and Horizontal Lines: (a) Graphs of line: (b) Equations of line: (c) Slopes: 6

11 12. Parallel or Perpendicular Lines: (a) Slopes of parallel lines are: (b) Slopes of perpendicular lines are: (c) Graphs of parallel and perpendicular lines: 13. Write the equation of line parallel to 5x + 3y = 1 and passing through the point (3, 5). 14. Find the equation of the line perpendicular to 5x 3y + 4 = 0 and passes through the point ( 4, 1). 7

12 Section 2.3: Models and Applications 1. Page 102 has a table of common verbal expressions and their equivalent mathematical expressions. Page 103 has a list of directions on how to model a linear equation to fit a real-word problem. 2. If a car rental agency charges $0.10 per mile plus a daily fee of $50. What is the daily car rental cost, C? 3. Find a linear equation that represents the situation and find the two numbers: One number exceeds another number by 17 and their sum is There are two cell phone companies that offer different packages. Company A charges a monthly service fee of $34 plus $0.05 per minute of talk-time. Company B charges a monthly service fee of $40 plus a $0.04 per minute of talk time. (a) Write a linear equation that models the packages offered by both companies. (b) If the average number of minutes used each month is 1,160 which company offers the better plan? (c) If the average number of minutes used each month is 420 which company offers the better plan? 1

13 (d) How many minutes of talk-time would yield equal monthly statements from both companies? 5. On Saturday morning, it took Jennifer 3.6h to drive to her mother s house for the weekend. On Sunday evening, it took Jennifer 4 h to return home. Her speed was 5mph slower on Sunday than on Saturday. What was her speed on Sunday? 6. The perimeter of a rectangular outdoor patio is 54ft. The length is 3 feet greaterthan the width. What are the dimensions of the patio? 2

14 7. A game room has a perimeter of 70ft. The length is five more than twice the width. How many square feet of new carpeting should be ordered? 8. Find the dimensions of a shipping box, given that the length is twice the width, the height is 8 inches, and the volume is 1600 cubic inches. 3

15 Section 2.4: Complex Numbers I believe you only need to memorize two things: i = 1 and how to divide complex numbers. Everything else should be natural. 1. Introduction: i = 1 (just memorize this, the rest should follow) i 2 = 1 2 = 1 i 3 = i 2 i = i i 4 = i 2 i 2 = ( 1)( 1) = 1 2. Rewrite: (Main goal is to get it in a form of ±i, ±1.) i 37 = i 58 = i 75 = i 80 = Note, there are other methods where people find a factor of Working with roots: (Note, when you have a square root of a negative number always pull out the i, then deal with it.) 7 = i 7 = 7i (Note, i is not under square root!, if so it would be i = 1 = 4 1. Not what you want. BE CLEAR!)

16 4. A word of caution: Notice: 2 5 = 2 5 = 10 you cannot do this for complex. ALWAYS first pull out the i. 2 5 = i 2 i 5 = (i) 2 10 = 10 Notice this is not the same as 2 5 = ( 2)( 5) = Complex Number: A number of the form a + bi where a, b are real numbers. a- real part b- imaginary part If a = 0 then the complex number is bi, called pure imaginary. 6. Graphing Complex Numbers: 7. Add/Subtract/Multiply (do what is natural ) (8 + 6i) + (3 + 2i) (4 + 5i) (6 3i) (1 + 2i)(1 + 3i) 2

17 (3 7i) 2 8. Conjugate of a + bi is a bi. The entire point of this is for division! (5 + 2i)(5 2i) (8i)( 8i) 9. Dividing: Do note, you want to write this as a complex number:a + bi, it is not of that form yet! 2 5i 1 6i 3

18 Section 2.5: Quadratic Equations 1. If the product of two numbers is zero what must be true about at least one of them? 2. Factor and solve the following equations: (a) x 2 5x 6 = 0 (b) x 2 + 8x + 15 = 0 (c) x 2 25 = 0 (d) 4x x + 9 = 0 1

19 (e) 3x 3 5x 2 2x = 0 3. Solve the following, using the square root property: (a) x 2 = 8 (b) 4x = 7 (c) 3(x 4) 2 = Quadratic Formula: Written in standard form, ax 2 + bx + c = 0, any quadratic equation can be solved using: x = b ± b 2 4ac, 2a where a, b and c are real numbers and a 0. 2

20 5. Solve, using the quadratic formula: (a) x 2 + 5x + 1 = 0 (b) 9x 2 + 3x = 2 6. Discriminant The discriminant is the expression under the radical in the quadratic formula: b 2 4ac. This tells you how many solutions the quadratic equation will have: If b 2 4ac > 0 then the quadratic equation has 2 real solutions If b 2 4ac = 0 then the quadratic equation has 1 real solution If b 2 4ac < 0 then the quadratic equation has 2 imaginary solutions (Note, if b 2 4ac > 0 and a perfect square there are two rational solutions. b 2 4ac > 0 and not a perfect square there are two irrational solutions.) 3

21 7. Use the discriminant to find the nature of the solutions to the following quadratic equations: (a) x 2 + 4x + 4 = 0 (b) 8x x = 3 (c) 5x = 2 3x 2 (d) 3x 2 10x + 15 = 0 8. Abercombie and Fitch stock had a price given as P = 0.2t 2 5.6t , wehre t is the time in months from 1999 to (t = 1 is January 1999). Find the two months in which the price of the stock was $ An epidemological study of the spread of a certain influenza strain that hit a small school population found that the total number of students, P, who contracted the flu t days after it broke out is given by the model P = t t + 130, where 1 t 6. Find the day that 160 students had the flu. Recall that the restriction on t is at most 6. 4

22 Section 2.6: Other Types of Equations 1. Rational Exponents.: A rational exponent indicates a power in the numerator adn a root in the denominator. There are multiple ways of writing an expression, a variable or a number with a rational exponent: ) m a m n = (a 1 n = (a m ) 1 n = n a m = ( n a ) m 2. Evaluate the following: (a) (b) Solve the following: (a) x 5 4 = 32 (b) 3x 3 4 = x 1 2 1

23 4. Polynomial Equation: A polynomial of degree n is an expression of the type a n x n + a n 1 x n a 2 x 2 + a 1 x + a 0 where n is a positive integer, a n,..., a 0 are called coefficients and are real numbers. The leading coefficient, a n 0. Setting the polynomial equal to zero gives a polynomial equation. The total number of solutions (real and complex) to a polynomial equation is equal to the highest exponent n. 5. Solve: (a) 5x 4 = 80x 2 (b) x 3 + x 2 9x 9 = 0 6. Rational Equations: These are equations that contain variables in the radicand (expression under a radical). They may have one or more radical terms, you solve by getting rid of one at a time. You will usually end up having extraneous solutions, answers that are not the actual answer. Therefore, you must always check the answer in the original equation when solving radical equations! Note, with square root you will square, cube root you will cube, etc.. 2

24 7. Solve: (a) 15 2x = x (b) x = 3x (c) 2x x 2 = 4 3

25 8. Solving Absolute Value Equation. (a) 6x + 4 = 8 (b) 3x + 4 = 9 (c) 3x 5 4 = 6 (d) 5x + 10 = 0 4

26 9. Solving Equations that can become Quadrtic: (a) 3x 4 2x 2 1 = 0 (b) (x + 2) (x + 2) = 12 5

27 10. Solving a Rational Equation that can become a Quadratic: (Note, you must always list and check the excluded values!) 4x x x + 1 = 8 x 2 1 6

28 Section 2.7: Linear Inequalities and Absolute Value Inequalities 1. For inequalities we can use different notation to describe the same situation. 2. (a) Use interval notation to indicate all real numbers greater than or equal to 2. (b) Express all real numbers less than 2 or greater than or equal to 3 in interval notation. 3. Solve each of the following: (a) x 15 < 4 (b) 6 x 1 (c) x + 7 > 9 (d) 3x < 6 (e) 2x 1 5 (f) 5 x > 10 1

29 (g) 13 7x 10x 4 (h) 3 4 x x 4. Solve the Compound Inequalities: (a) 3 2x + 2 < 6 (b) 3 + x > 7x 2 > 5x 10 2

30 5. Absolute Inequalities: Let X be an algebraic expression adn k > 0 then X < k is equivalent to k < X < k X > k is equivalent to X < k or X > k. 6. Describe all values x within a distance of 4 from the number Solve: (a) x 1 3 (b) 1 4x < 0 2 3

31 Chapter 3: Functions Section 3.1: Functions and Function Notation 1. A function is a correspondence between a first set (domain/inputs/independent variable) and a second set (range/outputs/dependent variable) such that each member of domain corresponds to exactly one member of range. A relation is a correspondence between domain and range such that each member of domain corresponds to at least one member of range. Note: there can be a relationship or correspondence but not a function. Note: Remember the phrasing y is a function of x. For instance Height is a funciton of age. 2. Considering the set, where the first number in each pair are the first five natural numbers, the second number in the pair is twice that of the first: Is this relationship a function? What is the domain, in set notation? What is the range, in set notation? {(1, 2), (2, 4), (3, 6), (4, 8), (5, 10)} 3. Considering the set of ordered pairs that relates the terms even and odd to the first five natural numbers: {(odd, 1), (even, 2), (odd, 3), (even, 4), (odd, 5)} Is this relationship a function? What is the domain, in set notation? What is the range, in set notation? 4. Considering the following three diagrams, which relationships are functions, which are not? And why? 1

32 5. A coffee shop has the following (limited!) menu. (a) Is price a funciton of the item? (b) Is the item a function of the price? 6. In a particular math class, the overall percent grade corresponds to a grade-point average. The below table shows the class rule for assigning grade points: (a) Is grade-point average a funciton of the percent grade? (b) Is the percent grade a function of the grade-point average? 2

33 7. The below table lists five greatest baseball players of all time in order of rank. (a) Is rank a funciton of the player name? (b) Is the player name a function of the rank? 8. Function notation: The notation y = f(x) defines a function named f. This is read as y is a function of x. The letter x is the independent variable and y or f(x) is the dependent. 9. Use function notation to represent a function whose input is the name of a month and output is the number of days in that month. 10. A function N = f(y) gives the number of police officers, N, in a town in the year y. What does f(2005) = 300 represent? 11. Do the following tables represent functions? 3

34 12. Using the function f(x) = x 2 + 3x 4 answer the following questions: (a) f(2) = (b) f(a) = (c) f(a + h) = (d) f(a + h) f(a) h 13. Given the function h(p) = p 2 + 2p solve for h(p) = 3. 4

35 14. Express the relationship 2n + 6p = 12 as a function p = f(n), if possible. 15. Express the relationship x 2 + y 2 = 1 as a function y = f(x), if possible. 16. Using the below table find g(3) and g(n) = Using the graph, evaluate f(2) and solve for f(x) = A one-to-one function is a function in which each output value corresponds to exactly one input value. There are no repeated x- or y- values. 19. Not all diagrams represent a function, but out of those, which one(s) are one-to-one? 5

36 20. Is the area of a circle a function of its radius? If yes, is the function one-to-one? 21. Are the following graphs of functions, and if so are they one to one? 6

37 Section 3.2: Domain and Range 1. Different type of notation: 1

38 2. Find the domain of the following functions: (a) {(2, 10), (3, 10), (4, 20), (5, 30), (6, 40)} (b) f(x) = x 2 1 (c) f(x) = x x (d) f(x) = 7 x 3. Fill out the below table to describe the numberline below: 4. Find the domain and range of the below graphs: 2

39 5. On the list of Toolkit functions Write down the domain/range of all. 6. A museum charges $5 per person for a guided tour with a group of 1 to 9 people, or a fixed $50 fee for a group of 10 or more people. Write a function relating the number of people, n, to the cost, C. 7. A cell phone company uses the function below to determine the cost, C, in dollars for g gigabytes of data transfer. { 25 if 0 < g < 2 C(g) = (g 2) if g 2 Find the cost of using 1.5 gigabytes of data, and cost of using 4 gygabytes of data. 8. Sketch a graph of the function: x 2 if x 1 f(x) = 3 if 1 < x 2 x if x >