Summer Math Packet (revised 2017)

Transcription

1 Summer Math Packet (revised 07) In preparation for Honors Math III, we have prepared a packet of concepts that students should know how to do as these concepts have been taught in previous math classes. This packet does not require ou to use a calculator; in fact ou should not use a calculator on some of the problem sets where it is noted. Honors Math III builds on the concepts in this packet, and we start teaching Math III concepts on the first da of school. We epect ou to know the concepts in the packet in order to help ou be successful in Honors Math III. Although ou do not have to complete ever problem in this packet, ou will be held responsible for knowing how to complete them all and the content/skills used. Answer kes are included at the end of this packet to check our understanding. If ou are struggling with an content from this packet, get help from a friend, parent, peer, or tutor. If ou can t find someone to help ou, there are tutors available. A list of tutors can be found b calling Reagan High School or from Reagan s Student Services website. Keep in mind these tutors ma charge a fee. You can also go to outube.com and tpe in the name of the concept and watch videos of math teachers and tutors eplain the concepts for FREE!!!! You will have a graded assignment within the first two weeks of school. The graded assignment will cover the concepts/skills in the packet, but will not have the eact same problems. The graded assignment will contain a calculator inactive part so be prepared b following instructions where calculators are not allowed. You will be able to use our packet with completed problems as a resource on the assessment at the beginning of the ear so it is in our best interest to complete as needed. The packet s answer ke section ma not be used on the graded assessment. TABLE OF CONTENTS: ) Vocabular & Important Things to Remember page (KEY p. 9-0) ) Operations with Fractions page (KEY p. ) ) Solving Equations & Inequalities page (KEY p. ) ) Linear Equations page (KEY p. ) ) Graphing Linear Equations & Inequalities page 6 (KEY p. ) 6) Sstems of Equations page 8 (KEY p. 7) 7) Laws of Eponents page 0 (KEY p. 9) 8) Polnomial Operations page (KEY p. 0) 9) Radicals & & Rational Eponents page (KEY p. ) 0) Using Function Notation (with numbers & variables; word problems) page (KEY p. ) ) Solving Quadratic Equations (Quadratic Formula, Factoring, page (KEY p. ) Square Root Propert & Completing the Square) ) Using Midpoint & Distance Formulas page (KEY p. ) ) Proving Congruent Triangles page 6 (KEY p. ) ) Transformations (using absolute value graphs) page 7 (KEY p. 6) ) ANSWER KEYS page 8

2 ) Vocabular/Know the Difference Review. Proportions vs. multipling fractions Proportion 8 vs.. * =? vs. + =? Multipling Fractions 8. order of operations: don t be tricked b these common mistaken identities a. * vs. ( * ) b. + ( + ) c. -8 vs. (-8). ( 9) vs. ( + 9) vs. ( 9)( + 9) a. ( 9) = b. ( + 9) = c. ( 9)( + 9) =. know the difference between a term, epression, equation, and inequalit term epression equation inequalit 6. know the difference between solve, evaluate and simplif simplif evaluate solve 7. know the difference between rational and irrational rational irrational 8. know the > and < smbols b name: 9. know coefficient 0. know factor. reduce factors not individual terms

3 ) Operations with Fractions DO NOT USE A CALCULATOR ) 9 ) 7 ) 6 ) 9 ) 7 6) 6 7) 9 8) 7 9) 6

4 ) Solve Equations and Inequalities DO NOT USE A CALCULATOR Clear out fractions. 7 ) 7 ) 7 ) a a a 8 6 Variables on both sides. ) ( 8 ) + ( + ) = ) 8 + ( a ) = ( a + ) 6) 6 + = Solve inequalities and graph our answer. When ou multipl or divide both sides b a negative remember to flip the inequalit. Get variable on the left. 6 8) 6 6 7) Word Problems - You must be able to write the equation or inequalit first. Then solve for the variable. 9) The greater of two numbers is 6 more than times the smaller. Their sum is. Find the numbers. 0) Find three consecutive integers whose sum is 0. ) Find three consecutive even integers whose sum is 8. ) The length of a rectangle is feet more than its width. If its perimeter is 0 feet, find the length & width. ) The second angle in a triangle is less than twice the first angle. The third angle measure 8 more than twice the first angle. Find each angle. ) Jeffer has grades of 9 and 8 on the st tests of the quarter. Progress reports go home after the rd test. If Jeffer does not have an A average on his progress report, he cannot go to the football game that week. Jeffer will have to make at least what grade on the third test to be allowed to go to the football game?

5 ) Linear Equations: Slope, Writing Linear Equations, Horizontal & Vertical, Parallel & Perpendicular Find the slope (rate of change) of the following problems., 8,.,. Da Temperature. ( F ) The cost of museum tickets is $8 for four people and $78 for 0 people. What is the cost per person? Write the equation of the line in slope-intercept form & standard form given the following.., m 6., 7 m 7., 6, 8. m =, b = 9. (, -) m = 0 0. (, -) m = undefined.. The cost for 7 dance lessons is $8. The cost for lessons is $. Write a linear equation to find the total cost C for L dance lessons. Then use the equation to find the cost of lessons.

6 . Write the equation for a vertical line that goes through the point (, ).. Write the equation for a horizontal line that goes through the point (-, ). Solve the equation for if necessar, and find the slope. Then, find the slope of a line parallel and perpendicular to the original line. SLOPE PARALLEL PERPENDICULAR. + 6 = 8 6. = 7. = - Write the answers for 8-0 in slope intercept and standard forms. 8. Write the equation for the line parallel to the given line = 9 and and through the point (, ). 9. Write the equation for the line perpendicular to the given line = 9 and through the point (8, ). 0. Given a line through (, ) and (8, ), find the equation of the line perpendicular to that line through the midpoint of those points.

7 ) Graphing Linear Equations and Inequalities DO NOT USE A CALCULATOR Graph each of the following lines. ) slope:,, through, ) slope:, through ) slope:, -intercept: ) slope:, -intercept: ) 6) 7) 6 8) 9) 7 0) ) 6 ) 0 6

8 ) ) 7 ) 6) 7) no slope; through (, ) 8) slope: 0 through (, 7) Write each of the following in slope-intercept form: ( m b) 9) A computer technician charges $7 for a consultation plus $ per hour. 0) The population of Pine Bluff is 679 and is decreasing at the rate of 7 per ear. ) A video store charges $0 for a rental card plus a $ per rental. Graph each inequalit. Remember to use either a solid or dotted line, then SHADE. ) < ) > ) < 7

9 8 6) Sstems of Equations IF THE LINES INTERSECT ONCE, ANSWER IS THE ORDERED PAIR. IF THE LINES DO NOT INTERSECT (PARALLEL), ANSWER IS. IF THE LINES ALWAYS TOUCH (ARE THE SAME LINE), ANSWER IS INFINITLY MANY. Solve b graphing. ) 6 ) 0 ) 8 ) ) 6) 8 6 9

10 9 Solve using Substitution or Elimination. 7) 8) 9) 0) ) 7 ) 6 ) 6 ) 0

11 7) Laws of Eponents (negative eponents should alwas be simplified) (Remember: a b ab, a b ab ( ), a b ab, 0 and a a ) a b a 7 b ) z z ) a b b c a c ) ) 0 ) c c 7) ac bc ab 6) 8) a b 9) 6 0 a b 0 0) ) ) 0 () ) a b ab 7 8 8a b ) 0 a b ) ) 0 7) 7 8) 9) 6 0) 6 ) a 9 8 ) 7 ) ) ( 0 ) 0

12 8) Polnomial Operations Simplif. ) (6 ) ) ab ab a b ) 6 ( 7 ) ) 8 ) 6) 6 7) 8) n 8n 8 9) 6w w 7w 0) 8 ) m mp 6 p m mp p

13 9) Radical & Rational Eponents Simplif a a a 9. z z... Rational Eponents See these as eamples of rational eponents: DO NOT USE A CALCULATOR = = 8 = 8 Simplif the following Write as a rational eponent:

14 0) Using Function Notation State whether each set is a function. Answer es or no. Find the domain and the range. ) {(, ), (, 6), (, -6), (, 8)} Domain: Range: ) {(, -), (8, -), (-, 8), (-, )} Domain: Range: Use the vertical line test to determine whether each graph is the graph of a function. Answer es or no. ) ) ) 6) f g Use and 7) f 8) to find each value. g 7 9) f f 8 0) ) f c ) gw 7 ) f m ) g ) The temperature of the atmosphere decreases about o F for ever 000 feet increase in altitude. Thus, if the temperature at ground level is 77 o F, the temperature t at a given altitude is found b using the equation t 77.00h, where h is the height in feet. a) Write the equation in function notation where t is a function of h [f() is meant as f is a function of ]. b) Find t(00) and eplain its meaning in this problem. g 6) The function 60. models the weight gain of a basketball plaer as he starts a workout program where g is the weight in pounds after weeks. a) Eplain the meaning of 60 in the contet of this problem. b) Eplain the meaning of. in the contet of this problem. c) Evaluate g 6 and eplain its meaning.

15 ) Solving Quadratic Equations Should be able to use: Square Root Propert Factoring (Remember when equation must be set = 0 before solving) Completing the Square [ a + b + c = 0 ] Quadratic Formula Solve using Square Root Propert: (best when onl term is present or polnomial is squared) ) ) ( ) 9 7 Solve using Factoring: (best when equation can be easil factored; equation must be set = 0 to begin) ) ( )( ) 9 ) ( 8)( ) 0 ) ( ) 7 6) ( ) Solve using Complete the Square: (best when b is even but a must be = before using) 7) 7 0 8) 6 6 Solve using Quadratic Formula: (this method alwas works but is time consuming; set equation =0) b b ac 9) 6 0 0) 0 a

16 ) Using Midpoint & Distance Formulas Midpoint Formula: ( +, + ) Find the midpoint of the segments with endpoints at the given coordinates. ), and 0, ), 8 and 0,6 ), 0 and 0, 0 ) 9,.7 and,. Distance Formula: d = ( ) + ( ) Find the distance between each pair of points with the given coordinates. Simplif radicals. ),7 and, 6), and, 7), and 6,9 8) 0, and,

17 ) Proving Congruent Triangles 6

18 ) Transformations f ( ) a h k DO NOT USE A CALCULATOR How does the graph of change to produce each of the following graphs? Write the equation of the absolute value functions with the following shifts: 0. Up 7 and left 0.. Down, Reflects over -ais, Vertical shrink of.. Right 6, Vertical stretch of. 7

19 ANSWER KEYS 8

20 ) Vocabular/Know the difference Review KEY. Proportions vs. multipling fractions (cross multipling for a proportion vs. multipling numerators and multipling denominators when multipling fractions) Proportion 8 = 8 = vs.. * = vs. + = Multipling Fractions 8 (8) 6 () 6. order of operations: don t be tricked b these common mistaken identities 9 a. * vs. ( * ) b. + ( + ) c. -8 vs. (-8) * 8 = 0 0 = not 8( + ) - * 6 = 6 (-8)(-8) = 6. ( 9) vs. ( + 9) vs. ( 9)( + 9) a. ( 9) = = b. ( + 9) = = c. ( 9)( + 9) = = 8. know the difference between a term, epression, equation, and inequalit term number, variable, or product of numbers and variables (e:,, or ) epression terms with mathematical smbols (e:,, +, ) equation epressions set equal to one another (e: =, + = ) inequalit epressions not equal to one another (e: >, + ) 6. know the difference between solve, evaluate and simplif simplif to rewrite an epression in simplest form possible where nothing else can be performed (includes no parentheses or negative eponents; all fractions have been reduced) evaluate to find the value of (once the value has been found, the final result should be written in simplest form) solve to work out the solution to the problem 7. know the difference between rational and irrational rational real number that can be written as a fraction (e:0,,,, 0.) [repeating or terminating decimals] irrational real number that can t be written as a fraction [nonrepeating, nonterminating decimals] (e:,, ) 8. know the smbols b name: > as greater than vs. < as less than 9

21 9. know coefficient numerical factor of a monomial [number being multiplied b a variable] 0. know factor two or more numbers that multipl to produce another number [ and are factors of 0]. reduce entire factors not parts of a factor (individual terms) - ou can reduce 0 = 0 b rewriting the numerator in factored form first: - ou can t reduce 0 b tring to reduce just 0 0

22 ) Operations with Fractions DO NOT USE A CALCULATOR KEY ) 9 = 0 9 ) 7 = 9 0 ) 6 = ) 9 = 8 7 ) 7 = 7 6) = 6 8 7) = 9 8) 7 = 6 9) = 6 9

23 ) Solve Equations and Inequalities DO NOT USE A CALCULATOR KEY Clear out fractions. 7 ) 7 ) 7 ) a a a 8 6 = 8 = 7 a = / Variables on both sides. ) ( 8 ) + ( + ) = ) 8 + ( a ) = ( a + ) 6) 6 + = a 7 Solve Inequalities and graph our answer. When ou multipl or divide both sides b a negative remember to flip the inequalit. Get variable on the left. 8) 6 6 7) Word Problems - You must be able to write the equation or inequalit first. Then solve for the variable. 9) The greater of two numbers is 6 more than times the smaller. Their sum is. Find the numbers. 6 7 and,,6 0) Find three consecutive integers whose sum is 0. 0, 6, 8 ) Find three consecutive even integers whose sum is 8. 8 ) The length of a rectangle is feet more than its width. If its perimeter is 0 feet, find the length and width. 0 w w Length: Width: 9 ) The second angle in a triangle is less than twice the first angle. The third angle measure 8 more than twice the first angle. Find each angle. 8 80, 67, 78 ) Jeffer has grades of 9 and 8 on the first two tests of the quarter. Progress reports will go home after the third test. If Jeffer does not have an A average on his progress report, he cannot go to the football game that week. Jeffer will have to make at least what grade on the third test to be allowed to go to the football game?

24 ) Linear Equations - Slope, Writing Linear Equations, Horizontal & Vertical, Parallel & Perpendicular KEY Find the slope (rate of change) of the following problems., 8,., 7 8. Da Temperature. ( F ) The cost of museum tickets is $8 for four people and $78 for 0 people. What is the cost per person? $ per person Write the equation of the line in slope-intercept form & standard form given the following.., m 6., 7 m 7., 6, 8. m =, b = (, -) m = 0 0. (, -) m = undefined.. The cost for 7 dance lessons is $8. The cost for lessons is $. Write a linear equation to find the total cost C for L dance lessons. Then use the equation to find the cost of lessons. C 0L $

25 . Write the equation for a vertical line that goes through the point (, ). =. Write the equation for a horizontal line that goes through the point (-, ). = Solve the equation for if necessar, and find the slope. Then, find the slope of a line parallel and perpendicular to the original line. SLOPE PARALLEL PERPENDICULAR. + 6 = 8 6. = undefined/none undefined/none 0 7. = undefined/none Write the answers for 8-0 in slope intercept and standard forms. 8. Write the equation for the line parallel to the given line = 9 and through the point (, ). 9. Write the equation for the line perpendicular to the given line = 9 and through the point (8, ). 0. Given a line through (, ) and (8, ), find the equation of the line perpendicular to that line through the midpoint of those points. 9 9

26 ) Graphing Linear Equations and Inequalities DO NOT USE A CALCULATOR KEY

27 9) 7 0) ) 0 ) ) ) 6

28 7 6) Sstems of Equations KEY ) 6 ) 0,, ) 8 ), 7, ) 6) infinitel man

29 7) 8) KEY,, 9) 0), ) 7 ) 6,, ) 6 ) 0 infinitel man 0, 8

30 7) Laws of Eponents KEY ) a b a 7 b ) z z ) a b b c a c ) 0 a b 6 z a 6 b c 6 ) c c 7) ac bc ab 6) 8) 0 8c 7 0 a b c 0 a b 8 0 a b 9) 6 0 0) ) ) 0 () 8b a ) a b ab 7 8 8a b ) 0 a b ) ) 0 a b 7 a b ) 7 8) 9) 6 0) ) a 9 8 ) 7 ) a 6 ) ( 6 0 ) 9

31 8) Polnomial Operations KEY Simplif. ) (6 ) ) ab ab a b ) 6 ( 7 ) 0 0 a b 0a b 6 6 ) 8 ) 6) 6 0 7) 8) n 8n 8 9) 6w w 7w n 7 8w 6w 0 0) 8 ) m mp 6 p m mp p 8 8m 7mp 7 p 0

32 9) Radicals & Rational Eponents KEY i i i a a a a 9. z z 0a 6a z 6 0 z... 6 ( ) Rational Eponents See these as eamples of rational eponents: = = 8 = 8 Simplif the following Write as a rational eponent:

33 0) Using Function Notation KEY State whether each set is a function. Answer es or no. Find the domain and the range. ) {(, ), (, 6), (, -6), (, 8)} no Domain:,, 6,,6,8 Range: ) {(, -), (8, -), (-, 8), (-, )} es Domain:,,,8 Range:,,,8 Use the vertical line test to determine whether each graph is the graph of a function. Answer es or no. ) no ) es ) es 6) no f g Use and 7) f 6 8) to find each value. g 7 9 9) f 9 f 8 0) 0 ) f c ) gw 7 ) f m ) g 9c 9 w m m ) The temperature of the atmosphere decreases about o F for ever 000 feet increase in altitude. Thus, if the temperature at ground level is 77 o F, the temperature t at a given altitude is found b using the equation t 77.00h, where h is the height in feet. a) Write the equation in function notation where t is a function of h. [f() is meant as f is a function of ] t( h) h 76. b) Find t(00) and eplain its meaning in this problem. 6) The function g 60. models the weight gain of a basketball plaer as he starts a workout program where g is the weight in pounds after weeks. a) Eplain the meaning of 60 in the contet of this problem. Starting/initial weight (-intercept) b) Eplain the meaning of. in the contet of this problem. # of pounds added per week (slope) c) Evaluate g 6 and eplain its meaning. 69; weight 6 weeks after starting

34 ) Solving Quadratic Equations Should be able to use: Square Root Propert KEY Factoring (Remember when equation must be set = 0 before solving) Completing the Square [ a + b + c = 0 ] Quadratic Formula Solve using Square Root Propert: (best when onl term is present or polnomial is squared) ) ) ( ) 9 7 =,, Solve using Factoring: (best when equation can be easil factored; equation must be set = 0 to begin) ) ( )( ) 9 ) ( 8)( ) 0 6,, ) ( ) 7 6) ( ) 9, 8 9, Solve using Complete the Square: (best when b is even but a must be = before using) 7) 7 0 8) 6 6 i i Solve using Quadratic Formula: (this method alwas works but is time consuming; set equation =0) b b ac 9) 6 0 0) 0 a i 8 7

35 ) Using Midpoint & Distance Formulas KEY Midpoint Formula: ( +, + ) Find the midpoint of the segments with endpoints at the given coordinates. ), and 0, ), 8 and 0,6 (7, 8) (6, ) ), 0 and 0, 0 ) 9,.7 and,. (, ) ( 0, ) Distance Formula: d = ( ) + ( ) Find the distance between each pair of points with the given coordinates. Simplif radicals. ),7 and, 6), and, 0 7), and 6,9 8) 0, and,

36 ) Proving Congruent Triangles KEY

37 ) Transformations f ( ) a h k DO NOT USE A CALCULATOR KEY How does the graph of change to produce each of the following graphs?.. up.. left. 9. right 9.. reflect over -ais; vertical stretch _.. down vertical shrink left ; down vertical stretch; right _ reflect over -ais; vertical shrink; up Write the equation of the absolute value functions with the following shifts: f ( ) 0. Up 7 and left Down, Reflects over -ais, Vertical shrink of. _ f ( ). Right 6, Vertical stretch of. f ( ) 6 6