1.6. Piecewise Functions. LEARN ABOUT the Math. Representing the problem using a graphical model
|
|
- Diane Cameron
- 7 years ago
- Views:
Transcription
1 . Piecewise Functions YOU WILL NEED graph paper graphing calculator GOAL Understand, interpret, and graph situations that are described b piecewise functions. LEARN ABOUT the Math A cit parking lot uses the following rules to calculate parking fees: A flat rate of $5. for an amount of time up to and including the first hour A flat rate of $.5 for an amount of time over h and up to and including h A flat rate of $3 plus $3 per hour for each hour after h? How can ou describe the function for parking fees in terms of the number of hours parked? EXAMPLE Representing the problem using a graphical model Use a graphical model to represent the function for parking fees. Solution Time (h) Parking Fee ($) Create a table of values.. Piecewise Functions
2 . Plot the points in the table of values. Use a solid dot to include a value in an interval. Use an open dot to eclude a value from an interval. Cost ($) 3 Time (h) 5 The domain of this piecewise function is $. The function is linear over the domain, but it is discontinuous at 5,, and. There is a solid dot at (, ) and an open dot at (, 5) because the parking fee at h is $.. There is a closed dot at (, 5) and an open dot at (,.5) because the parking fee at h is $5.. There is a closed dot at (,.5) and an open dot at (, 3) because the parking fee at h is $.5. The last part of the graph continues in a straight line since the rate of change is constant after h. piecewise function a function defined b using two or more rules on two or more intervals; as a result, the graph is made up of two or more pieces of similar or different functions Each part of a piecewise function can be described using a specific equation for the interval of the domain. EXAMPLE Representing the problem using an algebraic model Use an algebraic model to represent the function for parking fees. Solution if 5 if, # if, # if. Write the relation for each rule. Chapter 7
3 f () 5 μ, if 5 5, if, #.5, if, # 3 3, if. Combine the relations into a piecewise function. The domain of the function is $. The function is discontinuous at 5,, and because there is a break in the function at each of these points. Reflecting A. How do ou sketch the graph of a piecewise function? B. How do ou create the algebraic representation of a piecewise function? C. How do ou determine from a graph or from the algebraic representation of a piecewise function if there are an discontinuities? APPLY the Math EXAMPLE 3 Representing a piecewise function using a graph Graph the following piecewise function.,if, f () 5 e 3, if $ Solution Create a table of values. f () 5 f () 5 3 f() f() From the equations given, the graph consists of part of a parabola that opens up and a line that rises from left to right. Both tables include 5 since this is where the description of the function changes.. Piecewise Functions
4 . = f() Plot the points, and draw the graph. A solid dot is placed at (, 7) since 5 is included with f() 5 3. An open dot is placed at (, ) since 5 is ecluded from f() 5. f () is discontinuous at 5. EXAMPLE = f () Representing a piecewise function using an algebraic model Determine the algebraic representation of the following piecewise function. Solution f () 5,if #, if. The graph is made up of two pieces. One piece is part of the reciprocal function defined b 5 when #. The other piece is a horizontal line defined b 5 when.. The solid dot indicates that point Q, R belongs with the reciprocal function. Chapter 9
5 EXAMPLE 5 Reasoning about the continuit of a piecewise function Is this function continuous at the points where it is pieced together? Eplain., if # g() 5, if,, 3, if $ 3 Solution The function is continuous at the points where it is pieced together if the functions being joined have the same -values at these points. Calculate the values of the function at 5 using the relevant equations: () 5 Calculate the values of the function at 5 3 using the relevant equations: 5 5 (3) The function is discontinuous, since there is a break in the graph at 5 3. The graph is made up of three pieces. One piece is part of an increasing line defined b 5 when #. The second piece is an increasing line defined b 5 when,, 3. The third piece is part of a parabola that opens down, defined b 5 when $ 3. The two -values are the same, so the two linear pieces join each other at 5. The two -values are different, so the second linear piece does not join with the parabola at 5 3. Tech Support For help using a graphing calculator to graph a piecewise function, see Technical Appendi, T-. Verif b graphing. 5. Piecewise Functions
6 . In Summar Ke Ideas Some functions are represented b two or more pieces. These functions are called piecewise functions. Each piece of a piecewise function is defined for a specific interval in the domain of the function. Need to Know To graph a piecewise function, graph each piece of the function over the given interval. A piecewise function can be either continuous or not. If all the pieces of the function join together at the endpoints of the given intervals, then the function is continuous. Otherwise, it is discontinuous at these values of the domain. CHECK Your Understanding. Graph each piecewise function., if,,if # a) f () 5 e d) f () 5 e 3,if $, if., if, f () 5 e e), if $, if # c) f () 5 e f) f () 5,if,,if.,if $. State whether each function in question is continuous or not. If not, state where it is discontinuous. 3. Write the algebraic representation of each piecewise function, using function notation. a) = f () f () 5 e!,if,,if $ = f (). State the domain of each piecewise function in question 3, and comment on the continuit of the function. Chapter 5
7 PRACTISING 5. Graph the following piecewise functions. Determine whether each K function is continuous or not, and state the domain and range of the function., if, a) f () 5 e c) 3, if $ f () 5 e, if,, if $, if, f () 5 e,if # d) f () 5, if # # 3,if. 5, if. 3. Graham s long-distance telephone plan includes the first 5 min per A month in the $5. monthl charge. For each minute after 5 min, Graham is charged $.. Write a function that describes Graham s total long-distance charge in terms of the number of long distance minutes he uses in a month. 7. Man income ta sstems are calculated using a tiered method. Under a certain ta law, the first $ of earnings are subject to a 35% ta; earnings greater than $ and up to $5 are subject to a 5% ta. An earnings greater than $5 are taed at 55%. Write a piecewise function that models this situation.. Find the value of k that makes the following function continuous. T Graph the function. f () 5 e k,if,, if $ 9. The fish population, in thousands, in a lake at an time,, in ears is modelled b the following function:,if # # f () 5 e, if. This function describes a sudden change in the population at time 5, due to a chemical spill. a) Sketch the graph of the piecewise function. Describe the continuit of the function. c) How man fish were killed b the chemical spill? d) At what time did the population recover to the level it was before the chemical spill? e) Describe other events relating to fish populations in a lake that might result in piecewise functions. 5. Piecewise Functions
8 . Create a flow chart that describes how to plot a piecewise function with two pieces. In our flow chart, include how to determine where the function is continuous... An absolute value function can be written as a piecewise function that C involves two linear functions. Write the function f () 5 3 as a piecewise function, and graph our piecewise function to check it.. The demand for a new CD is described b D(p) 5 p,if, p # 5, if p. 5 where D is the demand for the CD at price p, in dollars. Determine where the demand function is discontinuous and continuous. Etending 3. Consider a function, f (), that takes an element of its domain and rounds it down to the nearest. Thus, f (5.) 5, while f (.7) 5 and f (3) 5 3. Draw the graph, and write the piecewise function. You ma limit the domain to P[, 5). Wh do ou think graphs like this one are often referred to as step functions?. Eplain wh there is no value of k that will make the following function continuous. 5, if, f () 5 k, if # # 3,if The greatest integer function is a step function that is written as f () 5 3, where f () is the greatest integer less than or equal to. In other words, the greatest integer function rounds an number down to the nearest integer. For eample, the greatest integer less than or equal to the number [5.3] is 5, while the greatest integer less than or equal to the number 35.3 is. Sketch the graph of f () a) Create our own piecewise function using three different transformed parent functions. Graph the function ou created in part a). c) Is the function ou created continuous or not? Eplain. d) If the function ou created is not continuous, change the interval or adjust the transformations used as required to change it to a continuous function. Chapter 53
9 .7 Eploring Operations with Functions GOAL Eplore the properties of the sum, difference, and product of two functions. A popular coffee house sells iced cappuccino for $ and hot cappuccino for $3. The manager would like to predict the relationship between the outside temperature and the total dail revenue from each tpe of cappuccino sold. The manager discovers that ever C increase in temperature leads to an increase in the sales of cold drinks b three cups per da and to a decrease in the sales of hot drinks b five cups per da. The function f () 5 3 can be used to model the number of iced cappuccinos sold. Number of iced cappuccinos sold 5 5 f() = Outside temperature (C) The function g() 55 can be used to model the number of hot cappuccinos sold. Number of hot cappuccinos sold g() = Outside temperature (C) In both functions, represents the dail average outside temperature. In the first function, f () represents the dail average number of iced cappuccinos sold. In the second function, g() represents the dail average number of hot cappuccinos sold.? How does the outside temperature affect the dail revenue from cappuccinos sold? A. Make a table of values for each function, with the temperature in intervals of 5, from to. B. What does h() 5 f () g() represent? 5.7 Eploring Operations with Functions
10 C. Simplif h() 5 (3 ) (5 )..7 D. Make a table of values for the function in part C, with the temperature in intervals of 5, from to. How do the values compare with the values in each table ou made in part A? How do the domains of f (), g(), and h() compare? E. What does h() 5 f () g() represent? F. Simplif h() 5 (3 ) (5 ). G. Make a table of values for the function in part F, with the temperature in intervals of 5, from to. How do the values compare with the values in each table ou made in part A? How do the domains of f (), g(), and h() compare? H. What does R() 5 f () 3g() represent? I. Simplif R() 5 (3 ) 3(5 ). J. Make a table of values for the function in part I, with the temperature in intervals of 5, from to. How do the values compare with the values in each table ou made in part A? How do the domains of f (), g(), and R() compare? K. How does temperature affect the dail revenue from cappuccinos sold? Reflecting L. Eplain how the sum function, h(), would be different if a) both f () and g() were increasing functions both f () and g() were decreasing functions M. What does the function k() 5 g() f () represent? Is its graph identical to the graph of h() 5 f () g()? Eplain. N. Determine the function h() 5 f () 3 g(). Does this function have an meaning in the contet of the dail revenue from cappuccinos sold? Eplain how the table of values for this function is related to the tables of values ou made in part A. O. If ou are given the graphs of two functions, eplain how ou could create a graph that represents a) the sum of the two functions the difference between the two functions c) the product of the two functions Chapter 55
11 In Summar Ke Idea If two functions have domains that overlap, the can be added, subtracted, or multiplied to create a new function on that shared domain. Need to Know Two functions can be added, subtracted, or multiplied graphicall b adding, subtracting, or multipling the values of the dependent variable for identical values of the independent variable. Two functions can be added, subtracted, or multiplied algebraicall b adding, subtracting, or multipling the epressions for the dependent variable and then simplifing. The properties of each original function have an impact on the properties of the new function. FURTHER Your Understanding. Let f 5 5(, ), (, ), (, 3), (3, 5), (, ) and g 5 5(, ), (, ), (, ), (, ), (, ), (, ). Determine: a) f g f g c) g f d) fg. Use the graphs of f and g to sketch the graphs of f g. a) f f g g 3. Use the graphs of f and g to sketch the graphs of f g. a) f g f g 5.7 Eploring Operations with Functions
12 .7. Use the graphs of f and g to sketch the graphs of fg. a) f g g f 5. Determine the equation of each new function, and then sketch its graph. a) h() 5 f () g(), where f () 5 and g() 5 p() 5 m() n(), where m() 5 and n() 57 c) r() 5 s() t(), where s() 5 and t() 5 d) a() 5 b() 3 c(), where b() 5 and c() 5. a) Using the graphs ou sketched in question 5, compare and contrast the relationship between the properties of the original functions and the properties of the new function. Which properties of the original functions determined the properties of the new function? 7. Let f () 5 3 and g() 5 5, PR. a) Sketch each graph on the same set of aes. Make a table of values for 3 # # 3, and determine the corresponding values of h() 5 f () 3 g(). c) Use the table to sketch h() on the same aes. Describe the shape of the graph. d) Determine the algebraic model for h(). What is its degree? e) What is the domain of h()? How does this domain compare with the domains of f () and g()?. Let f () 5 and g() 5, PR. a) Sketch each graph on the same set of aes. Make a table of values for 3 # # 3, and determine the corresponding values of h() 5 f () 3 g(). c) Use the table to sketch h() on the same aes. Describe the shape of the graph. d) Determine the algebraic model for h(). What is its degree? e) What is the domain of h()? Chapter 57
13 Chapter Review Stud Aid See Lesson., Eamples, 3, and. Tr Chapter Review Questions 7,, and 9. FREQUENTLY ASKED Questions Q: In what order are transformations performed on a function? Function Perform all stretches/compressions and reflections. Perform all translations. A: All stretches/compressions (vertical and horizontal) and reflections can be applied at the same time b multipling the - and -coordinates on the parent function b the appropriate factors. Both vertical and horizontal translations can then be applied b adding or subtracting the relevant numbers to the - and -coordinates of the points. Stud Aid See Lesson.5, Eamples,, and 3. Tr Chapter Review Questions to 3. Q: How do ou find the inverse relation of a function? A: You can find the inverse relation of a function numericall, graphicall, or algebraicall. To find the inverse relation of a function numericall, using a table of values, switch the values for the independent and dependent variables. f() f (, ) (, ) To find the inverse relation graphicall, reflect the graph of the function in the line 5. This is accomplished b switching the - and -coordinates in each ordered pair. To find the algebraic representation of the inverse relation, interchange the positions of the - and -variables in the function and solve for. 5 Chapter Review
14 Q: Is an inverse of a function alwas a function? A: No; if an element in the domain of the original function corresponds to more than one number in the range, then the inverse relation is not a function. Q: What is a piecewise function? A: A piecewise function is a function that has two or more function rules for different parts of its domain.,if, For eample, the function defined b f () 5 e, if $ consists of two pieces. The first equation defines half of a parabola that opens down when,. The second equation defines a decreasing line with a -intercept of when $. The graph confirms this. Stud Aid Chapter Review See Lesson.5, Eamples,, and 3. Tr Chapter Review Questions to 3. Stud Aid See Lesson., Eamples,, 3, and. Tr Chapter Review Questions to 7. Q: If ou are given the graphs or equations of two functions, how can ou create a new function? A: You can create a new function b adding, subtracting, or multipling the two given functions. This can be done graphicall b adding, subtracting, or multipling the -coordinates in each pair of ordered pairs that have identical -coordinates. This can be done algebraicall b adding, subtracting, or multipling the epressions for the dependent variable and then simplifing. Stud Aid See Lesson.7. Tr Chapter Review Questions to. Chapter 59
15 PRACTICE Questions Lesson.. Determine whether each relation is a function, and state its domain and range. a) c) d). A cell phone compan charges a monthl fee of $3, plus $. per minute of call time. a) Write the monthl cost function, C(t), where t is the amount of time in minutes of call time during a month. Find the domain and range of C. Lesson Graph f () 5 3, and state the domain and range.. Describe this interval using absolute value notation Lesson.3 5. For each pair of functions, give a characteristic that the two functions have in common and a characteristic that distinguishes them. a) f () 5 and g() 5 sin f () 5 and g() 5 c) f () 5 and g() 5 d) f () 5 and g() 5. Identif the intervals of increase/decrease, the smmetr, and the domain and range of each function. a) c) Lesson. f () 5 3 f () 5 f () 5 7. For each of the following equations, state the parent function and the transformations that were applied. Graph the transformed function. a) 5 5.5"3( 7) c) 5 sin (3), # # 3 d) 5 3. The graph of 5 is horizontall stretched b a factor of, reflected in the -ais, and shifted 3 units down. Find the equation that results from the transformation, and graph it. 9. (, ) is a point on the graph of 5 f (). Find the corresponding point on the graph of each of the following functions. a) 5f () 5 f (( 9)) 7 c) 5 f ( ) d) 5.3f (5( 3)) e) 5 f ( ) f) 5f (( )) Chapter Review
16 Lesson.5. For each point on a function, state the corresponding point on the inverse relation. a) (, ) d) f (5) 5 7 (, 9) e) g() 53 c) (, 7) f) h() 5. Given the domain and range of a function, state the domain and range of the inverse relation. a) D 5 5PR, R 5 5 PR,, D 5 5PR $ 7, R 5 5 PR,. Graph each function and its inverse relation on the same set of aes. Determine whether the inverse relation is a function. a) f () 5 g() 5 3. Find the inverse of each function. a) f () 5 g() 5 3 Lesson.. Graph the following function. Determine whether it is discontinuous and, if so, where. State the domain and the range of the function., if, f () 5 e, if $ 5. Write the algebraic representation for the following piecewise function, using function notation If f () 5 e, if, 3,if $ is f () continuous at 5? Eplain. 7. A telephone compan charges $3 a month and gives the customer free call minutes. After the min, the compan charges $.3 a minute. a) Write the function using function notation. Find the cost for talking 35 min in a month. c) Find the cost for talking min in a month. Lesson.7. Given f 5 5(, ), (, 3), (, 7), (5, ) and g 5 5(, ), (, ), (, 3), (, ), (, 9), determine the following. a) f () g() f () g() c) 3 f ()3g() 9. Given f () 5, # # 3 and g() 5, 3 # # 5, graph the following. a) f d) f g g e) fg c) f g. f () 5 and g() 5. Match the answer with the operation. Answer: Operation: a) 3 3 A f () g() B f () g() c) 3 C g() f () d) D f () 3 g(). f () 5 3 and g() 5, a) Complete the table. 3 f() g() (f g)() Chapter Review Use the table to graph f () and g() on the same aes. c) Graph (f g)() on the same aes as part. d) State the equation of (f g)(). e) Verif the equation of (f g)() using two of the ordered pairs in the table. Chapter
17 Chapter Self-Test Consider the graph of the relation shown. a) Is the relation a function? Eplain. State the domain and range.. Given the following information about a function: D 5 5PR R 5 5 PR $ decreasing on the interval (`, ) increasing on the interval (, `) a) What is a possible parent function? Draw a possible graph of the function. c) Describe the transformation that was performed. 3. Show algebraicall that the function f () 5 3 is an even function.. Both f () 5 and g() 5 have a domain of all real numbers. List as man characteristics as ou can to distinguish the two functions Describe the transformations that must be applied to 5 to obtain the function f () 5( 3) 5, then graph the function.. Given the graph shown, describe the transformations that were performed to get this function. Write the algebraic representation, using function notation. 7. (3, 5) is a point on the graph of 5 f (). Find the corresponding point on the graph of each of the following relations. a) 5 3f ( ) 5 f (). Find the inverse of f () 5( ). 9. A certain ta polic states that the first $5 of income is taed at 5% and an income above $5 is taed at %. a) Calculate the ta on $5. Write a function that models the ta polic.. a) Sketch the graph of where f () 5 e, if, f ()! 3, if $ Is f () continuous over its entire domain? Eplain. c) State the intervals of increase and decrease. d) State the domain and range of this function. Chapter Self-Test
18 Chapter Task Modelling with Functions In 95, a team of chemists led b Dr. W. F. Libb developed a method for determining the age of an natural specimen, up to approimatel ears of age. Dr. Libb s method is based on the fact that all living materials contain traces of carbon-. His method involves measuring the percent of carbon- that remains when a specimen is found. The percent of carbon- that remains in a specimen after various numbers of ears is shown in the table below. Years Carbon- Remaining (%) ? 573 How can ou use the function P(t) 5 (.5) to model this situation and determine the age of a natural specimen? t A. What percent of carbon is remaining for t 5? What does this mean in the contet of Dr. Libb s method? B. Draw a graph of the function P(t) 5 (.5), using the given table of values. C. What is a reasonable domain for P(t)? What is a reasonable range? D. Determine the approimate age of a specimen, given that P(t) 5 7. E. Draw the graph of the inverse function. F. What information does the inverse function provide? G. What are the domain and range of the inverse function? t Task Checklist Did ou show all our steps? Did ou draw and label our graphs accuratel? Did ou determine the age of the specimen that had 7% carbon- remaining? Did ou eplain our thinking clearl? Chapter 3
19 Answers Chapter Getting Started, p.. a) c) 5 d) a 5a. a) ( )( ) (5 )( 3) c) ( )( ) d) (a ( ) 3. a) horizontal translation 3 units to the right, vertical translation units up; horizontal translation unit to the right, vertical translation units up; c) horizontal stretch b a factor of, vertical stretch b a factor of, reflection across the -ais; d) horizontal compression b a factor of vertical stretch b a factor of,, reflection across the -ais; a) D 5 5PR # #, f) R 5 5PR # # D 5 5PR, R 5 5 PR $9 c) D 5 5PR, R 5 5 PR d) D 5 5PR, R 5 5 PR 3 # $ 3 e) D 5 5PR, R 5 5 PR. 5. a) This is not a function; it does not pass the vertical line test. This is a function; for each -value, there is eactl one corresponding -value. c) This is not a function; for each -value greater than, there are two corresponding -values. d) This is a function; for each -value, there is eactl one corresponding -value. e) This is a function; for each -value, there is eactl one corresponding -value.. a) about.7 7. If a relation is represented b a set of ordered pairs, a table, or an arrow diagram, one can determine if the relation is a function b checking that each value of the independent variable is paired with no more than one value of the dependent variable. If a relation is represented using a graph or scatter plot, the vertical line test can be used to determine if the relation is a function. A relation ma also be represented b a description/rule or b using function notation or an equation. In these cases, one can use reasoning to determine if there is more than one value of the dependent variable paired with an value of the independent variable. Lesson., pp. 3. a) D 5 5PR; R 5 5PR # #; This is a function because it passes the vertical line test. D 5 5PR # # 7; R 5 5 PR 3 # # ; This is a function because it passes the vertical line test. c) D 5 5,, 3, ; R 5 55,, 7, 9, ; This is not a function because is sent to more than one element in the range. d) D 5 5PR; R 5 5PR; This is a function because ever element in the domain produces eactl one element in the range. e) D 5 5, 3,, ; R 5 5,,, 3; This is a function because ever element of the domain is sent to eactl one element in the range. D 5 5PR; R 5 5PR # ; This is a function because ever element in the domain produces eactl one element in the range.. a) D 5 5PR; R 5 5 PR #3; This is a function because ever element in the domain produces eactl one element in the range. D 5 5PR 3; R 5 5 PR ; This is a function because ever element in the domain produces eactl one element in the range. c) D 5 5PR; R 5 5 PR. ; This is a function because ever element in the domain produces eactl one element in the range. d) D 5 5PR; R 5 5 PR # # ; This is a function because ever element in the domain produces eactl one element in the range. e) D 5 5PR 3 # # 3; R 5 5PR 3 # # 3; This is not a function because (, 3) and (, 3) are both in the relation. f) D 5 5PR; R 5 5PR # # ; This is a function because ever element in the domain produces eactl one element in the range. 3. a) function; D 5 5, 3, 5, 7; R 5 5,, function; D 5 5,,, 5; R 5 5, 3, c) function; D 5 5,,, 3; R 5 5, d) not a function; D 5 5,, ; R 5 5, 3, 5, 7 e) not a function; D 5 5,, ; R 5 5,,, 3 f) function; D 5 5,, 3, ; R 5 5,, 3,. a) function; D 5 5PR; R 5 5 PR $. not a function; D 5 5PR $ ; R 5 5 PR c) function; D 5 5PR; R 5 5 PR $.5 d) not a function; D 5 5PR $ ; R 5 5 PR e) function; D 5 5PR ; R 5 5 PR f) function; D 5 5PR; R 5 5 PR 5. a) 5 3 c) 5 3( ) 5 5 d) 5 5 Answers
20 . a). a) The length is twice the width. f () 5 ; f (7) 5 ; f () 5 5 f (l ) 5 3 l Yes, f (5) 5 f (3) 3 f (5) c) Yes, f () 5 f (3) 3 f () c) f(b) d) Yes, there are others that will work. f (a) 3 f ( 5 f (a 3 whenever a and b have no common factors other than. 3. Answers ma var. For eample: B independent numerical variable model d) length 5 m; width 5 m 7. a) Time (s) D 5 5,,,,,,,,,,,, c) R 5 5, 5, d) It is a function because it passes the vertical line test. e) Height (m) Time (s) Height (m) f) It is not a function because (5, ) and (5, ) are both in the relation.. a) 5(, ), (3, ), (5, ) 5(, ), (3, ), (5, ) c) 5(, ), (, 3), (5, ) 9. If a vertical line passes through a function and hits two points, those two points have identical -coordinates and different -coordinates. This means that one -coordinate is sent to two different elements in the range, violating the definition of function.. a) Yes, because the distance from (, 3) to (, ) is 5. No, because the distance from (, 5) to (, ) is not 5. c) No, because (, 3) and (, 3) are both in the relation.. a) g() 5 3 g(3) g() g(3 ) 5 g() 5 So, g(3) g() g(3 ).. domain range dependent variable The first is not a function because it fails the vertical line test: D 5 5PR 5 # # 5; R 5 5PR 5 # # 5. The second is a function because it passes the vertical line test: D 5 5PR 5 # # 5; R 5 5PR # # is a function of if the graph passes the horizontal line test. This occurs when an horizontal line hits the graph at most once. Lesson., p. FUNCTION function notation. 5,, 5,, 5. a) c) e) 35 d) f) 3. a). 3 c) $ # d) 5. a) mapping model algebraic model graphical model vertical line test c) The absolute value of a number is alwas greater than or equal to. There are no solutions to this inequalit. d) 5. a) # 3 c) $. d),. a) The graphs are the same. Answers ma var. For eample, 5( ), so the are negatives of each other and have the same absolute value. 7. a) c) d). When the number ou are adding or subtracting is inside the absolute value signs, it moves the function to the left (when adding) or to the right (when subtracting) of the origin. When the number ou are Answers Answers 3
21 adding or subtracting is outside the absolute value signs, it moves the function down (when subtracting) or up (when adding) from the origin. The graph of the function will be the absolute value function moved to the left 3 units and up units from the origin. 9. This is the graph of g() 5 horizontall compressed b a factor of and translated unit to the left.. This is the graph of g() 5 horizontall compressed b a factor of reflected over the -ais, translated, units to the right, and translated 3 units up. Lesson.3, pp Answers ma var. For eample, domain because most of the parent functions have all real numbers as a domain.. Answers ma var. For eample, the end behaviour because the onl two that match are and. 3. Given the horizontal asmptote, the function must be derived from. But the asmptote is at 5, so it must have been translated up two. Therefore, the function is f () 5.. a) Both functions are odd, but their domains are different. Both functions have a domain of all real numbers, but sin () has more zeros. c) Both functions have a domain of all real numbers, but different end behaviour. d) Both functions have a domain of all real numbers, but different end behaviour. 5. a) even d) odd odd e) neither even nor odd c) odd f) neither even nor odd. a), because it is a measure of distance from a number sin (), because the heights are periodic. c), because population tends to increase eponentiall d), because there is $ on the first da, $ on the second, $3 on the third, etc. 7. a) f () 5! c) f () 5 f () 5 sin d) f () 5. a) f () g() 5 sin 3 c) h() a) f () 5 ( ) There is not onl one function. f () 5 3 works as well. ( ) c) There is more than one function that satisfies the propert. f () 5 and f () 5 both work.. is a smooth curve, while has a sharp, pointed corner at (, ).. See net page. 3. It is important to name parent functions in order to classif a wide range of functions according to similar behaviour and characteristics. D 5 5PR, R 5 5 f ()PR; interval of increase 5 (`, `), no interval of decrease, no discontinuities, - and -intercept at (, ), odd, S `, S `, and S `, S `. It is ver similar to f () 5. It does not, however, have a constant slope. 5. No, cos is a horizontal translation of sin.. The graph can have,, or zeros. zeros: zero: 3 zeros: Mid-Chapter Review, p.. a) function; D 5 5, 3, 5, 7, R 5 5, 3, function; D 5 5PR, R 5 5PR c) not a function; D 5 5PR 5 # # 5, R 5 5PR 5 # # 5 d) not a function; D 5 5,,, R 5 5, 3,, 7. a) Yes. Ever element in the domain gets sent to eactl one element in the range. D 5 5,,, 3,, 5,, 7,, 9, c) R 5 5,, 5, 3, 35,, 5, 5 Answers
22 . Parent Function f() 5 g() 5 h() 5 k() 5 m() 5! p() 5 r() 5 sin Sketch. r() = sin Domain 5PR 5PR 5PR 5PR 5PR $ 5PR 5PR Range 5f() PR 5f()PR f() $ 5f()PR f() 5f()PR f() $ 5f()PR f() $ 5f()PR f(). 5f()PR # f() # Intervals of Increase (`, `) (, `) None (, `) (, `) (`, `) 39(k ), 9(k 3) KPZ Intervals of Decrease None (`, ) (`, ) (, `) (`, ) None None 39(k 3), 9(k ) KPZ Location of None None 5 None None 5 None Discontinuities and 5 Asmptotes Zeros (, ) (, ) None (, ) (, ) None k KPZ -Intercepts (, ) (, ) None (, ) (, ) (, ) (, ) Smmetr Odd Even Odd Even Neither Neither Odd End Behaviours S `, S ` S `, S ` S `, S S `, S ` S `, S ` S `, S ` Oscillating S `, S ` S `, S ` S `, S S `, S ` S `, S 3 Answers Answers 5
23 3. a) D 5 5PR, R 5 5 f ()PR; This if f () 5 sin translated down ; 5. a) f () 5, translated left function continuous 3 D 5 5PR 3 # # 3, R 5 5PR 3 # # 3; not a function c) D 5 5PR # 5, 3 3 R 5 5PR $ ; function d) D 5 5PR, R 5 5PR $; function c) This is f () 5 translated down ; 3. 3,, 3,, 5 continuous 5. a) f () 5, vertical stretch b c) f () 5 sin, horizontal compression 3 of translation up 3, c) 3 d). a) f () 5 f () 5 c) f () 5 " 7. a) even c) neither odd nor even even d) neither odd nor even. a) This is f () 5 translated right and up 3; discontinuous Lesson., pp a) translation unit down horizontal compression b a factor of, translation unit right c) reflection over the -ais, translation units up, translation 3 units right d) reflection over the -ais, vertical stretch b a factor of, horizontal compression b a factor of e) reflection over the -ais, translation 3 units down, reflection over the -ais, translation units left f) vertical compression b a factor of, translation units up, horizontal stretch b a factor of, translation 5 units right. a) a 5, k 5 d 5, c 5 3, a 5 3, k 5 d 5, c 5, 3. (, 3), (, 3), (, ), (, ), (, ), (, ). a) (, ), (, ), (, ), (, ) (5, 3), (7, 7), (, 5), (, ) c) (, 5), (, 9), (, 7), (, ) d) (, ), (3, ), (3, ), (5, 3) e) (, 5), (, ), (, 3), (, 7) f) (, ), (, ), (, ), (, 5) d) f () 5 translation up 3, e) f () 5, horizontal stretch b f) f () 5 ", horizontal compression b translation right, Answers
24 . a) c) d) e) f) 7. a). D 5 5PR,. R 5 5 f ()PR f () $ h() D 5 5PR, R 5 5 f ()PR f () $ g() D 5 5PR, R 5 5 f ()PR # f () # D 5 5PR, R 5 5 f ()PR f () 3 D 5 5PR, R 5 5 f ()PR f (). D 5 5PR $, f() R 5 5 f ()PR f () $ 3. a) a vertical stretch b a factor of a horizontal compression b a factor of c) () 5 5. Answers ma var. For eample: horizontal stretch or compression, based on value of k The domain remains unchanged at D 5 5PR. The range must now be less than : R 5 5 f ()PR f (),. It changes from increasing on (`, `) to decreasing on (`, `). The end behaviour becomes as S `, S, and as S `, S `. c) g() 5( 3() ) 53" 5 9. a) (3, ) d) (.75, ) (.5, ) e) (, ) c) (, 9) f) (, 7). a) D 5 5PR $, R 5 5 g()pr g() $ D 5 5PR $, R 5 5h()PR h() $ c) D 5 5PR #, R 5 5k()PR k() $ d) D 5 5PR $ 5, R 5 5 j()pr j() $3. 5 5( 3) is the same as 5 5 5, not vertical stretch or compression, based on value of a reflection in -ais if a, ; reflection in -ais if k, horizontal translation based on value of d vertical translation based on value of c 5. (, 5). a) horizontal compression b a factor of 3, translation units to the left because the are equivalent epressions: 3( ) 5 3 c) Lesson.5, pp a) (5, ) c) (, ) e) (, 3) (, 5) d) (, ) f) (7, ). a) D 5 5PR, R 5 5PR D 5 5PR, R 5 5PR $ c) D 5 5PR,, R 5 5PR $5 d) D 5 5PR 5,,, R 5 5PR, 3. A and D match; B and F match; C and E match. a) (, 9) (9, ) c) D 5 5PR, R 5 5PR d) D 5 5PR, R 5 5PR e) Yes; it passes the vertical line test. 5. a) (, ) (, ) c) D 5 5PR, R 5 5PR $ d) D 5 5PR $ R 5 5 PR e) No; (, ) and (, ) are both on the inverse relation.. a) Not a function Not a function c) Function p d) Not a function p p p p p p p 7. a) C 5 5 (F 3); this allows ou to 9 convert from Fahrenheit to Celsius. C 5 F. a) r 5 $ p; A this can be used to determine the radius of a circle when its area is known. A 5 5p cm, r 5 5 cm 9.. k 5 a) 3 c) e) 5 d) f) Answers Answers 7
25 . No; several students could have the same grade point average.. a) f () 5 ( ) 3 h () 5 c) g () 5 " 3 d) m () a) 5 ( 3) c) 5 Å d) (., 3.55), (.,.), (3.55,.), (3., 3.) e) $ 3 because a negative square root is undefined. f) g() 5 5, but g (5) 5 or ; the inverse is not a function if this is the domain of g.. For 5", D 5 5PR $ and R 5 5PR #. For 5, D 5 5PR and R 5 5PR $. The student would be correct if the domain of 5 is restricted to D 5 5PR #. 5. Yes; the inverse of 5 " is 5 so long as the domain of this second function is restricted to D 5 5PR $.. John is correct. Algebraic: ; ; ( ) 5 3 ; 5 " 3 ( ). Numeric: Let Graphical: 3 5 " 3 () 5 " ; 5 " 3 ( ) 5 " 3 ( ) The graphs are reflections over the line f () 5 k works for all kpr. 5 k Switch variables and solve for : 5 k 5 k So the function is its own inverse.. If a horizontal line hits the function in two locations, that means there are two points with equal -values and different -values. When the function is reflected over the line 5 to find the inverse relation, those two points become points with equal -values and different -values, thus violating the definition of a function. Lesson., pp a) c) d) e) f). a) Discontinuous at 5 Discontinuous at 5 c) Discontinuous at 5 d) Continuous e) Discontinuous at 5 f) Discontinuous at 5 and 5 3. a) 3 f() 5 e, if #, if., if, f () 5 e!, if $. a) D 5 5PR; the function is discontinuous at 5. D 5 5PR; the function is continuous. 5. a) The function is discontinuous at 5. D 5 5PR R 5 5, 3 The function is continuous. D 5 5PR R 5 5f ()PR f () $ Answers
26 c). Answers ma var. For eample: Plot the function for the left interval. Plot the function for the right interval.. Answers ma var. For eample: 3, if, a) f () 5, if # #!, if. 5 The function is continuous. D 5 5PR R 5 5f ()PR f () $ d) The function is continuous. D 5 5PR R 5 5 f ()PR # f () # 5 5, if # # 5. f () 5 e 5., if $ 5 7. f () 5.35, if # #.5, if, # 5.55, if. 5. Determine if the plots for the left and right intervals meet at the -value that serves as the common end point for the intervals; if so, the function is continuous at this point. Determine continuit for the two intervals using standard methods. 3, if $3 f () e 3, if,3. discontinuous at p 5 and p 5 5; continuous at, p, 5 and p c) The function is not continuous. The last two pieces do not have the same value for 5. 3, if, d) f () 5, if # #!, if. Lesson.7, pp a) 5(, ), (, 5), (, 5), (, ) 5(, ), (, 3), (, ), (, ) c) 5(, ), (, 3), (, ), (, ) d) 5(, ), (, ), (, ), (, ). a). k 5 9. a) The function is discontinuous at 5. c) 3 fish d) 5 ; 5 5; 5 e) Answers ma var. For eample, three possible events are environmental changes, introduction of a new predator, and increased fishing. 3., if #,, if #, f () 5 f, if #, 3 3, if 3 #,, if #, It is often referred to as a step function because the graph looks like steps.. To make the first two pieces continuous, 5() 5 k, so k 5. But if k 5, the graph is discontinuous at a) Answers Answers 9
27 c) 5. a) a) 5. a) 5 d) 5 3. a) Answers ma var. For eample, properties of the original graphs such as intercepts and sign at various values of the independent variable figure prominentl in the shape of the new function. 7. a) c) f() g() h() 5 f() 3 g() d) h() 5 ( )( ) 5 ; degree is e) D 5 5PR f() g() h() 5 f() 3 g() c) d) h() 5 ( 3)( 5) ; degree is 3 e) D 5 5PR; this is the same as the domain of both f and g. Chapter Review, pp.. a) function; D 5 5PR; R 5 5 PR function; D 5 5PR; R 5 5 PR # 3 c) not a function; D 5 5PR # # ; R 5 5 PR d) function; D 5 5PR. ; R 5 5 PR. a) C(t) 5 3.t D 5 5tPR t $, R 5 5C(t)PR C(t) $ 3 3. D 5 5PR, R 5 5 f ()PR f () $ Answers
28 ., 5. a) Both functions have a domain of all real numbers, but the ranges differ. Both functions are odd but have different domains. c) Both functions have the same domain and range, but is smooth and has a sharp corner at (, ). d) Both functions are increasing on the entire real line, but has a horizontal asmptote while does not.. a) Increasing on (`, `); odd; D 5 5PR; R 5 5 f ()PR Decreasing on (`, ); increasing on (, `); even; D 5 5PR; R 5 5 f ()PR f () $ c) Increasing on (`, `); neither even nor odd; D 5 5PR; R 5 5 f ()PR f (). 7. a) Parent: 5 ; translated left c) Parent: 5 sin ; reflected across the -ais, epanded verticall b a factor of, compressed horizontall b a factor of translated up b 3, 7 Parent: 5!; compressed verticall b a factor of.5, reflected across the -ais, compressed horizontall b a factor of and translated left 7 3, d) Parent: 5 ; reflected across the -ais, compressed horizontall b a factor of and translated down b 3, 5Q 3 R 9. a) (, ) (, ) c) (, 3) d) a 7 5,.3b e) f) (, ) (9, ). a) (, ) (9, ) c) (7, ) d) (7, 5) e) (3, ) f) (, ). a) D 5 5PR,,, R 5 5PR D 5 5PR,, R 5 5PR $ 7. a) The inverse relation is not a function. 3. a) f () 5 g () 5 " 3. The function is continuous; D 5 5PR, R 5 5PR 3, if # 5. f () 5 e, if. ; the function is discontinuous at 5.. In order for f () to be continuous at 5, the two pieces must have the same value when 5. When 5, 5 and The two pieces are not equal when 5, so the function is not continuous at 5. 3, if # 7. a) f () 5 e.3, if. $3.5 c) $3. a) 5(, 7), (, 5) 5(, ), (, ) c) 5(, ), (, 5) 9. a) Answers The inverse relation is a function. c) 3 Answers
29 d) e). a) D C c) A d) B. a) c) d) 3 e) Answers ma var. For eample, (, ) belongs to f, (, ) belongs to g and (, ) belongs to f g. Also, (, 3) belongs to f, (, 5) belongs to g and (, ) belongs to f g. Chapter Self-Test, p.. a) Yes. It passes the vertical line test. D 5 5PR; R 5 5PR $. a) f () 5 or f () f() 9 3 g() (f g)() 3 3 or PR. c) The graph was translated units down. 3. f () 5 3() () f (). has a horizontal asmptote while does not. The range of is while the range of is 5PR $. is increasing on the whole real line and has an interval of decrease and an interval of increase. 5. reflection over the -ais, translation down 5 units, translation left 3 units. horizontal stretch b a factor of, translation unit up; f () 5 if 7. a) (, 7) (5, 3). 9. f () 5 a) $9.5, if # 5 f () 5 e., if. 5. a) f () is discontinuous at 5 because the two pieces do not have the same value when 5. When 5, 5 and " c) Intervals of increase: (`, ), (, `); no intervals of decrease d) D 5 5PR, R 5 5PR,, or $ 3 Chapter Getting Started, p.. a) 3 7. a) Each successive first difference is times the previous first difference. The function is eponential. The second differences are all. The function is quadratic. 3. a) 3 c) 5, 5, d) 7, 9. a) vertical compression b a factor of vertical stretch b a factor of, horizontal translation units to the right c) vertical stretch b a factor of 3, reflection across -ais, vertical translation 7 units up d) vertical stretch b a factor of 5, horizontal translation 3 units to the right, vertical translation units down, 5. a) A 5 (.) t $59.7 c) No, since the interest is compounded each ear, each ear ou earn more interest than the previous ear.. a) 5 m; m s c) 5 m 7. Linear relations constant; same as slope of line; Lesson., pp a) 9 c) 3 e). 5 d) f).. a) i) 5 m> s ii) 5 ms > Nonlinear relations variable; can be positive, positive for negative, or Rates of Change lines that for slope up from left to different parts of the right; negative for same relation lines that slope down from left to right; for horizontal lines. Answers
1.6. Piecewise Functions. LEARN ABOUT the Math. Representing the problem using a graphical model
1. Piecewise Functions YOU WILL NEED graph paper graphing calculator GOAL Understand, interpret, and graph situations that are described b piecewise functions. LEARN ABOUT the Math A cit parking lot uses
More information1.6. Piecewise Functions. LEARN ABOUT the Math. Representing the problem using a graphical model
1.6 Piecewise Functions YOU WILL NEED graph paper graphing calculator GOAL Understand, interpret, and graph situations that are described by piecewise functions. LEARN ABOUT the Math A city parking lot
More information5.3 Graphing Cubic Functions
Name Class Date 5.3 Graphing Cubic Functions Essential Question: How are the graphs of f () = a ( - h) 3 + k and f () = ( 1_ related to the graph of f () = 3? b ( - h) 3 ) + k Resource Locker Eplore 1
More information135 Final Review. Determine whether the graph is symmetric with respect to the x-axis, the y-axis, and/or the origin.
13 Final Review Find the distance d(p1, P2) between the points P1 and P2. 1) P1 = (, -6); P2 = (7, -2) 2 12 2 12 3 Determine whether the graph is smmetric with respect to the -ais, the -ais, and/or the
More informationMATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60
MATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60 A Summar of Concepts Needed to be Successful in Mathematics The following sheets list the ke concepts which are taught in the specified math course. The sheets
More information6. The given function is only drawn for x > 0. Complete the function for x < 0 with the following conditions:
Precalculus Worksheet 1. Da 1 1. The relation described b the set of points {(-, 5 ),( 0, 5 ),(,8 ),(, 9) } is NOT a function. Eplain wh. For questions - 4, use the graph at the right.. Eplain wh the graph
More informationINVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1
Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.
More informationDownloaded from www.heinemann.co.uk/ib. equations. 2.4 The reciprocal function x 1 x
Functions and equations Assessment statements. Concept of function f : f (); domain, range, image (value). Composite functions (f g); identit function. Inverse function f.. The graph of a function; its
More informationExponential and Logarithmic Functions
Chapter 6 Eponential and Logarithmic Functions Section summaries Section 6.1 Composite Functions Some functions are constructed in several steps, where each of the individual steps is a function. For eample,
More informationLESSON EIII.E EXPONENTS AND LOGARITHMS
LESSON EIII.E EXPONENTS AND LOGARITHMS LESSON EIII.E EXPONENTS AND LOGARITHMS OVERVIEW Here s what ou ll learn in this lesson: Eponential Functions a. Graphing eponential functions b. Applications of eponential
More informationC3: Functions. Learning objectives
CHAPTER C3: Functions Learning objectives After studing this chapter ou should: be familiar with the terms one-one and man-one mappings understand the terms domain and range for a mapping understand the
More informationM122 College Algebra Review for Final Exam
M122 College Algebra Review for Final Eam Revised Fall 2007 for College Algebra in Contet All answers should include our work (this could be a written eplanation of the result, a graph with the relevant
More informationFunctions and Graphs CHAPTER INTRODUCTION. The function concept is one of the most important ideas in mathematics. The study
Functions and Graphs CHAPTER 2 INTRODUCTION The function concept is one of the most important ideas in mathematics. The stud 2-1 Functions 2-2 Elementar Functions: Graphs and Transformations 2-3 Quadratic
More information2.5 Library of Functions; Piecewise-defined Functions
SECTION.5 Librar of Functions; Piecewise-defined Functions 07.5 Librar of Functions; Piecewise-defined Functions PREPARING FOR THIS SECTION Before getting started, review the following: Intercepts (Section.,
More informationSolving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form
SECTION 11.3 Solving Quadratic Equations b Graphing 11.3 OBJECTIVES 1. Find an ais of smmetr 2. Find a verte 3. Graph a parabola 4. Solve quadratic equations b graphing 5. Solve an application involving
More informationMath 152, Intermediate Algebra Practice Problems #1
Math 152, Intermediate Algebra Practice Problems 1 Instructions: These problems are intended to give ou practice with the tpes Joseph Krause and level of problems that I epect ou to be able to do. Work
More informationFINAL EXAM REVIEW MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
FINAL EXAM REVIEW MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether or not the relationship shown in the table is a function. 1) -
More information1. a. standard form of a parabola with. 2 b 1 2 horizontal axis of symmetry 2. x 2 y 2 r 2 o. standard form of an ellipse centered
Conic Sections. Distance Formula and Circles. More on the Parabola. The Ellipse and Hperbola. Nonlinear Sstems of Equations in Two Variables. Nonlinear Inequalities and Sstems of Inequalities In Chapter,
More information5.2 Inverse Functions
78 Further Topics in Functions. Inverse Functions Thinking of a function as a process like we did in Section., in this section we seek another function which might reverse that process. As in real life,
More information5.1. A Formula for Slope. Investigation: Points and Slope CONDENSED
CONDENSED L E S S O N 5.1 A Formula for Slope In this lesson ou will learn how to calculate the slope of a line given two points on the line determine whether a point lies on the same line as two given
More information10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED
CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations
More informationI think that starting
. Graphs of Functions 69. GRAPHS OF FUNCTIONS One can envisage that mathematical theor will go on being elaborated and etended indefinitel. How strange that the results of just the first few centuries
More informationMore Equations and Inequalities
Section. Sets of Numbers and Interval Notation 9 More Equations and Inequalities 9 9. Compound Inequalities 9. Polnomial and Rational Inequalities 9. Absolute Value Equations 9. Absolute Value Inequalities
More informationColegio del mundo IB. Programa Diploma REPASO 2. 1. The mass m kg of a radio-active substance at time t hours is given by. m = 4e 0.2t.
REPASO. The mass m kg of a radio-active substance at time t hours is given b m = 4e 0.t. Write down the initial mass. The mass is reduced to.5 kg. How long does this take?. The function f is given b f()
More informationAlgebra II. Administered May 2013 RELEASED
STAAR State of Teas Assessments of Academic Readiness Algebra II Administered Ma 0 RELEASED Copright 0, Teas Education Agenc. All rights reserved. Reproduction of all or portions of this work is prohibited
More informationWhen I was 3.1 POLYNOMIAL FUNCTIONS
146 Chapter 3 Polnomial and Rational Functions Section 3.1 begins with basic definitions and graphical concepts and gives an overview of ke properties of polnomial functions. In Sections 3.2 and 3.3 we
More informationSolving Absolute Value Equations and Inequalities Graphically
4.5 Solving Absolute Value Equations and Inequalities Graphicall 4.5 OBJECTIVES 1. Draw the graph of an absolute value function 2. Solve an absolute value equation graphicall 3. Solve an absolute value
More informationFunctions and Their Graphs
3 Functions and Their Graphs On a sales rack of clothes at a department store, ou see a shirt ou like. The original price of the shirt was $00, but it has been discounted 30%. As a preferred shopper, ou
More informationSolving Systems of Equations
Solving Sstems of Equations When we have or more equations and or more unknowns, we use a sstem of equations to find the solution. Definition: A solution of a sstem of equations is an ordered pair that
More information4.9 Graph and Solve Quadratic
4.9 Graph and Solve Quadratic Inequalities Goal p Graph and solve quadratic inequalities. Your Notes VOCABULARY Quadratic inequalit in two variables Quadratic inequalit in one variable GRAPHING A QUADRATIC
More informationSECTION 2.2. Distance and Midpoint Formulas; Circles
SECTION. Objectives. Find the distance between two points.. Find the midpoint of a line segment.. Write the standard form of a circle s equation.. Give the center and radius of a circle whose equation
More informationQuadratic Equations and Functions
Quadratic Equations and Functions. Square Root Propert and Completing the Square. Quadratic Formula. Equations in Quadratic Form. Graphs of Quadratic Functions. Verte of a Parabola and Applications In
More informationSECTION 5-1 Exponential Functions
354 5 Eponential and Logarithmic Functions Most of the functions we have considered so far have been polnomial and rational functions, with a few others involving roots or powers of polnomial or rational
More informationLesson 9.1 Solving Quadratic Equations
Lesson 9.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with a. One -intercept and all nonnegative y-values. b. The verte in the third quadrant and no -intercepts. c. The verte
More informationEQUATIONS OF LINES IN SLOPE- INTERCEPT AND STANDARD FORM
. Equations of Lines in Slope-Intercept and Standard Form ( ) 8 In this Slope-Intercept Form Standard Form section Using Slope-Intercept Form for Graphing Writing the Equation for a Line Applications (0,
More informationGraphing Quadratic Equations
.4 Graphing Quadratic Equations.4 OBJECTIVE. Graph a quadratic equation b plotting points In Section 6.3 ou learned to graph first-degree equations. Similar methods will allow ou to graph quadratic equations
More informationTo Be or Not To Be a Linear Equation: That Is the Question
To Be or Not To Be a Linear Equation: That Is the Question Linear Equation in Two Variables A linear equation in two variables is an equation that can be written in the form A + B C where A and B are not
More informationMATH 185 CHAPTER 2 REVIEW
NAME MATH 18 CHAPTER REVIEW Use the slope and -intercept to graph the linear function. 1. F() = 4 - - Objective: (.1) Graph a Linear Function Determine whether the given function is linear or nonlinear..
More informationShake, Rattle and Roll
00 College Board. All rights reserved. 00 College Board. All rights reserved. SUGGESTED LEARNING STRATEGIES: Shared Reading, Marking the Tet, Visualization, Interactive Word Wall Roller coasters are scar
More informationPOLYNOMIAL FUNCTIONS
POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a
More informationLinear Equations in Two Variables
Section. Sets of Numbers and Interval Notation 0 Linear Equations in Two Variables. The Rectangular Coordinate Sstem and Midpoint Formula. Linear Equations in Two Variables. Slope of a Line. Equations
More informationIn this this review we turn our attention to the square root function, the function defined by the equation. f(x) = x. (5.1)
Section 5.2 The Square Root 1 5.2 The Square Root In this this review we turn our attention to the square root function, the function defined b the equation f() =. (5.1) We can determine the domain and
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION INTEGRATED ALGEBRA. Tuesday, January 24, 2012 9:15 a.m. to 12:15 p.m.
INTEGRATED ALGEBRA The Universit of the State of New York REGENTS HIGH SCHOOL EXAMINATION INTEGRATED ALGEBRA Tuesda, Januar 4, 01 9:15 a.m. to 1:15 p.m., onl Student Name: School Name: Print our name and
More informationUse order of operations to simplify. Show all steps in the space provided below each problem. INTEGER OPERATIONS
ORDER OF OPERATIONS In the following order: 1) Work inside the grouping smbols such as parenthesis and brackets. ) Evaluate the powers. 3) Do the multiplication and/or division in order from left to right.
More informationNorth Carolina Community College System Diagnostic and Placement Test Sample Questions
North Carolina Communit College Sstem Diagnostic and Placement Test Sample Questions 0 The College Board. College Board, ACCUPLACER, WritePlacer and the acorn logo are registered trademarks of the College
More informationZeros of Polynomial Functions. The Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra. zero in the complex number system.
_.qd /7/ 9:6 AM Page 69 Section. Zeros of Polnomial Functions 69. Zeros of Polnomial Functions What ou should learn Use the Fundamental Theorem of Algebra to determine the number of zeros of polnomial
More informationSECTION 2-2 Straight Lines
- Straight Lines 11 94. Engineering. The cross section of a rivet has a top that is an arc of a circle (see the figure). If the ends of the arc are 1 millimeters apart and the top is 4 millimeters above
More informationZero and Negative Exponents and Scientific Notation. a a n a m n. Now, suppose that we allow m to equal n. We then have. a am m a 0 (1) a m
0. E a m p l e 666SECTION 0. OBJECTIVES. Define the zero eponent. Simplif epressions with negative eponents. Write a number in scientific notation. Solve an application of scientific notation We must have
More informationHigher. Polynomials and Quadratics 64
hsn.uk.net Higher Mathematics UNIT OUTCOME 1 Polnomials and Quadratics Contents Polnomials and Quadratics 64 1 Quadratics 64 The Discriminant 66 3 Completing the Square 67 4 Sketching Parabolas 70 5 Determining
More informationCore Maths C2. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...
More informationPolynomial Degree and Finite Differences
CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial
More informationSection 1-4 Functions: Graphs and Properties
44 1 FUNCTIONS AND GRAPHS I(r). 2.7r where r represents R & D ependitures. (A) Complete the following table. Round values of I(r) to one decimal place. r (R & D) Net income I(r).66 1.2.7 1..8 1.8.99 2.1
More information2.3 TRANSFORMATIONS OF GRAPHS
78 Chapter Functions 7. Overtime Pa A carpenter earns $0 per hour when he works 0 hours or fewer per week, and time-and-ahalf for the number of hours he works above 0. Let denote the number of hours he
More informationSLOPE OF A LINE 3.2. section. helpful. hint. Slope Using Coordinates to Find 6% GRADE 6 100 SLOW VEHICLES KEEP RIGHT
. Slope of a Line (-) 67. 600 68. 00. SLOPE OF A LINE In this section In Section. we saw some equations whose graphs were straight lines. In this section we look at graphs of straight lines in more detail
More informationTHE POWER RULES. Raising an Exponential Expression to a Power
8 (5-) Chapter 5 Eponents and Polnomials 5. THE POWER RULES In this section Raising an Eponential Epression to a Power Raising a Product to a Power Raising a Quotient to a Power Variable Eponents Summar
More informationLinear Inequality in Two Variables
90 (7-) Chapter 7 Sstems of Linear Equations and Inequalities In this section 7.4 GRAPHING LINEAR INEQUALITIES IN TWO VARIABLES You studied linear equations and inequalities in one variable in Chapter.
More informationAlgebra II Notes Piecewise Functions Unit 1.5. Piecewise linear functions. Math Background
Piecewise linear functions Math Background Previousl, ou Related a table of values to its graph. Graphed linear functions given a table or an equation. In this unit ou will Determine when a situation requiring
More informationSummer Math Exercises. For students who are entering. Pre-Calculus
Summer Math Eercises For students who are entering Pre-Calculus It has been discovered that idle students lose learning over the summer months. To help you succeed net fall and perhaps to help you learn
More informationImagine a cube with any side length. Imagine increasing the height by 2 cm, the. Imagine a cube. x x
OBJECTIVES Eplore functions defined b rddegree polnomials (cubic functions) Use graphs of polnomial equations to find the roots and write the equations in factored form Relate the graphs of polnomial equations
More informationFlorida Algebra I EOC Online Practice Test
Florida Algebra I EOC Online Practice Test Directions: This practice test contains 65 multiple-choice questions. Choose the best answer for each question. Detailed answer eplanations appear at the end
More informationCore Maths C3. Revision Notes
Core Maths C Revision Notes October 0 Core Maths C Algebraic fractions... Cancelling common factors... Multipling and dividing fractions... Adding and subtracting fractions... Equations... 4 Functions...
More informationMathematical goals. Starting points. Materials required. Time needed
Level A7 of challenge: C A7 Interpreting functions, graphs and tables tables Mathematical goals Starting points Materials required Time needed To enable learners to understand: the relationship between
More informationSo, using the new notation, P X,Y (0,1) =.08 This is the value which the joint probability function for X and Y takes when X=0 and Y=1.
Joint probabilit is the probabilit that the RVs & Y take values &. like the PDF of the two events, and. We will denote a joint probabilit function as P,Y (,) = P(= Y=) Marginal probabilit of is the probabilit
More informationStart Accuplacer. Elementary Algebra. Score 76 or higher in elementary algebra? YES
COLLEGE LEVEL MATHEMATICS PRETEST This pretest is designed to give ou the opportunit to practice the tpes of problems that appear on the college-level mathematics placement test An answer ke is provided
More informationThe Slope-Intercept Form
7.1 The Slope-Intercept Form 7.1 OBJECTIVES 1. Find the slope and intercept from the equation of a line. Given the slope and intercept, write the equation of a line. Use the slope and intercept to graph
More informationSection V.2: Magnitudes, Directions, and Components of Vectors
Section V.: Magnitudes, Directions, and Components of Vectors Vectors in the plane If we graph a vector in the coordinate plane instead of just a grid, there are a few things to note. Firstl, directions
More informationGraphing Linear Equations
6.3 Graphing Linear Equations 6.3 OBJECTIVES 1. Graph a linear equation b plotting points 2. Graph a linear equation b the intercept method 3. Graph a linear equation b solving the equation for We are
More informationSection 2-3 Quadratic Functions
118 2 LINEAR AND QUADRATIC FUNCTIONS 71. Celsius/Fahrenheit. A formula for converting Celsius degrees to Fahrenheit degrees is given by the linear function 9 F 32 C Determine to the nearest degree the
More informationChapter 6 Quadratic Functions
Chapter 6 Quadratic Functions Determine the characteristics of quadratic functions Sketch Quadratics Solve problems modelled b Quadratics 6.1Quadratic Functions A quadratic function is of the form where
More informationDouble Integrals in Polar Coordinates
Double Integrals in Polar Coordinates. A flat plate is in the shape of the region in the first quadrant ling between the circles + and +. The densit of the plate at point, is + kilograms per square meter
More informationDISTANCE, CIRCLES, AND QUADRATIC EQUATIONS
a p p e n d i g DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS DISTANCE BETWEEN TWO POINTS IN THE PLANE Suppose that we are interested in finding the distance d between two points P (, ) and P (, ) in the
More informationStudents Currently in Algebra 2 Maine East Math Placement Exam Review Problems
Students Currently in Algebra Maine East Math Placement Eam Review Problems The actual placement eam has 100 questions 3 hours. The placement eam is free response students must solve questions and write
More information6706_PM10SB_C4_CO_pp192-193.qxd 5/8/09 9:53 AM Page 192 192 NEL
92 NEL Chapter 4 Factoring Algebraic Epressions GOALS You will be able to Determine the greatest common factor in an algebraic epression and use it to write the epression as a product Recognize different
More informationThe Big Picture. Correlation. Scatter Plots. Data
The Big Picture Correlation Bret Hanlon and Bret Larget Department of Statistics Universit of Wisconsin Madison December 6, We have just completed a length series of lectures on ANOVA where we considered
More informationWhy should we learn this? One real-world connection is to find the rate of change in an airplane s altitude. The Slope of a Line VOCABULARY
Wh should we learn this? The Slope of a Line Objectives: To find slope of a line given two points, and to graph a line using the slope and the -intercept. One real-world connection is to find the rate
More informationD.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review
D0 APPENDIX D Precalculus Review SECTION D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane An ordered pair, of real numbers has as its
More information2.7 Applications of Derivatives to Business
80 CHAPTER 2 Applications of the Derivative 2.7 Applications of Derivatives to Business and Economics Cost = C() In recent ears, economic decision making has become more and more mathematicall oriented.
More information2.3 Quadratic Functions
88 Linear and Quadratic Functions. Quadratic Functions You ma recall studing quadratic equations in Intermediate Algebra. In this section, we review those equations in the contet of our net famil of functions:
More informationSAMPLE. Polynomial functions
Objectives C H A P T E R 4 Polnomial functions To be able to use the technique of equating coefficients. To introduce the functions of the form f () = a( + h) n + k and to sketch graphs of this form through
More informationACT Math Vocabulary. Altitude The height of a triangle that makes a 90-degree angle with the base of the triangle. Altitude
ACT Math Vocabular Acute When referring to an angle acute means less than 90 degrees. When referring to a triangle, acute means that all angles are less than 90 degrees. For eample: Altitude The height
More informationSkills Practice Skills Practice for Lesson 1.1
Skills Practice Skills Practice for Lesson. Name Date Tanks a Lot Introduction to Linear Functions Vocabular Define each term in our own words.. function A function is a relation that maps each value of
More informationSECTION 7-4 Algebraic Vectors
7-4 lgebraic Vectors 531 SECTIN 7-4 lgebraic Vectors From Geometric Vectors to lgebraic Vectors Vector ddition and Scalar Multiplication Unit Vectors lgebraic Properties Static Equilibrium Geometric vectors
More informationTHE PARABOLA 13.2. section
698 (3 0) Chapter 3 Nonlinear Sstems and the Conic Sections 49. Fencing a rectangle. If 34 ft of fencing are used to enclose a rectangular area of 72 ft 2, then what are the dimensions of the area? 50.
More informationax 2 by 2 cxy dx ey f 0 The Distance Formula The distance d between two points (x 1, y 1 ) and (x 2, y 2 ) is given by d (x 2 x 1 )
SECTION 1. The Circle 1. OBJECTIVES The second conic section we look at is the circle. The circle can be described b using the standard form for a conic section, 1. Identif the graph of an equation as
More information3 e) x f) 2. Precalculus Worksheet P.1. 1. Complete the following questions from your textbook: p11: #5 10. 2. Why would you never write 5 < x > 7?
Precalculus Worksheet P.1 1. Complete the following questions from your tetbook: p11: #5 10. Why would you never write 5 < > 7? 3. Why would you never write 3 > > 8? 4. Describe the graphs below using
More informationPolynomial and Rational Functions
Polnomial and Rational Functions 3 A LOOK BACK In Chapter, we began our discussion of functions. We defined domain and range and independent and dependent variables; we found the value of a function and
More informationFunctions and their Graphs
Functions and their Graphs Functions All of the functions you will see in this course will be real-valued functions in a single variable. A function is real-valued if the input and output are real numbers
More information6.3 PARTIAL FRACTIONS AND LOGISTIC GROWTH
6 CHAPTER 6 Techniques of Integration 6. PARTIAL FRACTIONS AND LOGISTIC GROWTH Use partial fractions to find indefinite integrals. Use logistic growth functions to model real-life situations. Partial Fractions
More informationPre Calculus Math 40S: Explained!
Pre Calculus Math 0S: Eplained! www.math0s.com 0 Logarithms Lesson PART I: Eponential Functions Eponential functions: These are functions where the variable is an eponent. The first tpe of eponential graph
More informationChapter 4. Polynomial and Rational Functions. 4.1 Polynomial Functions and Their Graphs
Chapter 4. Polynomial and Rational Functions 4.1 Polynomial Functions and Their Graphs A polynomial function of degree n is a function of the form P = a n n + a n 1 n 1 + + a 2 2 + a 1 + a 0 Where a s
More informationLinear and Quadratic Functions
Chapter Linear and Quadratic Functions. Linear Functions We now begin the stud of families of functions. Our first famil, linear functions, are old friends as we shall soon see. Recall from Geometr that
More informationMATH 60 NOTEBOOK CERTIFICATIONS
MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5
More informationCRLS Mathematics Department Algebra I Curriculum Map/Pacing Guide
Curriculum Map/Pacing Guide page 1 of 14 Quarter I start (CP & HN) 170 96 Unit 1: Number Sense and Operations 24 11 Totals Always Include 2 blocks for Review & Test Operating with Real Numbers: How are
More informationSolving Special Systems of Linear Equations
5. Solving Special Sstems of Linear Equations Essential Question Can a sstem of linear equations have no solution or infinitel man solutions? Using a Table to Solve a Sstem Work with a partner. You invest
More information7.7 Solving Rational Equations
Section 7.7 Solving Rational Equations 7 7.7 Solving Rational Equations When simplifying comple fractions in the previous section, we saw that multiplying both numerator and denominator by the appropriate
More informationMath Review. The second part is a refresher of some basic topics for those who know how but lost their fluency over the years.
Math Review The Math Review is divided into two parts: I. The first part is a general overview of the math classes, their sequence, basic content, and short quizzes to see if ou are prepared to take a
More informationEllington High School Principal
Mr. Neil Rinaldi Ellington High School Principal 7 MAPLE STREET ELLINGTON, CT 0609 Mr. Dan Uriano (860) 896- Fa (860) 896-66 Assistant Principal Mr. Peter Corbett Lead Teacher Mrs. Suzanne Markowski Guidance
More informationEQUATIONS and INEQUALITIES
EQUATIONS and INEQUALITIES Linear Equations and Slope 1. Slope a. Calculate the slope of a line given two points b. Calculate the slope of a line parallel to a given line. c. Calculate the slope of a line
More informationPrentice Hall Mathematics: Algebra 2 2007 Correlated to: Utah Core Curriculum for Math, Intermediate Algebra (Secondary)
Core Standards of the Course Standard 1 Students will acquire number sense and perform operations with real and complex numbers. Objective 1.1 Compute fluently and make reasonable estimates. 1. Simplify
More informationAx 2 Cy 2 Dx Ey F 0. Here we show that the general second-degree equation. Ax 2 Bxy Cy 2 Dx Ey F 0. y X sin Y cos P(X, Y) X
Rotation of Aes ROTATION OF AES Rotation of Aes For a discussion of conic sections, see Calculus, Fourth Edition, Section 11.6 Calculus, Earl Transcendentals, Fourth Edition, Section 1.6 In precalculus
More information