Algebra II SOL Review

Transcription

1 Virginia each ity Public Schools lgebra II SOL Review Student Name lgebra II SOL Test ate

2 EO lgebra II Review 1

3 Epressions and Operations EO lgebra II Review

4 II.1 The student, given rational, radical, or polynomial epressions, will a) add, subtract, multiply, divide, and simplify rational algebraic epressions; Hints and Notes To add or subtract: Must have a common denominator To multiply: Factor numerator, factor denominator, cancel common factors PRTIE.1 1. Which is equivalent to ? To divide: Flip the fraction after the division sign and use multiplication rules To simplify: factor numerator, factor denominator, cancel common factors NO HOPPING!! omple Fractions: Simplify numerator, simplify denominator, then divide Which is equivalent to 6a 1 a 3 a a 6 a 6 a 6( a ) a 6a 4a 4 4 a 3. Which is equivalent to 3 5 y? y y y 14 EO lgebra II Review 3

5 4. Which equivalent to 7 9 7? 9 9 ( 7) Which is equivalent to 4y 5 y 1 4 y 5 y? y 4 y 5 y ( y 4)(y 5) y 3y 0 6. Which is equivalent to a b 9 ( a b) 9 ( a b) ( a b) 18 ( a b)? 18a 9 b II.1a SKILLS HEKLIST: I can dd, subtract, multiply, and divide rational algebraic epressions. Simplify a rational algebraic epression with common monomial or binomial factors. Recognize a comple algebraic fraction, and simplify it as a quotient or product of simple algebraic fractions. EO lgebra II Review 4

6 SOL II.1 The student, given rational, radical, or polynomial epressions, will b) add, subtract, multiply, divide, and simplify radical epressions containing rational numbers and variables, and epressions containing rational eponents; c) write radical epressions as epressions containing rational eponents and vice versa HINTS N NOTES a b a b Remember: enominator in ip To add or subtract radicals: Radicands must be the same. You may only add like radicals. lways simplify your radical completely. Pay attention to your root value. Everything is not a square root. PRTIE II.1bc 1. Which epression is equivalent to 3 a? 3 a 3 a 1 6 a 6 a. Which is equivalent to 3 8 6? 3 3. Which is equivalent to ? EO lgebra II Review 5

7 4. Which is equivalent to ab? 3 ab 3 ab 3 4 a b 4 3 a b 5. Which is equivalent to 1 3 3? What is the simplest form of 3 7 5? 7. What is the value of ? EO lgebra II Review 6

8 8. Which is equivalent to the epression II.1b, c SKILLS HEKLIST: I can Simplify radical epressions containing positive rational numbers and variables. onvert from radical notation to eponential notation, and vice versa. dd and subtract radical epressions. Multiply and divide radical epressions not requiring rationalizing the denominators. EO lgebra II Review 7

9 SOL II.1 The student, given rational, radical, or polynomial epressions, will d) factor polynomials completely HINTS and NOTES lways look for a greatest common factor first y + w = (y+w) Look for patterns: a b = (a + b)(a b) a + ab + b = (a+b) a 3 + b 3 = (a+b)(a ab + b ) a 3 b 3 = (a-b)(a + ab + b ) square-multiply-square-opposite-plus ***make sure you have that opposite sign in the second factor You can multiply or foil your choices to work backwards, if you want to work backwards. PRTIE II.1d 1. Which is a factored form of 9 5? Which is a factor of 16 1? Which is a factor of 15? Which is a factor of 6a 5ab 6 b? a 3 b a3 b 3a b 3a 3 b EO lgebra II Review 8

10 3 5. Which is the factored form of 8 1? Which is the factored form of 1 y 3? 1 y1 y y 1 y1 y y 1 y1 y y 1 y1 y y 7. Which represents the complete factorization of ( 1)( 4) ( 4)( 1) ( 1)( 4) ( 1)( 4) 8. Given the area of a rectangle being 51 Which of the following could represent the length of one side of the rectangle? II.1d SKILLS HEKLIST: I can Factor polynomials by applying general patterns including difference of squares, sum and difference of cubes, and perfect square trinomials. Factor polynomials completely over the integers. Verify polynomial identities including the difference of squares, sum and difference of cubes, and perfect square trinomials. EO lgebra II Review 9

11 II.3 The student will perform operations on comple numbers, epress the results in simplest form using patterns of the powers of i, and identify field properties that are valid for the comple numbers. HINTS and NOTES TI-83 alculator TIPS Remember Use your i button on your calculator. Remember to include your i If 3 i i 3 i parentheses. then 3 4, i 1 or just i i i i 1 1i 1 This sequence repeats itself. In other words i i i i i i 1 i i i i i 1 i 1 i i i i i lways change i to -1 when working out by hand. ny # i # 1 i 1 PRTIE.c 1. Which epression is equivalent to i i 5i 10 i 6 i 4 3 i. Which is equivalent to 4 i 53 i i 6 i 3. Which is equivalent to 43 i? i 4i 4. Which is equivalent to i i 4 19i 16 19i 6 9i 6 10i 3 5? EO lgebra II Review 10

12 5. Which is equivalent to 5 i 1 3 i? 4 8i 5 4 7i 5 1 7i 5 1 7i 4 6. Which is equivalent to i 3 i 9 9 i 7. What number does i4 equal? i -1 -i 1 EO lgebra II Review 11

13 Put your answer in the bo. These are open-ended questions. Work them out write your answer in the bo (on the computer you would type your answer in the bo being sure to put it in appropriate form, simplest fraction, decimal, etc.) For our purposes, you will write your answer in the bo. 8. Simplify the following epression, i)( i) 4 4. Type your answer in the bo. II.3 SKILLS HEKLIST: I can Recognize that the square root of 1 is represented as i. etermine which field properties apply to the comple number system. Simplify radical epressions containing negative rational numbers and epress in a+bi form. Simplify powers of i. dd, subtract, and multiply comple numbers. Place the following sets of numbers in a hierarchy of subsets: comple, pure imaginary, real, rational, irrational, integers, whole, and natural. Write a real number in a+bi form. Write a pure imaginary number in a+bi form. EO lgebra II Review 1

14 Equations and Inequalities EO lgebra II Review 13

15 SOL II.4 The student will solve, algebraically and graphically, a) absolute value equations and inequalities Graphing calculators will be used for solving and for confirming the algebraic solutions. HINTS and NOTES n absolute value equation or inequality makes TWO statements. Shading of Graphs: absolutevalue number N sentence (Less than or equal to) Look for graph shaded between numbers and closed circle PRTIE II.4 1. Which of the following represents the solution to 7? or 7. Which inequality describes the solution set graphed absolutevalue number N sentence (Less than only) Look for graph shaded between numbers and open circles absolutevalue number OR sentence (Greater than or equal to) Look for graph shaded to the left and to the right, closed circles absolutevalue number OR sentence (Greater than only) Look for graph shaded to the left and to the right, open circles What is the solution to 3 1 3? or 1 7 or EO lgebra II Review 14

16 4. Which inequality describes the solution set graphed? What is the solution set for 6 or 1 1 or Which represents the graph of 1 10 EO lgebra II Review 15

17 7. Which of the following inequalities best represents SKILLS HEKLIST: I can Solve absolute value equations and inequalities algebraically and graphically. EO lgebra II Review 16

18 SOL II.4 The student will solve, algebraically and graphically, b) quadratic equations over the set of comple numbers Graphing calculators will be used for solving and for confirming the algebraic solutions. HINTS and NOTES Ways to solve a quadratic equation.: Put equation in a + b + c = 0 form 1. Factor, set each factor equal to zero, find solutions. Use quadratic formula: b b 4ac a 3. Square Root both sides if a squared term is isolated on one side 4. Use your calculator: Sketch in y= Zoom 6 Look for zeros(-intercepts) 5. Work backwards Plug it in!!!! Substitute given answer choices into your calc and see what works Know terminology: Solutions, zeros, roots, -intercepts all mean the same thing PRTIE 4b 1. What is the solution set for 04,,,, Which is the solution set for i 3 4, 4, 3. Which is the solution set for i 1 i 1 i 4. What are the solutions to 616 0? ? y 1 or y 6 y 1 or y 6 y 1 or y 6 y 1 or y 6 ( y 3) 81 0 EO lgebra II Review 17

19 5. What are the solutions to 1 or or 4 3 i What are the solutions to 4i or 4 or 4 or i 4 or Which graph represents a quadratic equation with no real solutions? II.4b SKILLS HEKLIST: I can Solve a quadratic equation over the set of comple numbers using an appropriate strategy. alculate the discriminant of a quadratic equation to determine the number of real and comple solutions. EO lgebra II Review 18

20 SOL II.4c The student will solve, algebraically and graphically, c) equations containing rational algebraic epressions Graphing calculators will be used for solving and for confirming the algebraic solutions. HINTS and NOTES To solve a rational equation: Practice. 4c 1. What is the solution to Eliminate your denominators ross-multiply if possible Multiply both sides of equation by your Least ommon enominator Plug it in! (See what answer satisfies your equation) What value of q is the solution to the equation 11 q 8 q 6 31 q 9 q What is the solution to 6 0 7q9 6q EO lgebra II Review 19

21 4. What is the value of y is the solution to the 4 y30 6 y8 equation y 5 45 y 13 8 y 5 3 y 4 EO lgebra II Review 0

22 Put your answer in the bo. These are open-ended questions. Work them out write your answer in the bo (on the computer you would type your answer in the bo being sure to put it in appropriate form, simplest fraction, decimal, etc.) For our purposes, you will write your answer in the bo. 5. Solve the following equation for. Place your answer in the bo II.4c SKILLS HEKLIST: I can... Solve equations containing rational algebraic epressions with monomial or binomial denominators algebraically and graphically. EO lgebra II Review 1

23 SOL II.4d The student will solve, algebraically and graphically, d) equations containing radical epressions Graphing calculators will be used for solving and for confirming the algebraic solutions. HINTS and NOTES To solve radical equation: If a squared equation, square root both sides If a cubed equation, cube root both sides Plug it in!!!! (see what answer satisfies your equation) PRTIE II.4d 1. What is the solution set for What is the solution set for 77,, What is the solution set for y y3 y y1 1 y y 3 1 y y 9 3y 4 5 EO lgebra II Review

24 4. What is the solution for 16 3? What is the solution to The length, s, (in feet) of the skid mark left by an automobile traveling at r miles per hour can be approimated by the relation r 5 s. If a car is going 80 miles per hour when the brakes are applied, about how many feet long would the skid mark be? 30 ft 410 ft 640 ft 180 ft II.4d SKILLS HEKLIST: I can Solve an equation containing a radical epression algebraically and graphically. Verify possible solutions to an equation containing rational or radical epressions. EO lgebra II Review 3

25 SOL II.5 The student will solve nonlinear systems of equations, including linear-quadratic and quadratic-quadratic, algebraically and graphically. Graphing calculators will be used as a tool to visualize graphs and predict the number of solutions. HINTS and NOTES Remember your solution is your point or points of intersection. If given a graph, look at your choices and approimate where the solutions are (Read your graph) If given equations: 1. Plug it in!! Plug in choices to see which point satisfies both equations. You are working backwards and checking. Use your calculator. Make sure your equations. are in Y= form Put equation one in y 1 Put second equation in y Use Zoom 6, then adjust your window as needed Then nd alc #5, put blinker on point, Enter, Enter, Enter e sure to check your answer(s). ***If your curves do not intersect at all There is no solution to your system! Practice II.5 1. What is the solution set to the following system of equations? y 3 y ( 8, 18) and (, ) ( 8, ) and ( 18, ) (, ) and ( 18, 8) ( 8, 14) and (, 6). Which set of ordered pairs is the solution to the system of equations ( 0, 3) and ( 1, ) ( 0, 1) and ( 3, ) ( 0, 1) and ( 6, 3) ( 3, ) and ( 6, 11) y 1 y Which of the following is the solution to the system of equations below? y 6 9 y 1 ( 5, 7) and ( 1, 1) ( ( 1, 1) and ( 5, 3) ( 1, 7) and ( 5, 3) 1 (, ) EO lgebra II Review 4

26 4. Which is most likely the solution set for the system of equations? ( 0, 1) and ( 3, ) ( 1, ) and (, 3) ( 1, ) and (, 7) ( 3. 7, 0) and ( 0. 4, 0) 5. What is most likely the solution to the system ( 1. 5,. 5) and ( 3, ) (. 5, 1. 5) and (, 3) (, 3) and (. 5, 1. 5) ( 3, ) and ( 1. 5, 3) II.5 SKILLS HEKLIST: I can Predict the number of solutions to a nonlinear system of two equations. Solve a linear-quadratic system of two equations algebraically and graphically. Solve a quadratic-quadratic system of two equations algebraically and graphically. EO lgebra II Review 5

27 Functions and Statistics EO lgebra II Review 6

28 SOL II.6 The student will recognize the general shape of function (absolute value, square root, cube root, rational, polynomial, eponential, and logarithmic) families and will convert between graphic and symbolic forms of functions. transformational approach to graphing will be employed. Graphing calculators will be used as a tool to investigate the shapes and behaviors of these functions. HINTS and NOTES Functions: 1. bsolute Value. y Looks like a V. If positive, goes up. If negative goes down.. Square Root y Looks like the path of a Rocket Has a definite starting point. 3. Quadratic y U shape an go up or down 4. Eponential y in eponent 5. Logarithmic y log Inverse of eponential equation p ( ) 6. Rational Function y q ( ) Look for Horizontal and Vertical asymptotes (In calculator p( ) q( ) use parentheses) 7. Polynomial Function n n1 n y a b c... decreasing powers equation: up/down or down/up Two movements (Parabola) 3 equation: up/down/up or down/up/down Three movements (cubic) 4 equation: up/down/up/down or down/up/down/up Four movements ( W ) PRTIE II.6 1. The graph below is an eample of which type of function? bsolute Value Eponential Linear Quadratic. Which most likely represents the equation of the graph y 4 y 4 y 4 y 4 EO lgebra II Review 7

29 Transformations: Remember horizontal shifts are the opposite of the sign y Shifts units to the right y Shifts units to the left 3. Which of the following is most likely the equation graphed Vertical shift is at the end of your function and moves the same y Shift verte up y Shift verte down alculator: You can sketch every function in your calculator using Y= and zoom 6 or zoom 4 y 1 y 5 1 y y 1 4. Which sketch could represent the graph of the function f ( ) a EO lgebra II Review 8

30 5. This is a portion of the graph of a polynomial function. If written in order of descending powers, which could be the first term of the polynomial? Which could be the graph of f ( ) a b c d if a, b, c, and d are real numbers and a 0? II.6 SKILLS HEKLIST: I can Recognize graphs of parent functions. Given a transformation of a parent function, identify the graph of the transformed function. Given the equation and using a transformational approach, graph a function. Given the graph of a function, identify the parent function. Given the graph of a function, identify the transformations that map the preimage to the image in order to determine the equation of the image. Using a transformational approach, write the equation of a function given its graph. EO lgebra II Review 9

31 SOL II.7 The student will investigate and analyze functions algebraically and graphically. Key concepts include a) domain and range, including limited and discontinuous domains and ranges; b) zeros; c) - and y-intercepts; d) intervals in which a function is increasing or decreasing; e) asymptotes; f) end behavior; g) inverse of a function; and h) composition of multiple functions. Graphing calculators will be used as a tool to assist in the investigation of functions. HINTS and NOTES 9. omain: values Range: y values THINK: alphabetical order To be a function: No two s are the same or it passes the vertical line test Zero (root, solution, -intercept) of a function: Where your function has a value of 0 (where your graph crosses the -ais) TI-83 alculator: Put eqt. in y= Zoom 6 or 4 nd alc # zero Left ound/right ound/guess Work backwards! Substitute the answers given to see which answer gives you a 0 To find value of a function: Plug the number into your epression for every To find composite functions: f(g()) Substitute the entire inner function into the outer function in the position This is a portion of the graph of a polynomial function. pparently the function has a turning point at 1, 1, 3 01, 1 0 4, 10. Which describes the end behavior of the function f ( ) as approaches infinity? y approaches y approaches 0 y approaches y approaches EO lgebra II Review 30

32 To find inverse of a function: y = switch and y Solve for y Inverse (f -1 ()) will be what y = 3. On a graph: intercept where your function crosses the -ais y intercept where your function crosses the y-ais turning point point where your function turns the corner and switches direction interval increasing region on graph where y values are increasing interval decreasing region on the graph where y values are decreasing symptotes: Vertical & Horizontal Vertical: Look at denominator (whatever makes = 0) =? Horizontal: same power leading coefficient y leading coefficient higher power below no horizontal asymptote higher power above y 0 End ehavior: Look at your graph and your question. Read carefully. It will ask you what y value you are approaching as approaches a given value. If it is a rational function, you are always approaching your horizontal asymptote. Use your calculator!! 4. This is a portion of the graph of a polynomial function. pparently the function has a domain of 0 y y This is a portion of the graph of a polynomial function. This function apparently has the following types of roots 5 different real roots 4 different real roots real roots and imaginary roots no real roots 5. Which value is not a zero of 3 P EO lgebra II Review 31

33 6. This is portion of a polynomial function. pparently the function has a double zero of between - and -1 between - and 1 between 1 and between 3 and 4 7. The polynomial function f ( ) has a zero between -4 and -3 - and -1-1 and 0 3 and 4 8. If the domain of f ( ) 3 is limited to 3, 1, 1, 3, what is the range? 1, 5, 1, 15 1, 15 1, 15 1, 5, 15, 1 9. If f and g 3, which if the following epressions represents g f? EO lgebra II Review 3

34 10. If f 5 7 what is f 3? Which function represents the inverse of the function f f 1 ( ) f 1 ( ) f 1 ( ) f 1 ( ) ( ) EO lgebra II Review 33

35 Highlight each correct answer. These questions give you choices. You must click on each correct answer and make sure you mark every answer that is correct. If you forget one, it will be incorrect. ircle each correct answer. More than one answer may be selected. 1. The graph of a polynomial function is graphed below. Highlight each epression that represents an increasing interval? EO lgebra II Review 34

36 lick and rag. These questions give you the choices for your answer or answers. You must click on each correct answer and drag it to the appropriate bo. You must get all of them correct to get the answer correct. For our purposes, just write the correct answers in the boes. 13. lick and drag the horizontal and vertical asymptotes for the function g ( ) 1 4 Horizontal Vertical 1 y 1 4 y 4 4 y 4 II.7 SKILLS HEKLIST: I can Identify the domain, range, zeros, and intercepts of a function presented algebraically or graphically. escribe restricted/discontinuous domains and ranges. Given the graph of a function, identify intervals on which the function is increasing and decreasing. Find the equations of vertical and horizontal asymptotes of functions. escribe the end behavior of a function. Find the inverse of a function. Graph the inverse of a function as a reflection across the line y =. Investigate eponential and logarithmic functions, using the graphing calculator. onvert between logarithmic and eponential forms of an equation with bases consisting of natural numbers. Find the composition of two functions. Use composition of functions to verify two functions are inverses. EO lgebra II Review 35

37 SOL II.8 The student will investigate and describe the relationships among solutions of an equation, zeros of a function, -intercepts of a graph, and factors of a polynomial epression. HINTS and NOTES Solutions of equation, zeros of a function, roots of the equation, and -intercepts all refer to the same thing: Where is the value of my function equal to zero? If zeros are a,b,c Then factors are ( a)( b)( c) **** Factors will contain opposite sign of roots PRTIE II.8 1. Which is a zero of f ( ) 6? polynomial function has a zero at 4. Which epression must be a factor of the polynomial? Which of the following functions has -intercepts of - and 1 f () f () f () 1 f () 1 4. Which of the following sets contains all the apparent zeroes for the function shown? 1, 0,, 1, 1, 1, 0, 1 EO lgebra II Review 36

38 5. section of the graph of the polynomial function with integral roots is shown. Which of the following sets most likely contain only elements that are factors of the polynomial? ( ),( 1. 5 ) ( ),( 1),( 1 ) ( ),( 1),( 1 ),( ),( 1),( 1 6. Which of the following sets contains all the apparent zeroes for the function, 0 10,,, 1, 0, 1. 5, II.8 SKILLS HEKLIST: I can escribe the relationships among solutions of an equation, zeros of a function, -intercepts of a graph, and factors of a polynomial epression. efine a polynomial function, given its zeros. etermine a factored form of a polynomial epression from the -intercepts of the graph of its corresponding function. For a function, identify zeros of multiplicity greater than 1 and describe the effect of those zeros on the graph of the function. Given a polynomial equation, determine the number of real solutions and nonreal solutions. EO lgebra II Review 37

39 SOL II. The student will investigate and apply the properties of arithmetic and geometric sequences and series to solve real-world problems, including writing the first n terms, finding the n th term, and evaluating summation formulas. Notation will include n X i and a n. i1 HINTS and NOTES Formulas will be on your Formula Sheet, but not they are used for!! Pay attention to whether you have an arithmetic (dding) or a geometric (Multiplying)sequence If you get stuck with the formulas you can always just write all of your numbers out and find a term or add the numbers up to get the sum rithmetic Term a ( n ) d n 1 1 rithmetic Sum n ( a ) 1 an S n Geometric Term a r n 1 n1 Geometric Sum ( 1 n a1 r ) Sn 1 r Infinite Geom. Sum a1 S 1 r PRTIE II. 1. What is the sum of the series defined by n n0. Two geometric means between and 54 are 3. If 4 and 1 6 and 1 6 and 18 1 and 18 a? a n1 n, which number represents What are two arithmetic means between 3 and 4? 8 and 1 8 and 16 9and and 17 EO lgebra II Review 38

40 5. What is the value of 6 n? n If a n 1 1, then what is a 9? n EO lgebra II Review 39

41 Put your answer in the bo. These are open-ended questions. Work them out write your answer in the bo (on the computer you would type your answer in the bo being sure to put it in appropriate form, simplest fraction, decimal, etc.) For our purposes, you will write your answer in the bo. 7. riving a piling into a harbor bottom, a pile driver sinks the piling 4 inches on the 1 first stroke, 18 inches on the second stroke, and 13 inches on the third stroke. If the sequence is continued, how far will the piling be driven down on the 5th stroke? II. SKILLS HEKLIST: I can istinguish between a sequence and a series. Generalize patterns in a sequence using eplicit and recursive formulas. Use and interpret the notations, n, n th term, and a n. Given the formula, find a n (the n th term) for an arithmetic or a geometric sequence. Given formulas, write the first n terms and find the sum, S n, of the first n terms of an arithmetic or geometric series. Given the formula, find the sum of a convergent infinite series. Model real-world situations using sequences and series. EO lgebra II Review 40

42 SOL II.9 The student will collect and analyze data, determine the equation of the curve of best fit, make predictions, and solve real-world problems, using mathematical models. Mathematical models will include polynomial, eponential, and logarithmic functions. HINTS and NOTES If given a graph, LOOK at the shape of your graph to determine if the est Model would be Linear, Quadratic, ubic, Eponential, or Logarithmic Linear line Quadratic Parabola U shape ubic Egyptian Eponential One side is increasing rapidly Logarithmic Inverse of Eponential If given a set of data, read carefully. If it states the type of model, then you are lucky and you use it. If it states to determine the best model for the data, then do a rough sketch on your paper and you look to see which model is best TI-83 LULTOR Remember to put s in List 1 and y s in List. lear anything in your lists. STT-L #4 Linear #5 Quadratic #6 ubic #9 Ln (Logarithmic) #0 (10) Eponential ENTER PRTIE.8 1. In 1940, the life epectancy at birth in the general public was 6.9 years. y 1980 it had risen to 73.7 years. ssuming a linear relation, which is the best prediction of life epectancy in the year 000? Which is the best quadratic model for the data ( 1, ),( 0, 1),(, 5) y y 1 y 1 y 1 3. Which type of function would best model the data y Quadratic ubic Eponential Logarithmic EO lgebra II Review 41

43 4. What is the best model equation for the data y y y. 1( 1. 0) y 1. 0(. 1) y What is your best prediction for y when 30? y The table shows the number of students enrolled in the lgebra/trig program at a high school the first 5 years the course was offered. No. of Year students () (y) Which of the following equations most closely describes the relationship between the number of students enrolled and the number of years the class has eisted? y 13 y13 43 y10 3 y1343 EO lgebra II Review 4

44 7. What is the best prediction for year #10? No. of Year students () (y) II.9 SKILLS HEKLIST: I can ollect and analyze data. Investigate scatterplots to determine if patterns eist and then identify the patterns. Find an equation for the curve of best fit for data, using a graphing calculator. Models will include polynomial, eponential, and logarithmic functions. Make predictions, using data, scatterplots, or the equation of the curve of best fit. Given a set of data, determine the model that would best describe the data. EO lgebra II Review 43

45 SOL II.10 The student will identify, create, and solve real-world problems involving inverse variation, joint variation, and a combination of direct and inverse variations. HINTS N NOTES irect Variationy k or y k is the form of a direct variation equation, where is the constant of variation y-intercept is always zero The constant of the function is always the slope Graphs of irect Variation equations always cross through the origin The constant can be negative or positive Inverse Variationk y k or y is the form of an inverse variation equation, where k is the constant s input values increase, output values decrease and vice versa Graphs of Inverse Variations are not linear Joint Variationk yz k or y is the form z of a joint variation equation, where k is the constant You cannot graph a joint variation because it involves more than one variable 1. The time it takes to travel a given distance varies inversely as the average rate of travel. veraging 4 miles per hour, it takes John 5 hours to drive to Pittsburgh. If it took him 4 hours and 0 minutes to reach Pittsburgh on his last trip, what was his average rate of travel? 36.4 mph 46.7 mph 48.5 mph 49.4 mph. The volume (V) of a sphere varies directly with the cube of its radius (r). If k is the constant of proportionality, which is the formula for this relationship? V kr V kr 3 V k r 3 r kv 3 3. Hooke s law states that the force required to stretch a spring varies directly with the distance the spring is stretched. If a 10 pound force stretches a spring inches, what force is required to stretch 5 inches? 15 pounds 0 pounds 5 pounds 30 pounds EO lgebra II Review 44

46 4. The amount of interest (I) owed on a loan varies directly with the length of time (t) of the loan. If k is the constant of proportionality, which formula represents this relationship? I kt I k t t ki t k I 5. oyle s Law states that, for a fied amount of gas, the volume of the gas at a constant temperature is inversely proportional to the pressure. If a certain gas occupies 9.84 liters at a pressure of 50 centimeters of mercury (cm Hg), what is the approimate pressure when the volume is increased to 1 liters? 39.8 cm Hg 41.0 cm Hg 43. cm Hg 45.1 cm Hg 6. In which of the following is z directly proportional to and inversely proportional to the square of y? z k y z ky z k y z k y II.10 Skills hecklist: I can Translate y varies jointly as and z as y = kz. Translate y is directly proportional to as y = k. Translate y is inversely proportional to as y k. Given a situation, determine the value of the constant of proportionality. Set up and solve problems, including real-world problems, involving inverse variation, joint variation, and a combination of direct and inverse variations. EO lgebra II Review 45

47 SOL II.11 The student will identify properties of a normal distribution and apply those properties to determine probabilities associated with areas under the standard normal curve. HINTS and NOTES Normal istribution hart You must be able to draw your standard normal curve and know your percentages. If your z-score is an integer, then you can use your normal curve for the percentage. If your z-score is a decimal, then your will have to use your standard normal probability table. Use your formula sheet!!! PRTIE t Thomas Nelson ommunity ollege, the Pre-Test for Mathematics has 00 points on the test. The mean is 10 and the standard deviation is 0. What is the approimate percentage of students which score below 100 points? 50% 34% 16% 7%. t Thomas Nelson ommunity ollege, the Pre-Test for Mathematics has 00 points on the test. The mean is 10 and the standard deviation is 0. In order to pass the Mathematics Pre-Test, a student must score 140 points. If 360 freshmen took the pretest last fall, how many passed? On his midterm eam, Jimmy scored 75 points, which was eactly standard deviations above the mean. If the standard deviation for the test is 4, what is the mean for the test? Susie s test grades in lgebra II for the third quarter are 100, 80, 60, 88, 9, and 90. How many scores are within one standard deviation of the mean? all 6 scores 5 scores 4 scores 3 scores EO lgebra II Review 46

48 5. normally distributed set of 968 values has a standard deviation of 11 and a mean of 108. Which is closest to the number of values epected to be above 15? Put your answer in the bo. These are open-ended questions. Work them out write your answer in the bo (on the computer you would type your answer in the bo being sure to put it in appropriate form, simplest fraction, decimal, etc.) For our purposes, you will write your answer in the bo. 5. Roanoke, Virginia had the folowing amounts of snowfall last January 4. ". 3" 6" 7. 8" 10" 5. 5" 1. 5" 0. 8 " Find the mean and the standard deviation. Type your answers in the appropriate bo Mean Standard eviation 6. survey of 0 colleges found that seniors graduated with an average $1,000 in debt from student loans. The debt was normally distributed with a standard deviation of $300. Find the probability that a senior graduated owing more than $16,000. II.11 SKILLS HEKLIST: I can Identify the properties of a normal probability distribution. escribe how the standard deviation and the mean affect the graph of the normal distribution. ompare two sets of normally distributed data using a standard normal distribution and z-scores. Represent probability as area under the curve of a standard normal probability distribution. Use the graphing calculator or a standard normal probability table to determine probabilities or percentiles based on z-scores. EO lgebra II Review 47

49 SOL II.1 The student will compute and distinguish between permutations and combinations and use technology for applications. HINTS and NOTES Permutation: Order matters Formula on formula sheet Eamples are combinations on locks, combination lock should be called a permutation lock! ombination: Order does not matter Formula on formula sheet Eamples are lotteries or the items in a fruit salad PRTIE If the digits can be repeated, how many 3-digit numbers can be formed using the digits 1,, 3, and 4? Meredith has 14 girls on her softball team. She wants to have co-captains. How many different choices does she have? 14! 14 P If numbers and letters can be repeated, how many different 6-digit license plates can be made if the first two positions are letters and the last four are digits? 49, ,000 6,760, ,65 EO lgebra II Review 48

50 Put your answer in the bo. These are open-ended questions. Work them out write your answer in the bo (on the computer you would type your answer in the bo being sure to put it in appropriate form, simplest fraction, decimal, etc.) For our purposes, you will write your answer in the bo. 1. committee of 3 teachers and 3 students is to be formed to judge a contest. If there are 7 students and 5 teachers to choose from, how many different committees could be formed? Type your answer in the bo provided. 5. How many permutations can be formed using the letters in the word MTHEMTIS? Type your answer in the bo provided. II.1 SKILLS HEKLIST: I can ompare and contrast permutations and combinations. alculate the number of permutations of n objects taken r at a time. alculate the number of combinations of n objects taken r at a time. Use permutations and combinations as counting techniques to solve real-world problems. EO lgebra II Review 49

51 nswer Key for Functions and Statistics Equation ost 8 Graphs highlighted 9 oes highlighted ML NFL S = S = V = V = M = M = omain Range oes Highlighted EO lgebra II Review 50