Bungee Jumper KEY. Suppose that the function. can be used to model the bungee jumpers height in terms of time for 0 t 6.

Transcription

1 Questions: Bungee Jumper KEY Suppose that the function h ( t) 5t 8t 8t 89 can be used to model the bungee jumpers height in terms of time for t 6. Complete the table to show the height (h, in feet) of a bungee jumper at given values of time (t, in seconds). t h (sec) (ft) ) Describe a good graphing window for this function. Sample: Xmin =, Xma = 6 (Xscl = ) Ymin =, Yma = (Yscl = ) ) How tall was the platform from which the daredevil jumped? How do you know? 89 feet. This is the value of h at t =. ) The bungee jumper went up a little into the air before she started to fall. How many feet above the platform did she jump? When did the peak of this jump occur? At t =.4 seconds, the jumper was ft high. This means she jumped = 5.44 feet above the platform. 4) What is the closest that the bungee jumper came to the ground? After how many seconds into the jump did this point occur? At t = 4.67 seconds, the jumper was only.6 ft high from the ground. 5) At t = 5 seconds, was the bungee jumper falling down or bouncing back up? Eplain. Bouncing back up. At t = 5, the height is 4 feet (higher than the low point described in #4). 6) Since h(t) passes through the point (4, ), it is correct to say: At t = 4 seconds, the jumper is falling. However, it is incorrect to say: At h = feet, the jumper is falling. Why? There are two times, at t = 4 and approimately at t = 5.7, at which the height of the jumper is feet. One point occurs the height is decreasing, and the other point occurs it is increasing. 7) Evaluate the function at t = 7. Does this value make sense in the contet of the problem? Eplain your reasoning. h(7) = 8 ft. This doesnt make sense because the bungee jumper could not bounce back higher than the original platform. Also, as stated in the problem, the domain is restricted to values of t from to 6, inclusive ( t 6). 9, TESCCC 8//9 page 7 of 98

2 Bungee Jumper Questions: Suppose that the function h( t) 5t 8t 8t 89 can be used to model the bungee jumpers height in terms of time for t 6. Complete the table to show the height (h, in feet) of a bungee jumper at given values of time (t, in seconds). t (sec) h (ft) ) Describe a good graphing window for this function. ) How tall was the platform from which the daredevil jumped? How do you know? ) The bungee jumper went up a little into the air before she started to fall. How many feet above the platform did she jump? When did the peak of this jump occur? 4) What is the closest that the bungee jumper came to the ground? After how many seconds into the jump did this point occur? 5) At t = 5 seconds, was the bungee jumper falling down or bouncing back up? Eplain. 6) Since h(t) passes through the point (4, ), it is correct to say: At t = 4 seconds, the jumper is falling. However, it is incorrect to say: At h = feet, the jumper is falling. Why? 7) Evaluate the function at t = 7. Does this value make sense in the contet of the problem? Eplain your reasoning. 9, TESCCC 8//9 page 8 of 98

3 Heart Medicine KEY A dose of a specific medication is used in emergencies to quickly elevate a persons heart rate and then stabilize it to a normal rhythm. After being given such a drug, one patients heart rate behaved according to the following function: ht () (5t 6t ) Here, t stands for time in minutes after administration of the medicine, and H is the patients heart rate in beats per minute. Complete the table and sketch the graph to show how the patients heart rate changed over time. t t h(t) Then, answer the questions that follow. ) Use a graphing calculator to find the maimum rate at which the patients heart was beating. After how many minutes did this occur? beats per minute,.87 minutes after the medicine was given ) Describe how the patients heart rate behaved after reaching this maimum. Sample: The heart rate starts decreasing, but it also levels off. In other words, the heart rate never drops below a certain level. ) According to this function model, what would be the patients heart rate hours after the medicine was given? After 4 hours? hours = 8 minutes h(8) 6.4 bpm 4 hours = 4 minutes h(4) 6. bpm 4) This function has a horizontal asymptote. Where does it occur? How can its presence be confirmed using a graphing calculator? Asymptote occurs at 6 (or, in this case, h() = 6). Etend the table or the graph to include values such as = 4. The y-values in the table will approach 6, and the graph will flatten out. 9, TESCCC 8//9 page of 98

4 Heart Medicine A dose of a specific medication is used in emergencies to quickly elevate a persons heart rate and then stabilize it to a normal rhythm. After being given such a drug, one patients heart rate behaved according to the following function: ht () (5t 6t ) Here, t stands for time in minutes after administration of the medicine, and H is the patients heart rate in beats per minute. Complete the table and sketch the graph to show how the patients heart rate changed over time. Then, answer the questions that follow. t t h(t) ) Use a graphing calculator to find the maimum rate at which the patients heart was beating. After how many minutes did this occur? ) Describe how the patients heart rate behaved after reaching this maimum. ) According to this function model, what would be the patients heart rate hours after the medicine was given? After 4 hours? 4) This function has a horizontal asymptote. Where does it occur? How can its presence be confirmed using a graphing calculator? 9, TESCCC 8//9 page 4 of 98

5 How Many Hundreds? When school started, Jamie wasnt ready for the fact that his College Algebra teacher was going to take up homework. His first two homework grades were a and a 45. ) At this point, what was Jamies homework average? ) After the first two homework grades, Jamie shapes up and starts getting grades of on all his remaining homework assignments. Complete the table to see how his average changes with each additional grade. # s Grades (list) Total of Grades # of Grades Average, 45, 45, 45,,, 45,,, 4 5 If = the number of consecutive s Jamie makes (after bombing the first two assignments), write epressions for the total, number, and average. ) On the grid provided, plot the points from the table above to show Jamies homework average in terms of. Then use a calculator to help you complete the graph of the function. 4) How many consecutive one hundreds will Jamie have to make before his average is A) at least an 85? B) at least a 9? C) a 95? 5) What happens to the average as? Eplain, in terms of the function and the situation. 9, TESCCC 8//9 page 46 of 98

6 Moving on Up (pp. of ) The enrollment of an urban high school is increasing over time, and so is the number of its students who continue with a post-secondary education. The table shows information about these groups. Year 588 Urban High School Statistics Number Students that Total Continue with Enrollment Post-Secondary (t) Education (n) Percent Continuing with Post- Secondary Education (p) ) In 6, students who continue with a postsecondary education made up what percentage of the total enrollment? Round to the nearest tenth. (Place this answer in the appropriate spot in the table.) In the following problems, let = the number of years since. Also, assume that the students for both groups are increasing at a constant rate. ) Write a linear function can be used to relate and t, the total enrollment of the school. ) Use this function to predict the total enrollment of the school in the year. 4) Write a linear function can be used to relate and n, the number of students who continue with a post-secondary education at the school. 5) Use this function to predict the number of students who continue with a post-secondary education at the school in the year. 6) Use your answers from # and #5 to predict the percentage of students who continue with a post-secondary education at the school in the year. 9, TESCCC 8//9 page 49 of 98

7 Moving on Up (pp. of ) 7) Write function rule to find p, the percentage of students who continue with a post-secondary education at the school, in terms of (the number of years since ). p() = 8) Use a calculator to complete the table and sketch the graph of this relationship. Label the independent and dependent variables in the correct column. p 4 5 9) According to this model, in what year will the percentage first reach 6%? 7%? ) Use the model to estimate the percentage of students who continue with a post-secondary education at the urban school in the year 995. Does the answer make sense? ) Use the model to estimate the percentage of students who continue with a post-secondary education at the urban school in the year 98. Does the answer make sense? ) Use the model to estimate the percentage of students who continue with a post-secondary education at the urban school in the year. Does the answer make sense? ) Write a few sentences to compare the domain of the problem situation with the domain of the function rule. 9, TESCCC 8//9 page 5 of 98

8 Boost (pp. of ) ft Lil Bro (4 ft) Fence (6 ft) y While playing Frisbee, Dylans little brother accidentally threw the disc over their neighbors fence. However, a mean dog lives in the neighbors back yard, and the siblings are not sure how near or far the Frisbee landed from the fence. Dylan decides to give his brother (who is 4 feet tall) a boost to look over the fence (which is 6 feet tall). Here, let = the number of feet Dylan is able to lift his little brother into the air, and let the closest distance (in feet) the brother can see on the ground past the fence. ) Eplain why Dylan must boost his brother more than feet into the air (or, > ). ) Suppose Dylans brother can only see 5 feet past the fence ( 5). Draw a diagram of the situation. Set up and solve a proportion to determine how much of a boost () he is getting. ) When Dylan lifts his brother all the way up to his chin, the brother is 5 feet above the ground ( = 5). Draw a diagram of the situation. Set up and solve a proportion to determine how far past the fence (y) he can see. 4) Draw a diagram and set up a proportion that relates the two variables in this situation. Then solve the equation for y. 9, TESCCC 8//9 page 5 of 98

9 Boost (pp. of ) 5) Complete the table and sketch the graph of this function. Label the independent and dependent variables in the correct column y 6) When Dylan lifts his little brother as high as he possibly can, he finally sees the Frisbee, just. feet from the fence. Lil Bro At this moment, how high was Dylan able to boost his brother? ft Frisbee. 7) Use the answers from the previous questions to describe the domain and range of the problem situation. EXTRA!! When the brother spots the Frisbee, how far is it away from his eyes? 9, TESCCC 8//9 page 54 of 98

10 Fan Club (pp. of ) When Sam found out that teen pop sensation Callie Colorado was going to be the guest of honor in their towns fall parade, he almost freaked out! (He was Callie Colorados number one fan.) On the night of the parade, Sam arrived early and staked out a spot on the street right net to the parade route. Sam first spotted Callie she was 87 feet down the street, and her car was moving at a speed of 6 feet per second. When she passed in front of him, Sam was only feet away from her. (He let out a big scream!) ft v Sam distance down the street C.C. ) When Callie Colorado was 87 feet down the street, her visual distance (v) from Sam was actually about 89.7 feet. Eplain why. ) Complete the table to describe the given distances with respect to the time in seconds since Sam first spotted Callie Colorado. Sketch the graph of visual distance as a function of time. Time (sec) Distance down the street (ft) Visual distance between (ft) t d V * ) Write a linear function to relate time in seconds (t) and the distance (d) down the street. Then use this function to determine * Callie Colorado was directly across the street from Sam. 9, TESCCC 8//9 page 57 of 98

11 Fan Club (pp. of ) 4) What function rule can be used to find the visual distance (v) between Sam and Callie in terms of t (time in seconds)? 5) Use this function rule to complete the following table for selected values of t. Time (sec) Distance down the street (ft) Visual distance between (ft) t d v A -5 B C 6 D 4 6) Do your answers for point A (above) make sense in the contet of the problem situation? What could these numbers represent? 7) Do your answers for point D (above) make sense in the contet of the problem situation? What could these numbers represent? 8) What restrictions, if any, must be placed on the domain of this function? 9) Are there any restrictions on the domain and range of the problem situation? 9, TESCCC 8//9 page 58 of 98

12 Row & Cone (pp. of ). Row A person in a kayak is 9m north of a buoy and rowing toward it at a rate of 7 meters per second. At the same time, a sailboat is 5m east of the buoy moving west at a rate of m/s. A) How far is each vessel from the buoy after 5 seconds? B) At this moment, how far apart are the kayak and the sailboat? C) Write the two linear epressions that can be used to find the distance from each vessel to the buoy in terms of time (t, in seconds). D) What function rule can be used to find the distance between the kayak and the sailboat (d, in meters) in terms of time, t (in seconds)? E) Graph this function rule in a calculator using an appropriate window. What is the closest that the two boats get to each other? After how many seconds does this occur? F) Discuss any restrictions on the domain of the problem situation. 9, TESCCC 8//9 page 6 of 98

13 Row & Cone (pp. of ). Cone A drain is in the shape of an inverted cone with a height of cm and a diameter of 6 cm. However, the drain is usually only partially filled with water. A) Use proportions from similar triangles to find the radius (r) of the waters surface in the cone it is filled to a height of h = cm and it is filled to a height of h = 7 cm. h = height of liquid in cone 8 cm r = radius of liquid surface in cone B) What linear function gives values of r in terms of h? h 8 cm r (cross section) C) The volume of a cone is given as V r h. Find the volume of the water in the drain at the two levels described in part (A). D) What function rule can be used to find the volume of water in the drain in terms of the height (h) of the water inside? E) Discuss any restrictions on the domain and range of the problem situation. F) At what depth is the drain filled to half its volume capacity in cubic centimeters? (The answer is not h = 5.) 9, TESCCC 8//9 page 6 of 98

14 Tet Message Mayhem Veronica has a cell phone plan where she pays $9.9 per month for unlimited calls and up to 5 tet messages. However, if she goes over her allowance of tet messages, she is charged and additional fee of $. apiece. Complete the table to compute Veronicas monthly cell phone bill for various numbers of tet messages that she could make. Label the independent and dependent variables in the correct column. Use these numbers to sketch the graph of this relationship. OMG! TXT ME LTR. BRB. Process 5 (included in plan) $ $ Questions: ) Describe the domain and range of this relationship. ) Is this relationship a function? Why or why not? ) What function can be used to relate the monthly bill to the number of tet messages is greater than 5? 4) Eplain why this function cannot be used for values of between and 5. 9, TESCCC 8//9 page 64 of 98

15 Tanks A Lot (pp. of ) A water tank is comprised of a cylinder and an inverted cone, as shown in the diagram. The volume, V (in cubic feet), of water in the tank is a function of the depth (or height, h, in feet) of water inside. ) Find the volume of water in the tank it is filled to a height of 5 feet. 5 ft 4 ft 5 ft Cylinder-Cone Tank 4 ft h = h = ft r = ft h = 5 h =.5 ) Find the volume of water in the tank it is full (or, filled to a height of feet). Volume of a Cylinder: V r h Volume of a Cone: V r h ) Write a piecewise function that gives the volume of water in the tank, V(h), in terms of the height. Then, sketch the graph of the function over an appropriate domain. Formula Interval Hint V(h) =,, h < 5 Its NOT linear. It is linear. 4) On a separate sheet of paper, write four questions that relate to this function, and then provide answers for each. 9, TESCCC 8//9 page 85 of 98

16 Tanks A Lot (pp. of ) A different water tank is comprised of a square-based prism and an inverted pyramid, as shown in the diagram. The volume, V (in cubic feet), of water in the tank is a function of the depth (or height, h, in feet) of water inside. 5) Find the volume of water in the tank it is filled to a height of 4 feet. 6 ft 4 ft Prism-Pyramid Tank 6 ft 4 ft h = h = 7 4 ft a = ft h = 4 h = 6) Find the volume of water in the tank it is full (or, filled to a height of feet). Volume of a Prism: V B h Volume of a Pyramid: V B h 7) Write a piecewise function that gives the volume of water in the tank, V(h), in terms of the height. Then, sketch the graph of the function over an appropriate domain. 8) On a separate sheet of paper, write four questions that compare the volume function for both tanks, and then provide answers for each. 9, TESCCC 8//9 page 86 of 98

17 Absolutely in Pieces (pp. of ) y f ( ) y < > Although it is given as one single function, the absolute value function has two linear branches that meet at the origin. On the left branch of the graph (or < ), the function follows the function -. On the right branch of the graph (or ), the function follows the function. Because they use absolute value, the functions that follow also graph into linear pieces. For each, use a calculator to sketch the graphs and complete the tables. Then see if you can determine the functions and intervals for each branch. ) Function: f ( ) Graph Sketch Table Branches y Rule - - Restriction ) Function: f ( ) Graph Sketch Table Branches y - Rule - Restriction ) Function: f ( ) Graph Sketch Table Branches y - Rule Restriction 9, TESCCC 8//9 page 67 of 98

18 Absolutely in Pieces (pp. of ) 4) Function: f ( ) 5 Graph Sketch Table Branches y Rule 5 7 Interval 5) Function: f ( ) Graph Sketch Table Branches y -5 Rule - 5 Interval 6) Function: f ( ) 6 Graph Sketch Table Branches y - Rule Interval 9, TESCCC 8//9 page 68 of 98

19 Piecewise-Defined Functions (pp. of ) KEY A piecewise-defined function is a function that uses different formulas or rules depending on what values of are being used. The notation looks like this: st Epression, st Interval f ( ) 4, 4, f () nd Epression, nd Interval rd Epression, rd Interval g ( ) 5,, 4 Notice: The intervals used can be open, or closed, or infinite (but they should not overlap). Graphically, the pieces of the function can connect, but they dont have to. Part One: Evaluating Evaluate the function at the given values by first determining which formula to use. p ( ) 5, 5, 5 A) p () () + 5 = 8 B) p (5) (5) + 5 = C) p () () = 97 D) p ( 6) undefined f ( ) 8,, E) f () + = F) f () () 6 = G) f ( ) 8 H) f () () 6 = 94 Part Two: Writing Define a piecewise function based on the description provided. A) You determine your ta credit, C, based on your annual salary, a. If your annual salary is $4, or below, your ta credit is based on 5% of the salary. If the salary is between $4, and $5,, the percent drops to %. And if you make $5, or more, the credit is 4% of the annual salary. c(a) =.5a, a 4,.a, 4, < a < 5,.4a, a 5, 9, TESCCC 8//9 page 69 of 98

20 Piecewise-Defined Functions (pp. of ) KEY B) Define a piecewise function for this graph using linear functions. +, f() = 5, < < 5 - +, 5 Part Three: Graphing Graph the given piecewise functions on the grids provided. A) g ( ) 7,,, B) f ( ) 4,, 9, TESCCC 8//9 page 7 of 98

21 Piecewise-Defined Functions (pp. of ) A piecewise-defined function is a function that uses different formulas or rules depending on what values of are being used. The notation looks like this: st Epression, st Interval f ( ) 4, 4, f () nd Epression, nd Interval rd Epression, rd Interval g ( ) 5,, 4 Notice: The intervals used can be open, or closed, or infinite (but they should not overlap). Graphically, the pieces of the function can connect, but they dont have to. Part One: Evaluating Evaluate the function at the given values by first determining which formula to use. p ( ) 5, 5, 5 A) p () B) p (5) C) p () D) p ( 6) f ( ) 8,, E) f () F) f () G) f ( ) H) f () Part Two: Writing Define a piecewise function based on the description provided. A) You determine your ta credit, C, based on your annual salary, a. If your annual salary is below $4,, your ta credit is based on 5% of the salary. If the salary is between $4, and $5,, the percent drops to %. And if you make $5, or more, the credit is 4% of the annual salary. c(a) = 9, TESCCC 8//9 page 7 of 98

22 Piecewise-Defined Functions (pp. of ) B) Define a piecewise function for this graph using linear functions. f() = Part Three: Graphing Graph the given piecewise functions on the grids provided. A) g ( ) 7,,, B) f ( ) 4,, 9, TESCCC 8//9 page 7 of 98

23 The greatest integer function is defined by the rule that f() equals the greatest integer that is less than or equal to. This function is often studied because, unlike many other functions, it lacks continuity. Continuity (pp. of ) KEY (dotted) f( ) Greatest Integer Function Nicknames: Step function round down function Floor function Continuity can be referred to as connectedness. In general, a function has continuity if you can draw its graph without picking up your pencil. f f(( ) Other notation: f () = int() (calculator) f( ) (sometimes called the floor function) To graph this function, you would have to pick up your pencil many times. To see if a function is continuous at a point, check to see what happens just to the left and right. ) Complete the table and answer the questions that follow. Consider f( ) at = f () Here, as approaches from the left, Here, as approaches from the right, the function values equal. the function values equal. In math symbols, we can write: In math symbols, we can write: As -, f () =. As +, f () =. When these two do not match up (or, they are not equal), we say that a discontinuity occurs at =. At what other values of does this occur? = {-, -, -,,,...} (all integral values) 9, TESCCC 8//9 page 77 of 98

24 Continuity (pp. of ) KEY ) Graph each function using a decimal window (Zoom #4) to observe the different ways in which functions can lack continuity. Types: Jump Discontinuity Removable Discontinuity Infinite Discontinuity Description: Sample: Left and right sides dont match up f ( ) The left and right match up, but theres a hole f ( ) A vertical asymptote occurs f ( ) Graph: Where do discontinuities occur? Intervals on which f () is continuous = = = (-, ) (, ) (-, ) (, ) (-, ) (, ) ) Graph each function to determine where any discontinuities occur. Classify each by type. f( ).5 f ( ) f ( ),, Discontinuities: Jump discontinuities: = {-4, -,,, 4, 6 } (even integers) { = n, n J} Discontinuities: Infinite discontinuity: = - Removable discontinuity: = Discontinuities: No discontinuities Continuous on (-, ) 9, TESCCC 8//9 page 78 of 98

25 Closing Gaps, Filling Holes ) Graph each piecewise function to determine whether any discontinuities occur. A), f ( ) B) 5, f ( ), 5, Is the function continuous at =? Eplain. Is the function continuous at =? Eplain. ) In each case, find the value of c that makes the function continuous for all values of., 5, A) f ( ) B) g ( ) c, c, ) The following function has a removable discontinuity at = 4. Use the table to help you determine a method for rewriting the function as a piecewise function that is continuous at this point. (In other words, what point would fill the hole?) Function Graph Table Rewrite y y Error f () = , TESCCC 8//9 page 8 of 98