Precalculus Notes: Unit P Prerequisite Skills

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1 Syllabus Objective Note: Because this unit contains all prerequisite skills that were taught in courses prior to precalculus, there will not be any syllabus objectives listed. Teaching this unit within the school year is optional. A pretest may be given to students to check for mastery of prerequisite skills to determine if this unit should be taught. Another option would be to have students complete this unit as a summer assignment. Real Numbers Real Numbers ( ) : Numbers that can be written as decimals. The braces { } are used to enclose the elements (objects) of a set. The real number system contains several subsets: Whole Numbers : 0 and the set of the natural (counting) numbers ( ) { 0,,,,... } Integers ( ): the whole numbers and their opposites {...,,,0,,,,... } Rational Numbers ( ) : numbers that can be written as the ratio a/b of two integers, such that b 0. We can use set builder notation to describe the rational numbers: a ab, are integers, and b 0 b Irrational numbers: real numbers that are decimals that do not terminate and do not repeat. E: Label each bo with the five subsets above and place these elements (objects) in the appropriate boes. Indicate the rational numbers that terminate/repeat: 0 Real Rational: 7, 4 9,.4 Irrational: π, 0, Integers:,.4 0 π Whole: Natural:, 54 6 Page of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter P

2 If a and b are real numbers with a < b then we can use the following notation: Inequality Notation: a < < b Interval notation: (a, b] Bounded Closed [a, b] Unbounded [a, ) (, b] Open (a, b) (a, ) (, b) Half-open [a, b) (a, b] Unbounded intervals are never considered half-open We can describe inequalities by using words, graphs and interval notation. For eamples -4 also note whether the interval is bounded or unbounded and open, closed or half-open. E. Describe in words and graph the interval. a) (, ) is greater than or equal to -4 - ( Unbounded, open b) 4 < < all real numbers between and including 4 & [ ] Bounded, closed E. Use inequality and interval notation to describe the interval of real numbers. [ < < [, ) ) Bounded, half-open E.4 Use interval notation to describe the interval of real numbers and graph. is between 0 and 4. (0, 4) -4 - ( ) Page of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter P

3 Fill in the? of the equation for each property: Solutions:. Commutative + y =? y +. Associative (ab)c =? a(bc). Identity m +? = m 0 n(?) = n 4. Inverse z +? = 0 z w(?) =, w 0 w 5. Distributive? = rs rt r(s t) What misconception might a student have about n? They might think it is negative when in fact, it represents the opposite of n and therefore is dependent on the value of n. What is the difference between 4 and ( 4)? The bases are 4 and 4. So 4 = 6 and ( 4) = = 000. = 00. = 0. =. =. = =.0 = 00 = =.00 = 000 = 0 We define a 0 = & a -n = n a, a is indeterminate. Page of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter P

4 E.5 Eamine the properties of eponents: 4 = = 6 ( 4 ) = 4 4 = = 8 (4) +4 ( 4) = (8) = = 5 4 = 8 64 = 5 5 = = = 7 = 5 5 = 5 5 = 9 5 = 5 E.6 5 y y y y ( ) ( ) 5 y y ( ) y y = 5 5 y 8 = 40y 8 E.7 What is half of 40? What is one third of 8? 40 = 40 = 9 8 = 8 = 7 E.8 A pile of gravel contains 0 0 stones. Take ten stones from the original pile and throw nine onto a pile on the left and one onto a pile on the right. When the original pile is gone, how many stones are in each of the new piles? (9 + )(0) 9 = 9(0) 9 and (0) 9 Scientific Notation: One non-zero digit left of the decimal. 4 E.9 (.5 0 )( 6 0 )( 5 0 ) ( 0 ).5(6)(5) 0 4++( ) ( ) = = 5 0 = = Page 4 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter P

5 You Try: Simplify using scientific notation. ( 4,000,000)( 6,000,000,000),000,000 QOD: Describe the four types of unbounded intervals and give eamples of each. Sample SAT Question(s): Taken from College Board online practice problems.. When 70,000 is written as n, what is the value of n? (A) (B) (C) (D) 4 (E) 5. If P and Q are two sets of numbers, and if every number in P is also in Q, Which of the following CANNOT be true? (A) 4 is in both P and Q. (B) 5 is in neither P nor Q. (C) 6 is in P, but not in Q. (D) 7 is in Q, but not in P. (E) if 8 is not in Q, then 8 is not in P.. For all numbers a and b, let a b be defined a b = ab + a + b. For all numbers, y, and z, which of the following must be true? I. y = y II. ( ) ( ) ( ) + = III. ( y+ z) = ( y) + ( z) (A) I only (B) II only (C) III only (D) I and II only (E) I, II, and III only 4. What is the result when 46,9 is rounded to the nearest thousand and then epressed in scientific notation? (A) (B) (C) (D) (E) Page 5 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter P

6 5. If n is an odd integer, which of the following must be an odd integer? (A) n (B) n + (C) n (D) n + (E) 4n + 6. If n is an integer and if integer? (A) (B) (C) (D) (E) n n n n + n n n n + n + n n is a positive integer, which of the following must also be a positive Page 6 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter P

7 Cartesian Coordinates Coordinate (Cartesian) Plane: a plane in which an ordered pair can be located by reference to two perpendicular number lies, a horizontal (-ais) and vertical (y-ais) Origin: the intersection of the - and y- aes II I The coordinate aes separate the plane into four quadrants, I IV. III IV E. The table lists the percent of graduates taking the SAT and their average Math score. a. State the independent variable. b. Make a scatter plot for the data. c. Draw a trend line for the data. d. What type of association is there? e. Predict the mean SAT score for a state where 5% of graduates take the test. Arizona % 50 California 44% 484 Colorado 9% 5 Idaho 6% 50 Nevada 4% 486 New Meico % 54 Oregon 50% 486 Teas 45% 46 Utah 6% 56 Washington 7% 494 Solutions: a. percent of graduates taking the SAT b. c. d. negative correlation e. 505 Page 7 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter P

8 Absolute Value: a a, a 0 = a a < 0 E.: Rewrite without an absolute value symbol. a > 0, so by the definition, 4 7 = 4 7 b. 4π 7 4π 7 < 0, so by the definition, 4π 7 = 4π + 7 a+ b Number Line Formulas: Distance between a and b: a b Midpoint: E.: Find the distance between the points and the midpoint of the line segment that is formed by them. a.,5.4 Distance = 5.4 = 8.4 = 8.4 Midpoint = b., 6 4 Distance = = 7 = 7 Midpoint = = = = = 4 E. 4: Write the statement using absolute value notation: The distance between and 5 is less than 4. 5 < 4 or + 5 < 4 ( ) Coordinate Plane Formulas: Distance Formula (derived from the Pythagorean Theorem) (, y ) (4, ) (0, 0) (4, 0) (, y ) (, y ) 0 4= 4& 0 = & y y 4 + = c + y y = d c = 4 + = = 5 = 5 d = ( ) + ( y y) Page 8 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter P

9 E.5: Determine if the points form an isosceles triangle: (, ), (,0 ), ( 5,) Use the distance formula to find the lengths of the three sides: ( ) ( ) ( ) ( ) + 0 = 4 + = = 5 = 5 ( ( )) ( ) ( ) ( ) 5 + = + 4 = = 5 = 5 ( ) ( ) ( ) ( ) = 7 + = 49 + = 50 = 5 Yes the triangle has two congruent sides, therefore it is isosceles. Midpoint Formula: average of the -coordinates, average of the y-coordinates Midpoint of the line segment connecting y y : +, y + M y = (, ) & (, ) E. 6: The midpoint of AB is M (,4), and the coordinates of A are (, 9). Find B. + B = + = 4 = 5 B B = ( 5, ) B 9 + y B = 4 9+ y = 8 y = B B A circle is the locus of points ( y, ) equidistant (r) from a given point. Use the distance formula to find the equation of a circle centered at the origin. Then find the equation of a circle centered at ( hk, ). ( 0) ( 0) ( ( ) ( ) ) r = + y r = + y + y = r ( ) ( ) ( ( ) ( ) ) r = h + y k r = h + y k ( ) ( ) h + y k = r E.7: Find the equation of a circle that has (, ) and ( 5, ) as the endpoints of one of its diameters. The center will be at the midpoint of the diameter: ( hk) The radius is equal to half the length of the diameter: ( ) ( ) d = 5 + = = , =, =, 6 r = d = y = + y = 4 Equation of the circle: ( ) ( ) Page 9 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter P

10 9 + y = 4 You Try ( ) : Find the center and diameter of the circle. QOD: Write the standard equation of a circle, label the variables, and describe how it is derived. Sample SAT Question(s): Taken from College Board online practice problems.. If s, t, u, and v are the coordinates of the indication points on the number line above, which of the following is greatest? (A) s+ t (B) s+ v (C) s t (D) s v (E) s+ u. At a snack bar, a customer who orders a small soda gets a cup containing c ounces of soda, where c is at least but no more than. Which of the following describes all possible values of c? (A) (B) (C) (D) (E) c c c 4 c 4 c 4 4 Page 0 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter P

11 Linear Equations and Inequalities Linear Equation: an equation that can be written in the form a + b = 0, a 0 Equations are equivalent if they have the same solutions. E.: Show that solution. z = is not a solution of ( ) ( ) 4z z+ = z. Then find the Substitute z = into the equation: ( ( ) ) ( ) ( ) ( )?? ( ) ( ) 4 + = 5 = z = is not a solution. Solve for z: 8z 6 z = z 5z 9= z z = 8 z = 4 t + + = 4 6 E.: Solve the equation for t. ( t 5) Wipe out the fractions by multiplying by the least common denominator, in this case,. t + ( t+ 5) = t + 6t+ 80 = 9t = 75 t = Solving a Linear Inequality: E.: Solve and graph the solution on the number line. Epress the solution in interval notation. 5+ 8< ( ) 5+ 8 < 6 7< 4 > or 5+ 8 < 6 4 < 7 > Note: When multiplying or dividing by a negative number, you must flip the inequality sign. Interval notation: (, ) Solving a Compound Inequality: E.4: Solve and graph. Epress in interval notation < < > < Interval notation:, Page of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter P

12 E.5: Find the volume of material that makes up the Earth s crust, which is ten miles thick. 4 (Earth s radius 960 miles) Volume of a sphere: V = π r Volume of the crust = volume of the earth minus volume of the earth not including the crust 4 4 V = π ( 960) π ( 950),965,65,880cubic miles You Try: Solve the inequality and graph the solution on a number line. > QOD: Eplain why the inequality sign must be flipped when multiplying or dividing by a negative number. Sample SAT Question(s):. If Taken from College Board online practice problems. + k = and p( + k) = 6, what is the value of p? (A) (B) 4 (C) 6 (D) 9 (E). If < t 6 < 8, which of the following must be true? (A) t = 5 (B) 5< t < 6 (C) 4< t + < 9 (D) < t < 4 (E) t < or t > 8 Page of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter P

13 Lines in the Plane Slope: rate of change; steepness of a line rise change in y y y y y y d m = = = = = = run change in d ( ) E.: Find y so that m = on a line through the points ( 5, 8) and ( ) y 8 y + 8 = = y+ 8 = 4 y = 5 Equations of a Line, y. Point-Slope Form A line with slope m that passes through the point (, y) y y = m( ) Slope-Intercept Form A line with slope m and y-intercept b y = m + b C Standard Form A & B are not both 0; -intercept =, y-intercept = C A B A + By = C Find the slope: E.: Write an equation of the line that passes through the points (, ) and ( 5, 4) standard form. 4 m = = 5 4 ( ) in Use the slope and one of the given points to write the equation in point-slope form: y = ( + ) Multiply by the LCD and write in standard form: 4 y = + 4 y+ = + 4 y = + 4 y = ( ) 4 E.: Graph the line 4 y =. 4 = Find the - and y-intercepts: -int. = (, 0 ) y-int. y = y = 4 ( 0, 4) Page of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter P

14 Slope of Horizontal Lines: m = 0 Slope of Vertical Lines: m is undefined 4 m = = 0 E.4: Write an equation of the line that passes through the points (,4 ) & (, ). undefined Equation of the vertical line: = Parallel Lines( ) : lines in the same plane that do not intersect; Parallel lines have the same slope. Perpendicular Lines ( ) : lines in the same plane that intersect at a right angle; Perpendicular lines have opposite reciprocal slopes. E.5: Write an equation of the line parallel to 4+ y = 4 that passes through the point ( 0, ). 4 4 Find the slope of the given line: y = 4+ 4 y = + 8 m = Parallel lines have the same slope, so use this slope and the given point to write the equation in pointslope form. 4 4 y+ = ( 0) y = 4+ y = 6 Rate of Change E.6: A 5-minute phone call costs $0.8, and a 9-minute call costs $0.6. What is the rate of change? Find a linear function to represent the total cost (C) of a call to the duration in minutes (m). Then use this function to find the cost of a call that lasts hour and 5 minutes. (, ) : ( 5,0.8 ) &( 9,0.6) mc Rate of Change = = 0.4 = $0.06per minute Linear function: C ( m 5) hour and 5 minutes: 75 = or C = 0.06m C 75 = = $4.58 m = ( ) ( ) You Try: Write an equation of the line perpendicular to the line that passes through the points ( 5, ) & (, ) in point-slope form. QOD: Eplain why the slope of a horizontal line is zero and the slope of a vertical line is undefined graphically and using the definition of slope. Page 4 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter P

15 Sample SAT Question(s): Taken from College Board online practice problems.. Which of the following equations epresses y in terms of for each of the four pairs of values shown in the table above? (A) y = (B) y = 5.5+ (C) y = (D) y = 7.5 (E) y = In the y-coordinate plane, how many points are a distance of 4 units from the origin? (A) One (B) Two (C) Three (D) Four (E) More than four Grid-Ins. The price of a certain item was $0 in 990 and it has gone up by $ per year since 990. If this trend continues, in what year will the price be $00? t + y =. The equation above is the equation of a line in the y-plane, and t is a constant. If the slope of the line is 0, what is the value of t? Page 5 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter P

16 Solving Equations Solving Absolute Value Equations E.: Find the solution set for the equation. + 8 = 6 Isolate the absolute value epression. + = 4 + = 8 There are two values that have an absolute value of 8, 8 and 8. So + = 8 or + = 8. + = 8 + = Solve each equation for. 7 9 Solutions:, = = Quadratic Equation: an equation of the form a + b + c = 0, with a 0 Methods for Solving a Quadratic Equation Square Roots E.: Solve the equation ( + ) = 5 by etracting square roots. + = Isolate the epression that is squared: ( ) 5 + = + = Reminder: Square root both sides of the equation: ( ) 5 5 = Solve for : + = 5 5, + = = + 5 5, Set Notation: , + or, + (rationalized) Completing the Square Note: a must equal in order to complete the square. E.: Solve the equation 8+ = 0 by completing the square. Move c to the other side of the equation: 8 = Complete the square by adding b to both sides: = = + 6 Rewrite the perfect square trinomial as a binomial squared: ( 4) = 5 Solve by square roots: ( ) 4 = 5 4 = 5 4 = 5, 4 = 5 = 4 ± 5 or { 4 5, 4 + 5} Page 6 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter P

17 Quadratic Formula o Discriminant: b ± = 4ac b b 4ac a E.4: Solve the equation using the quadratic formula. Rewrite in standard form and determine a, b, and c. b b b 4ac > 0 two real solutions 4ac < 0 no real solutions ( comple solutions) 4ac = 0 one real solution = = 0 a =, b= 8, c = 6 Find the discriminant to determine the number of solutions. b 4ac = = 64 4 = 40 > 0 two real solutions ( ) ( )( ) Use the quadratic formula to solve. Solution set: { 4 0, 4 + 0} ( ) ( ) 8 ± 40 8 ± 0 = = = 4 ± 0 Factoring: use the zero product property E.5: Solve the equation by factoring. 0 = 7 Set the equation equal to zero. 0 7 = 0 Factor the quadratic. (ac method is shown) ac = = b = + = 0 Split the middle term: Factor by grouping: + 7 = 0 ( ) ( ) ( + )( 7) = = 0 + = 0 7= 0 Set each factor equal to zero and solve: =,7 Solving Rational Equations E.5: Find the value(s) of that make the equation true. 8 Factor and find the LCD. + = 7 ( 7)( ) Multiply each term by the LCD. ( 7) ( ) ( )( ) = ( ) ( ) ( ) ( ) + ( )( ) = = LCD: ( 7)( ) 8 ( 7) ( ) Page 7 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter P

18 Simplify and solve the resulting equation = 8 6 = 0 ( ) 6 = 0 = 0,6 Solving an Equation Graphically Method : Find the zeros (roots). E.6: Solve the equation graphically. Set the equation equal to zero. Graph the function and find the zero(s). 4 = 4 + = 0 4 y = + Solutions: { 0.5,.750 } Method : Finding the point(s) of intersection. E.7: Solve the equation graphically. 4 = + 6 Graph both sides of the equation as two separate functions. Then find the point(s) of intersection. Solution:.07 Think About It: What does the Y-value represent in the point of intersection? You Try: Solve for in the equation a + b + c = 0 by completing the square. QOD: True or False: An absolute value equation always has two solutions. Eplain your answer. Page 8 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter P

19 Sample SAT Question(s): Taken from College Board online practice problems.. If a 0 and = a, what is the value of? + a (A) 5 (B) (C) (D) (E) 5. If = + 6, which of the following must be true? (A) = 6 (B) < (C) > 0 (D) (E) < > Page 9 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter P

20 Solving an Absolute Value Inequality If If Solving Inequalities < a, then a < < a. Or is between a and a. > a, then < a or > a. Teacher Note: Remind students that an absolute value is a distance from the origin. This should make the inequalities above make sense. E.: Solve the inequality. 4 < Since the absolute value is less than, the value of the epression inside the absolute value must be between and. < 4 < 8 < < 6 Solve the compound inequality. 4< < 8 *Have students pick a value within the interval to verify that it is a solution to the original inequality. E.: Solve the inequality and verify your solution graphically. 7 Since the absolute value is greater than or equal to 7, the value of the epression inside the absolute value must be less than or equal to 7 or greater than or equal to 7. 7 or 7 7 or 7 Solve the compound inequality. 4 0 or 5 We can see that the graph of y = is greater than or equal to the graph of y = 7 when or 5. Solving Quadratic Inequalities Algebraic Method (sign chart) E.: Solve the inequality > 0 Solve the equation to find the zeros = 0 ( + 4)( + ) = 0 = 4, Make a sign chart using test values between and outside of the zeros. + + Page 0 of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter P

21 The solutions to the inequality are the -values that make the epression positive (greater than zero). Set notation: { 4 or } Graphing Method E.4: Solve the inequality graphically. 5 < 0 Because the graph is below the -ais ( y < 0) between = 0.5 and =.85, the solution, written in interval notation is: ( 0.5,.85) Projectile Motion: When an object is launched vertically from an initial height of s0 feet and an initial velocity of v 0 feet per second, then the vertical position s of the object t seconds after it is launched is s 6t vt 0 s0 = + +. E.5: A ball is thrown straight up from ground level with an initial velocity of 59 ft/sec. When will the ball s height above the ground be more than 0 ft? Write the equation of the height of the ball. Write an inequality to model the question. Solve by graphing. s = 6t + 59t + > 6t 59t 0 The ball s height will be more than 0 ft when < t <.078 seconds. You Try: Solve the quadratic inequality algebraically. Then verify your answer graphically. QOD: Can a quadratic and/or absolute value inequality have no solutions or one solution? Eplain your answer with an eample. Page of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter P

22 Sample SAT Question(s): Taken from College Board online practice problems.. If y = + and <, which of the following represents all the possible values for y? (A) y < 7 (B) y > 7 (C) y < 5 (D) y > 5 (E) 5< y < 7 Grid-In. The figure above shows the graph of a quadratic function in the y-plane. Of all the points ( y, ) on the graph, for what value of is the value of y greatest? Page of Precalculus Graphical, Numerical, Algebraic: Pearson Chapter P

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