Quadratic equations Questions and Answers

Use properties of logarithms to expand the logarithmic expression as much as possible Where possible evaluate logarithmic expressions without using a calculator 1083 81 4log 3 log x 1 log381 log x 1 4 log x 1 4 log x 1 4log33 log3 x 1

Show ALL work in order to receive any credit Prove the following identity csc 0 cos 0 cot 0 sin 0

The bar graph shows the average annual spending per person on selected items in 2010 The combined spending on housing yehicles gas and health care was 13 762 The difference between spending on housing and spending on vehicles gas was 3870 The difference between spending on housing and spending on health care was 593 Find the average per person spending on housing vehicles gas and health care in 2010 Find the average per person spending on housing in 2010 Annual Spending per Person Adjusted fo Annual spending per person S 8000Housing 7000 P 6000 5000 4000 3000 2000 1000 0 Vehicles 2010 Health care V

The population of a certain country from 1970 through 2007 is shown in the table to the right Use your graphing utility s logarithmic regression option to obtain a model of the form y a b In x that fits the data How well does the correlation coefficient r indicate that the model fits the data Obtain a model of the form y a binx that fits the data y Round to three decimal places as needed GIIB x Number of Years after 1969 1 1970 11 1980 21 1990 31 2000 38 2007 y Population millions 194 3 218 2 235 7 290 2 350 8

The days when something happens and it is not captured on camera appear to be over The bar graph shows that by 2007 68 of cellphones sold in a country were equipped with a camera The data can be modeled by the function f x 8 36 In x where f x is the percentage of new cellphones with cameras x years after 2002 A Underestimates B Overestimates GILB By how much 80 70 60 50 40 Percentage of New Cellphones with Cameras 50 34 63 68 II 8 2003 2004 2005 2006 2007 a Use the function to determine the percentage of new cellphones with cameras in 2005 Does this overestimate or underestimate the percent displayed by the graph By how much Determine the percentage of new cellphones with cameras in 2005 using the given function 48 Round to the nearest percent as needed Does this overestimate or underestimate the percent displayed by the graph

y exponential equation by taking the natural logarithm on both sides Express the solution in terms of natural logarithms Then use a calculator to obtain a decimal approximation for the solution 2e3x 1556 What is the solution in terms of natural logarithms XX The solution set is In 778 Use a comma to separate answers as needed Simplify your answer Use integers or fractions for any numbers in the expression What is the decimal approximation for the solution A The solution set is 1x 2 21 Use a comma to separate answers as needed Round to two decimal places as needed

Solve the exponential equation Express the solution in terms of natural logarithms Then use a calculator to obtain a decimal approximation for the solution ex 13 9 w What is the solution in terms of natural logarithms The solution set is Use a comma to separate answers as needed Simplify your answer Use integers or decimals for any numbers in the expression

appear to be over The bar graph shows that by 2007 68 of cellphones sold in a country were equipped with a camera The data can be modeled by the function f x 8 36 In x where f x is the percentage of new cellphones with 90 cameras x years after 2002 50 80 70 OA Underestimates OB Overestimates 40 30 20 10 0 centage of New Cellphones with Cameras 34 50 63 68 8 2003 2004 2005 2006 2007 a Use the function to determine the percentage of new cellphones with cameras in 2005 Does this overestimate or underestimate the percent displayed by the graph By how much Determine the percentage of new cellphones with cameras in 2005 using the given function 48 Round to the nearest percent as needed Does this overestimate or underestimate the percent displayed by the graph

The function P t 145e 0 092t models a runner s pulse P t in beats per minute t minutes after a race where Ost 15 Graph the function using a graphing utility TRACE along the graph and determine after how many minutes the runner s pulse will be 77 beats per minute Verify your observation algebraically Choose the correct graph below All graphs are 0 15 1 by 0 150 10 A B L Q ON OC The runner s pulse will be 77 beats per minute after about minutes Round to the nearest tenth of a minute as needed Q Q G OD U

1 point The matrix A 2 1 1 det 2 1 4 2 2 2 1 14 2 2 4 2 1 1 a a 1 a 4 2 b 0 which is the system 2 has characteristic equation The eigenvalue is For this eigenvalue the eigenvector v has components a and b that satisfy 22 a 1 b 0 0 and has multiplicity k b 0 b 0 Choosing a 1 for the eigenvector is v

eigenvectors In Exercises 1 6 a matrix A and one of its are given Find the eigenvalue of A for the given eigenvector 1 A x 2 A X 3 A X 4 A 7 4 3 1 3 9 19 48 2 1 3 6 1 2 11 6 12 8 5 6 15 2 4 19 8 22 14 6 15

b 0 1 68 0 0 0 1 The corresponding The corresponding The solution set is system has no solution system has one solution 0 The corresponding system has infinitely many solution The solution set is z is any real number er

Perform the elementary row operation R R on the given matrix 1531 6 I 4 341 3 I 2 5 10 1 Resulting matrix X

Graph the line y 5x 3 8 X

2 Sketch a graph of a fifth degree polynomial f x such that f x 0 when x 4 4 x 2 and 2 x 5 Explain how you came up with the solution y

5 For each of the following a use the leading terms to find a simpler function with the same end behaviour an b determine the equation of the horizontal asymptote of each function if it exists 3x4 x 12 i w x x6 2 a a ii k x iii d x 5x8 3x4 x 5 17x 12x 1 x6 3x 5 b b

Use the exponential growth model A Apekt to find the time it takes a population to triple to grow from A to 3A t a Simplify your answer FFF

In 1964 paleontologists discovered the bones of a new species of dinosaur The age of the dinosaur was estimated using potassium 40 dating of rocks surrounding the bones Analysis of these rocks indicated that 70 8 of th amount of the potassium 40 was still present The decay model for potassium 40 is A A e e 0 52912t where t is in billions of years Let A 0 708A in this decay model and estimate the age of the bones of the dinosaur The bones are approximately billion years old Type an integer or decimal rounded to four decimal places as needed CCCCS

transformations of the graph of f x 5 raph the function g x given below Graph d give the equation of the asymptote Use graph to determine the domain and range g x g x 5x 3 Graph the function g x 5x 3 and its asymptote Graph the asymptote as a dashed line Use the graphing tool to graph the function Click to enlarge graph 10

Given that the polynomial f x has degree 9 which of the following most accurately describes the number of turning points of f x Select the correct answer below The graph of f x has at most 10 turning points The graph of f x has at most 9 turning points has at most 8 turning points The graph of f x has at least 8 turning points The graph of f x has at least 10 turning points The graph of f x has at least 9 turning points The graph of f x

A Zookeeper has a walrus with a nutritional deficiency She wants to make sure the walrus consumes at least 2100 mg of iron 2700 mg of vitamin B 1 and 1400 mg of vitamin B 2 One Maxivite pill contains 500 mg of iron 300 mg of vitamin B 1 and 100 mg of vitamin B 2 and costs 0 07 One Healthovite pill provides 100 mg of iron 150 mg of vitamin B 1 and 175 mg of vitamin B 2 and costs 0 08 Complete parts a c below a What combination of Maxivite and Healthovite pills will meet the walrus s requirement at lowest cost What is the lowest cost Set up the linear programming problem The best combination of pills that meets the walrus s nutritional requirements is Maxivite pill s and Healthovite pill s at the lowest cost of b In the solution for part a does the walrus receive more than the minimum amount he needs of any vitamin If so which vitamin is it c Is there a way for the zookeeper to avoid having the walrus receive more than the minimum needed still meet the other constraints and minimize the cost Explain

x y A x y 11 X 2 1 1 y D x y 1 y 1 2 X B x yM y shane 1 E x y C x y

Number of integers in the solution set of a satisfying inequation 65 9 is Your Answer

For each of the following determine whether the statement is true or false and justify your answer a If U and V are subspaces of R and B is a basis for U and V then U b Let B be an ordered basis for R If R span i then R span i 2

Let y be the solution of IVP a e O b e c 2e d 2e 3y 3y y 0 y 0 1 y 0 0 y 0 1 Then y 1

A truck 3 00 m tall drives under a parabolic arch which is 5 00 m high and 8 00 m wide at road level Given the diagram below find a b What is the maximum horizontal distance at the base of the tunnel between the side of the truck and the wall What is the width of the arch at the level of the top of the truck

Write the equation in its equivalent exponential form Then solve for x log2 x 1 3 log 2 x 1 3 in its equivalent exponential form is Do not simplify

An amount of 37 000 is borrowed for 13 years at 3 5 interest compounded annually If the loan is paid in full at the end of that period how much must paid back Use the calculator provided and round your answer to the nearest dollar

Solve the compound inequality 4w 5 7 and 2w 4 10 Write the solution in interval notation If there is no solution enter 0

11 less than Janelle s height is 42 Use the variablej to represent Janelle s heigh 1 C

Use the properties of logarithms to rewrite the expression Simplify the result if possible Assume all variables represent positive real num log 09 5 10 Choose the correct answer OA 1 log 45 log410 log 3 OB This cannot be simplified O C log 45 log 10 log 43 OD log 45 log 10 log 43

3 Apply the Gram Schmidt orthonormalization process to transform the following basis into an orthonormal basis B 0 1 2 2 0 0 1 1 1

A ranger in tower A spots a fire at a direction of 355 A ranger in tower B located 50 mi at a direction of 49 from tower A spots the fire at a direction of 303 How far from tower A is the fire How far from tower B7 The fire is from tower A Round to the nearest whole number 355 499 30 mi

The diagram shows the cross section of a circular tunnel with diameter 3m The road AB is 2m wide A large truck is shown fitting exactly into the tunnel so that it touches at its top corners The truck is 1 2 m wide at the top Find PQ the height of the top of the truck from the road AB P n

a Use the formula for compound interest to write a function for the balance in each bank at any time t Write a function for the balance in the first bank at any time t Choose the correct answer below OA A 15 000 1 06 6t OC A 15 450 E A 15 000 1 03 2t Write a function for the balance in the second bank at any time t Choose the correct answer below OA A 15 000 0 996 OC A 15 060 OE A 15 000 1 008 2t Balance thousands 25 10 6t 0 b Use a graphing utility to graph both functions in an appropriate viewing rectangle Based on the graphs which bank offers the better return on your money Graph both functions Choose the correct graph below The graph for the first bank is blue and the graph for the second bank is pink O A OB O C 10 Q Q Balance thousands 25 10 0 10 GLULE Q B A 15 000 1 01 OD A 15 000 0 03 t OF A 15 000 0 97 6t B A 15 000 1 004 12t OD A 15 000 1 048 OF A 15 000 0 004 24t Balance thousands 25 10 0 NHH 10 Q 12t Q OD Balance thousands 25 10 0 10 Q Q G

Earth Horizon A satellite S in circular orbit around the earth is sighted by a tracking station 7 see the figure The distance TS is determined by radar to be 1 034 mi and the angle of elevation above the horizon is 32 4 Note that the radius and hon line form a right angle How high is the satellite above the earth at the time of the sighting The radius of the earth is r 3 964 mi mi

11 Determine if the following polynomials are reducible or irreducible over Z7 a x 3 5x 2 4x 1 b x 3 x 2 1 c x 4 x 2 2 6 points

Rewrite the following quadratic function in standard vertex form f x 7x 4x 6 Enter exact values and use improper fractions if necessary Provide your answer below

An injection contains 5 mg of medication in each milliliter mL If a physician prescribes a dose of 2 75 mg to be administered to a hospital patient three times daily how many milliliters of injection will be required over a 3 day period Round to three significant figures AV Question 10 Mandatory 1 point

If one dose of a cough syrup is 1 milliliters mL for a small child how many milliliters will be needed for 12 doses of the syrup N Question 4 Mandatory 1 point If a clinical study of a new drug demonstrated that the drug met the effectiveness criteria in 323 patients of the 471 patients enrolled in the study express these results as a percent Round answer to three significant figures A Question 5 Mandatory 1 point Round the following number to three significant figures

Use the trigonometric substitution to write the algebraic expression as a trigonometric function of 0 where 0 0 36x x 6 cos 0

Solve correct to 2 decimal places X 9 5 184 X

a Use the Leading Coefficient Test to determine the graph s end behavior O A The graph of f x rises to the left and falls to the right OB The graph of f x rises to the left and rises to the right OC The graph of f x falls to the left and rises to the right OD The graph of f x falls to the left and falls to the right iswer the parts b Find the x intercept s State whether the graph crosses the x axis or touches the x axis and turns around at each intercept The x intercept s is are Type an exact answer using radicals as needed Use a comma to separate answers as needed Type each answer only once At which x intercept s does the graph cross the x axis Select the correct choice below and if necessary fill in the answer box to complete your choice OA The x intercept s at which the graph crosses the x axis is are Type an exact answer using radicals as needed Use a comma to separate answers as needed Type each answer only once OB There are no x intercepts at which the graph crosses the x axis At which x intercept s does the graph touch the x axis and turn around Select the correct choice below and if necessary fill in the answer box to complete your choice OA The x intercept s at which the graph touches the x axis and turns around is are Type an exact answer using radicals as needed Use a comma to separate answers as needed Type each answer only once OB There are no x intercepts at which the graph touches the x axis and turns around c Find the y intercept The y intercept is Simplify your answer d Determine whether the graph has y axis symmetry origin symmetry or neither Choose the correct answer below OA The graph of f is symmetric about the y axis OB The graph of f is symmetric about the origin OC The graph of f is neither symmetric about the y axis nor symmetric about the origin e If necessary find a few additional points and graph the function Use the maximum number of turning points to check whether it is drawn correctly Choose the correct graph below O A Q 2 G C Ay 5 D Q 5

The graph to the right shows the heart rate of a person watching a short romantic film where x represents time in minutes and y represents heart rate in beats per minute Answer the following questions 80 70 60 50 40 0 A 4 a For which time periods was the person s heart rate increasing O A from 1 through 5 minutes and from 9 through 12 minutes OB from 5 through 15 minutes OC from 1 through 12 minutes OD from 5 through 9 minutes and from 12 through 15 minutes b For which time periods was the person s heart rate decreasing O A from 1 through 5 minutes and from 9 through 12 minutes OB from 1 through 12 minutes OC from 5 through 9 minutes and from 12 through 15 minutes O D from 5 through 15 minutes 12 7 c How many turning points from increasing to decreasing or from decreasing to increasing occurred for the person s heart rate during the 15 minute film d Suppose that a polynomial function is used to model the data displayed by the graph What is the degree of the polynomial function of best fit Use the number of turning points to determine the degree e For the model in part d should the leading coefficient of the polynomial function be positive or negative Why O A positive because the graph falls to the left and falls to the right OB positive because the graph falls to the left and rises to the right C negative because the graph falls to the left and falls to the right graph rises to the left and to the right O negative bec f Use the graph to estimate the person s maximum heart rate during the 15 minute film bpm After how many minutes did this maximum heart rate occur minute s g Use the graph to estimate the person s minimum heart rate during the 15 minute film bpm After how many minutes did this minimum heart rate occur minute s

K f x 2 x 4 1 Determine the symmetry of the function Choose the correct answer below O The graph off has neither y axis symmetry nor origin symmetry O The graph of f is symmetric with respect to the y axis O The graph of f is symmetric with respect to the origin 2 Find the y intercept Select the correct choice below and fill in any answer boxes within your choice OA The y intercept is OB There is no y intercept 3 Find the x intercept s Select the correct choice below and fill in any answer boxes within your choice O A The x intercept s is Use a comma to separato swers as needed OB There is no x intercept 4 What is the equation of the vertical asymptote s of f x Select the correct choice below and fill in any answer boxes within your choice X A x OB There is no vertical asymptote 5 What is the equation of the horizontal asymptote s of f x Select the correct choice below and fill in any answer boxes within your choice f x the book to graph the following rational function Type an integer or a simplified fraction O A y Use a comma to separate answers as needed Type an integer or a fraction OB There is no horizontal asymptote 6 Plot points between and beyond the x intercept and the vertical asymptotes Type the y coordinate for each of the following points 2 Use a comma to separate answers as needed Type an integer or a fraction 3 1 N 2 H 1 F 3 J

2x 9x 5 0 Choose the correct graph of 2x 9x 5 below A Q Q B N Q O C N What is the solution set Select the correct choice below and if necessary fill in the answer box to complete yo choice Q Q A OA The solution set is Type your answer in interval notation Use integers or decimals for any numbers in the expression B The solution set is the empty set G

Use the graph of the rational function to complete the following statement As x1 f x As x1 f x O LV 3

Lucy earns 75 000 and is filing her taxes as a single woman What amount of taxes does she owe Type in the number and do not include commas decimals dollar sign

Find the lengths of the sides of the right triangle Show that these lengths satisfy the Pythagorean Theorem Enter your answers from smallest to larg d d ds Show Transcribed Text d 1 1 1 d d3 Z These lengths satisfy the Pythagorean Theorem because Od d d 6 2 44 5 4 0 1234 3 45 36 62 3 5 1 These lengths satisfy the Pythagorean Theorem because Od d d3 021 12

Write the equation of the ellipse that has a center at 3 4 a vertex at 3 8 and a covertex at 1 4