Quadratic equations Questions and Answers

8 A student is determining if two functions are inverses of one another The student completes the following computation and concludes that yes the functions are inverses Is the student correct f x 3x 5 g x 5 f g x 3 35 5 x g f x 3x 5 5 3 x A The student is correct because f x is found by adding 5 and multiplying by 3 B The student is correct because the function composition f g x x and the function composition g f x x C The student is incorrect because the functions are not opposites of each other D The student is incorrect because the function composition f g x x and the function composition g f x x proves nothing

5 Matthew claims that the function g x x 2 5 is the inverse of the function of f x 2x 5 Which statement provides a correct analysis of Matthew s claim A Matthew s claim is correct because if you add 5 to g x and then multiply it by 2 you get f x B Matthew s claim is correct because when g x is substituted into every occurrence of x in f x f g x x and when f x is substituted into every occurrence of x in g x g f x x C Matthew s claim is incorrect because when g x is substituted into every occurrence of x in f x f g x x 5 therefore g x is not the inverse of f x D Matthew s claim is incorrect because when g x is substituted into every occurrence of x in f x f g x x therefore g x is not the inverse of f x

1 3 ut a ftware System by Elimination Solve each system by elimination 1 0 6x 8 10y 10y 19 3x 0 A 3 1 C 6 1 3 1 3 y 1 x 4 x 28 2y A 8 10 C 8 10 5 x 15 9 y A 7 10 C 4 10 2 2 044 8x 36y 7 8x 4y 4 9 12y 3x A 8 1 C 1 1 B 3 1 D No solution 9 1 5x 2y 10y 8x 46 A 3 7 C 1 7 B 10 8 D 8 9 B No solution D 7 10 B 1 1 D 1 1 B 3 7 D 3 7 All rights reserved 2 2 x 5 4 4 4 9 x y 5 5 6 8 9 y 0 44 0 2x 8y 4 1 y x 0 2 10 A 1 2 B 1 1 C 10 2 D Infinite number of solutions A 2 2 C 6 2 6 20 9y 5x 6x24 18y 0 A 0 6 C 4 0 8 9y 7x18 6x 9 7y A 9 9 C 9 9 B 2 2 D 4 2 A 2 3 C 2 9 B 0 8 D 0 8 B No solution D 9 9 10 24 9x 2y 0 5y 29 7x B 9 2 D 2 9

1 7 ut a System by Elimination Solve each system by elimination 1 0 6x 8 10y 10y 19 3x 0 A 3 1 C 6 1 1 3 3 3 1 4 x 28 2y A 8 10 C 8 10 5 x A 7 10 C 4 10 15 9 2 2 0 44 8x 36y 7 8x 4y 4 9 12y 3x ftware LLC All rights reserved A 8 1 C 1 1 9 1 5x 2y 10y 8x 46 A 3 7 C 1 7 B 3 1 D No solution B 10 8 D 8 9 B No solution D 7 10 B 1 1 D 1 1 B 3 7 D 3 7 2 6 8 5 4 4 4 9 x y 5 5 A 1 2 B 1 1 10 9 y 0 2 x 44 0 2x 8y 4 1 1 y x 0 2 C 10 2 D Infinite number of solutions A 2 2 C 6 2 6 20 9y 5x 6x24 18y 0 A 0 6 C 4 0 8 9y 7x18 6x 9 7y A 9 9 C 9 9 B 2 2 D 4 2 A 2 3 C 2 9 B 0 8 D 0 8 B No solution D 9 9 10 24 9x 2y 0 5y 29 7x B 9 2 D 2 9

1 3 ut a ftware System by Elimination Solve each system by elimination 1 0 6x 8 10y 10y 19 3x 0 A 3 1 C 6 1 3 1 3 y 1 x 4 x 28 2y A 8 10 C 8 10 5 x 15 9 y A 7 10 C 4 10 2 2 044 8x 36y 7 8x 4y 4 9 12y 3x A 8 1 C 1 1 B 3 1 D No solution 9 1 5x 2y 10y 8x 46 A 3 7 C 1 7 B 10 8 D 8 9 B No solution D 7 10 B 1 1 D 1 1 B 3 7 D 3 7 All rights reserved 2 2 x 5 4 4 4 9 x y 5 5 6 8 9 y 0 44 0 2x 8y 4 1 y x 0 2 10 A 1 2 B 1 1 C 10 2 D Infinite number of solutions A 2 2 C 6 2 6 20 9y 5x 6x24 18y 0 A 0 6 C 4 0 8 9y 7x18 6x 9 7y A 9 9 C 9 9 B 2 2 D 4 2 A 2 3 C 2 9 B 0 8 D 0 8 B No solution D 9 9 10 24 9x 2y 0 5y 29 7x B 9 2 D 2 9

1 3 1 3 7 ut a System by Elimination Solve each system by elimination 1 0 6x 8 10y 10y 19 3x 0 A 3 1 C 6 1 3 1 x 4 x 28 2y A 8 10 C 8 10 5 x A 7 10 C 4 10 15 9 2 2 0 44 8x 36y 7 8x 4y 4 9 12y 3x ftware LLC All rights reserved A 8 1 C 1 1 9 1 5x 2y 10y 8x 46 A 3 7 C 1 7 B 3 1 D No solution B 10 8 D 8 9 B No solution D 7 10 B 1 1 D 1 1 B 3 7 D 3 7 2 6 8 5 4 4 4 9 x y 5 5 A 1 2 B 1 1 10 9 y 0 2 x 44 0 2x 8y 4 1 1 y x 0 2 C 10 2 D Infinite number of solutions A 2 2 C 6 2 6 20 9y 5x 6x24 18y 0 A 0 6 C 4 0 8 9y 7x18 6x 9 7y A 9 9 C 9 9 B 2 2 D 4 2 A 2 3 C 2 9 B 0 8 D 0 8 B No solution D 9 9 10 24 9x 2y 0 5y 29 7x B 9 2 D 2 9

Which two triangles are congruent by the SSS Theorem Complete the congruence statement H A 111 G Z X Q

1 7 ut a System by Elimination Solve each system by elimination 1 0 6x 8 10y 10y 19 3x 0 A 3 1 C 6 1 3 1 3 3 1 x 4 x 28 2y A 8 10 C 8 10 5 x A 7 10 C 4 10 15 9 2 2 0 44 8x 36y 7 8x 4y 4 9 12y 3x ftware LLC All rights reserved A 8 1 C 1 1 9 1 5x 2y 10y 8x 46 A 3 7 C 1 7 B 3 1 D No solution B 10 8 D 8 9 B No solution D 7 10 B 1 1 D 1 1 B 3 7 D 3 7 2 6 8 5 4 4 4 9 x y 5 5 A 1 2 B 1 1 10 9 y 0 2 x 44 0 2x 8y 4 1 y x 0 2 C 10 2 D Infinite number of solutions A 2 2 C 6 2 6 20 9y 5x 6x24 18y 0 A 0 6 C 4 0 8 9y 7x18 6x 9 7y A 9 9 C 9 9 B 2 2 D 4 2 A 2 3 C 2 9 B 0 8 D 0 8 B No solution D 9 9 10 24 9x 2y 0 5y 29 7x B 9 2 D 2 9

C E G 3x 16 H 2x 50 F

Find the measure of ZN Ko L 61 x M P 75 3x N

In Exercises 1 and 2 complete the proof Example 1 1 Given PQ RS Prove PR QS P Q 2 Given Z1 is a complement of 22 22 23 R Prove 1 is a complement of 23 X In Exercises 3 6 name the property that the statement illustrates Example 2 3 If PQ ST and ST UV then PQ UV 4 LF F 5 If XY UV then UV XY 6 If ZL ZM and ZM ZN then ZL ZN PROVING A THEOREM In Exercises 7 9 write a two column proof for the property Example 3 7 Reflexive Property of Segment Congruence 8 Transitive Property of Angle Congruence 9 Transitive Property of Segment Congruence Because MNLQ and LQPN then MNPN by the Reflexive Property of Segment Congruence STATEMENTS 1 PQ RS 2 PQ QR RS QR 3 4 RS QR QS 5 PR QS 10 ERROR ANALYSIS In the diagram MN LQ and LQPN Describe and correct the error in the reasoning L Q STATEMENTS 1 Z1 is a complement of 22 2 22 23 3 mZ1 m2 90 4 m 2 m23 5 6 Z1 is a complement of 23 M REASONS PN 1 2 3 Segment Addition Postulate 4 Segment Addition Postulate 5 E REASONS 1 Given 2 PROOF In Exercises 11 and 12 write a two column proof Example 4 11 Given ZGFH ZGHF 3 4 Definition of congruent angles 5 Substitution Property of Equality 6 Prove ZEFG and ZGHF are supplementary F D GO DIGITAL 12 Given ABFG BF bisects AC and DG Prove BCDF 13 MAKING AN ARGUMENT In the figure SRCE and ACQR Your friend claims that because of this CB AC by the Transitive Property of Segment Congruence Is your friend correct Explain your reasoning B Q R

Elizabeth brought a box of donuts to share There are two dozen 24 donuts in the box all identical in size shape and color Two are jelly filled 10 are lemon filled and 12 are custard filled You randomly select one donut eat it and select another donut Find the probability of selecting two lemon filled donuts in a row Type an integer or a simplified fraction

cos sin 0 cos e Rewrite sine and or cosine cos 0 sin e sin e cos 8 sin e cos 0 sin e cos e sin 0 over a common denominator Type your answer in terms www Simplify your answer

5 5 2x 2y 16 x 6 8 A 8 2 B 8 0 C Infinite number of solutions D 8 3 7 y 6 6x 4y 6 A 5 6 C 5 6 B 5 6 D 5 2 6 8 6 4x 3y 21 x 5y 12 A 3 5 G 3 3 8 81 5y 22 5x 3y 23 A 1 6 C 1 6 B 5 3 D 5 3 B 1 6 D 1 6

In bungee jumping a bungee jump is the amount the cord will stretch at the bottom of the fall The stiffness of the 2W S L where W is the weight of the jumper s cord is related to the amount of stretch by the equation K in pounds K is the cord s stiffness in pounds per foot L is the free length of the cord in feet and S is the stretch in feet Use this equation to answer parts a and b O B exactly 25 feet O C less than 26 feet O D more than 25 feet O E between 25 and 26 feet b If safety requirements will not permit the jumper to get any closer than 6 feet to the ground what is the minimum height required for the ledge in part a

Determine where the graph of f is on or below the graph of g by solving the inequality f x g x Graph f and g together 4 f x x 1 2 g x 8x 8 The graph of f is on or below the graph of g over Type your answer in interval notation Choose the graph which correctly shows f and g together O A Q M B g x Q 37 O C Q G D Q o Ly

5 10 4 4 7x x 6x 5 A 4 3 2 012 3 6 11 11 7 4 x 4 18 7x 10 8 6 2 0

10 9 X g x 4 3 J 4 10 2 A 7 6 5 7 7 4 A 2 4 4 S 6 7 4 10 12 ve 15 56 17 18 12 30 25 22 22 set 2 1 0 1 0 1 2 2 2 3 4 2 7 2 2 20 From the graph of f x and the table of g x find fog 2

Extra Credit Verify the identity 1 1 cos 0 1 1 cos 0 2 2 cot 0

You are dealt one card from a 52 card deck Find the probability that you are not dealt a red picture card The probability is Type an integer or a simplified fraction

552x 2y 16 x 6y 8 7 A 8 2 B 8 0 C Infinite number of solutions D 8 3 7 y 6 6x 4y 6 A 5 6 C 5 6 B 5 6 D 5 2 6 8 6 4x 3y 21 x 5y 12 A 3 5 C 3 3 8 8x 5y 22 5x 3y 23 A 1 6 C 1 6 B 5 3 D 5 3 B 1 6 D 1 6

3 3 x 7y 10 3x 7y 2 A 2 4 C 4 2 B 4 2 D 4 2 4 4 3x 24y 3 x 8y 4 A No solution C 5 5 B 5 6 D 5 6

P 6 male children Type an integer or a simplified fraction Points 0 of 1 You select a family with three children If M represents a male child and F represents a female child the set of equally likely outcomes for the children s genders is MMM MMF MFM MFF FMM FMF FFM FFF Find the probability of selecting a family with 6 male children Save

3 If there are 20 students who are late to school on Monday 15 students who are late to school on Tuesday and 25 who are late to school on Wednesday what is the percent increase in tardiness from Tuesday to Wednesday

2 If there are 20 students who are late to school on Monday 15 students who are late to school on Tuesday and 25 who are late to school on Wednesday what percent of lateness oc curred on Tuesday

A 10 sided die is rolled Find the probability of rolling an odd number The set of equally likely outcomes is shown below 1 2 3 4 5 6 7 8 9 10 The probability of rolling an odd number on a 10 sided die is Type an integer or a simplified fraction

A 12 sided die is rolled The set of equally likely outcomes is 1 2 3 4 5 6 7 8 9 10 11 12 Find the probability of rolling a 7 The probability of rolling a 7 is Type an integer or a simplified fraction

Use the graph of the function f to solve the inequality a f x 0 b f x 0 X x 2 Q G C a The solution set for f x 0 is 1 0 U 2 00 Type your answer in interval notation b The solution set for f x 0 is Type your answer in interval notation

Solve the inequality R x 0 where R x using the graph of the function X 4 0 Q x 3 by x x 4 The solution set for R x 0 is Type your answer in interval notation

Solve the inequality f x 0 where f x x x 2 by using the graph of the function N Q The solution set for f x 0 is Type your answer in interval notation

Use the graph of the function f to solve the inequality a f x 0 b f x 0 X x 2 a The solution set for f x 0 is Type your answer in interval notation

Determine whether the following statement makes sense or does not make sense and explain the reasoning 1 I find it easier to multiply and than to add them 5 3 4 Choose the correct answer below SOA 3 The statement does not make sense Adding and only requires adding the numerators and denominators Multiplying requires rewriting the numbers with the least common denominator 1 5 4 O B The statement does not make sense The fractions cannot be added because they have different denominators 1 3 OC The statement makes sense Multiplying and only requires multiplying the numerators and denominators Adding requires rewriting the fractions with the least common denominator 5 4 OD The statement makes sense The fractions cannot be added because they have different denominators

Analyze the graph of the function 4 2 R x X X 8 2 x 49 a What is the domain of R x O A xlx 7 and x 7 O B x x 0 O C xlx 0 and x 7 and x 7 O D All real numbers b What is the equation of the vertical asymptote s of R x Select the correct choice below and fill in any answer boxe A x 7 7 Use a comma to separate answers as needed Type an integer or a fraction B There is no vertical asymptote

2x 7x 3 x 9 defining the rational function R using the following piecewise defined function Use this approach as a model to redefine the rational functions in parts a and b The graph of the rational function R x R x 2x 7x 3 x 9 5 6 R x if x 3 if x 3 a Redefine the function R x x 3x 18 x 6 x 3x 18 X X 6 3 37 Therefore the graph of R is discontinuous at has a hole at the point C so that the discontinuity at x 3 is removed This discontinuity could be removed

Simplify the complex rational expression using Method I or Method II x 10 x 10 x 13 X 5 1 5 X 5 X 10 x 10 x 13 X 5 5 X 5 1

Simplify the complex rational expression using Method II X 1 X X 6 X 4 5 x 1 X 6 1 X x 4 5 X 6

Let f x 3x 4 and g x 2x 7 Find f g x f g x fg x and 6 f g x Simplify your answer x Give the domain of each

If f x graph g x f x 3 5 Show ALL Intermediate graph s X Show your vertical and horizontal asymptotes utr 13 2 stat bet3 2 3 Domain Range 9 2 2413 15 K Complete the Chart for f x F x 18 10 1 XER 2 4 YEP Y 1 expe x 3 5 2 De Complete the Chart for g x f x 3 5 XER 2 2 1YER 38221 Domain Range Vert Asymptote

le write an inequality to represent the scenerio and solve Write a complete sentence to describe your solution 1 Keith has 500 in a savings account at the beginning of the summer He wants to have at least 200 at the end of the summer He withdraws 25 per week for food clothing and movie tickets How many weeks can Keith withdraw money from his account 2 A taxi charges a flat rate of 1 75 plus an additional 0 65 per mile If Erica has at most 10 to spend on the cab ride how far could she travel 2

Which stem and leaf plot shows a data set WILI median D Press to hear a reminder or hint for this problem r 8 1 2 17 3 3 3 3 39 408 216 26 0 00 0 08 Speed 1x 889 2 2 3 5 1433 1234 Because each data set has 10 values you can write the data sets one on top of the other and find all three medians at once 1892 216 26 100 Paused 48 11 2 2 4 5 2246 26 26

Bonus Give an example of a function with domain 0 1 and range consists of 1 1 and 2 3

What is the solution to the compound inequality in interval notation 4 x 1 4 or 2x 4 10 O 3 or 2 0 O 3 or 2 O 3 2 O 2 or 3

Solve for n 3 n 6 3n 8 O no solution all real numbers

Find each quotient 1 1 2 3 4 15 7 5 25 10 13 0 25 4 00 w 3 14 1 12 2 6 5 24 3 2 8 3 3 Fill in the blank with a number to make a true statement 1 0 25 14 2 4 2 11 2 4 3 4 7 3 5 3 6 10 10 6 0 125 0 5 6 9 Write a division problem for each situation Then solve it 16 How many quarter pound 4 packets of plant food can a garden shop make out of 8 pounds of the plant food 12 34 5100 8 1 15 7 3 17 The assembly of a machine takes 4 hour There are twelve steps in the assembly process What is the average time for each step 3 18 A 35 meter length of cable is cut into pieces that measure 1 25 meters each Into how many pieces is the cable cut 4 2 19 8 tons of gravel is spread evenly across 6 acres How many tons of gravel are on each acre 5 16

The light vehicle market share in percent in a particular country for pickup trucks is shown in the graph Let x 0 represent 1995 x 4 represent 1999 and so on a Use the points on the graph to write equations for the line segments in the intervals 0 4 and 4 8 b Define this graph as a piecewise defined function f x Percent Ay Pickup Truck Market Share 45 43 0 42 4 41 39 37 35 33 31 1995 1997 4 39 7 8 32 1 1999 2001 2003 Year a The equation for the line segment in the interval 0 4 is y Simplify your answer Use integers or decimals for any numbers in the expres

Graph the following function f x 3 x 5 2 Select all that apply A Vertical stretch or shrink B Horizontal stretch or shrink C Reflect across y axis D Reflect across x axis E Vertical translation up F Horizontal translation to the right G Horizontal translation to the left H Vertical translation down Use the graphing tool to graph the function Click to enlarge graph

Use the piecewise defined function to find the following values for f x f x 5 4x 3x 5x 3 f 2 13 f 1 if x 1 if 1 x 9 if x 9

For the given equation and graph of the quadratic function do the following Complete parts a through e f x x 4 4 The domain is Type your answer in interval notation The range is Type your answer in interval notation b Give coordinates of the vertex The vertex is Type an ordered pair c Give the equation of the axis of symmetry The axis of symmetry is x d Find the y intercept s The y intercept s is are Type an ordered pair Use comma to separate answers as needed e Find the x intercept s The x intercept s is are Type an ordered pair Use a comma to separate answers as needed b 10 b

M8 n 1 n4 Verges or diverges Give a reason for your answer Does the series converge or diverge 1 OA The Direct Comparison Test with shows that the series converges n OB The Direct Comparison Test with Oc The Direct Comparison Test with n 1 80 1 n 1 n 1 40 n 1 shows that the series converges shows that the series diverges 1 OD The Direct Comparison Test with E shows that the series diverges n

Solve the inequality Write the solution set in interval notation t2 8t 16 20 Select one a 00 00 b 4 U 4 c 0 4 U 4 d 4 U 4 s