Warm Up /6. Describe the pattern of the graph of each of the following situations as the graphs are read from left to right as increasing, decreasing, increasing and then decreasing, or decreasing and then increasing. a. The height of a child at birth and on each birthda from age to age 6 b. The height of a ball that is thrown upward from the top of a building from the time it is thrown until it hits the ground
Agenda /6. Warm Up. Finish Func9ons. Domain and Range Review. Piecewise Func9ons
Situation Graph Formula A. Plumber A plumber charges a fied fee for coming to our house, then charges a fied amount per hour on top of this. = the time the job takes in hours. = the total cost of the plumber s time in dollars. G 00" 0" 00" 0" 00" 0" 00" 0" 0" 0" " " " " " 6" 7" 8" 9" Step function is better A = 0 + 60 Plumber charges $80 for a - hour job. B. Ccling A cclist travels along a direct route from town A to town B. = the distance of the cclist from town A in miles. = the distance of the cclist from town B in miles. G0 0" 00" 80" 60" 0" 0" 0" 0" 0" 0" 60" 80" 00" 0" A = +00 Towns are 00 miles apart. C. Movie subscription You get two movies free, but then ou get charged at a fied rate per movie. = the number of movies seen. G 60" 0" 0" 0" 0" A = 0 The fied rate per movie is $. = the total mone spent in dollars. 0" 0" 0" " " 6" 8" 0" " " Graph to be drawn on b student. Discrete points would be better.
D. Internet café An internet café charges a fied amount per minute to use the internet. = the number of minutes spent on the internet. = the cost of using the internet in dollars. E. Cooling kettle A kettle of boiling water cools in a warm kitchen. = the time that has elapsed in minutes. = the temperature of the kettle in degrees Celsius. G7 G9 7000" 6000" 000" 000" 000" 000" 000" 0" 0" 9" 8" 7" 6" " " " " " 0" 0" " " 6" 8" 0" " " 0" " " " " " 6" 7" 8" 9" 0" A = $8 will bu 6 minutes. A = 0 + 80 (0.7) As the water cools it approaches the room temperature of 0 C.
F. Ferris wheel A ferris wheel turns round and round. G )!" (!" '!" A0 = 0 + 0sin(8) = the time that has elapsed in seconds. = the height of a seat from the ground in meters. &!" %!" $!" #!"!"!" '" #!" #'" $!" $'" %!" %'" &!" The starting point is = 0 and = 0. After one complete turn = 0.The Ferris wheel is at this height at = 0 (half a turn) and = 0 (complete turn). It takes 0 seconds for the Ferris wheel to turn once. G. Folding paper A piece of paper is folded in half. It is then folded in half again, and again. = the number of folds. = the thickness of the paper in inches. G 00" 0" 00" 0" 00" 0" 0" 0" " " " " " 6" 7" 8" 9" 0" The student ma change the shape of the curve. Paper can be folded up to 8 times. Discrete points would be better. A = 000 Folding it ten times results in thickness of about an inch. This is impossible! 8 in practice (tr it!).
H. Speed of golf shot A golfer hits a ball. = the time that has elapsed in seconds. = the speed of the ball in meters per second. I. Test drive A car drives along a test track. = the average speed of the car in meters per second. = the time it takes to travel the length of the track in seconds. J. Balloon A man blows up a balloon. = volume of air he has blown in cubic inches. = diameter of the balloon in inches. G8 G G6 better. '#" '!" &#" &!" %#" %!" $#" $!" #"!" 00" 80" 60" 0" 0" 00" 80" 60" 0" 0"." 0" "." "." "." " 0." 0"!" $" %" &" '" #" (" 0" " 0" " 0" " 0" 0" " 0" " 0" " 0" If puffs are allowed the graph will show steps. At some point the balloon will pop. A7 = 0 ( ) + 7 Speed is a minimum after seconds. A = 00 Distance = speed time =. The track is 00 meters long. A6 = =,000. Diameter of balloon is about foot (. inches).
K. Height of golf shot. A golfer hits a ball. = the time that has elapsed in seconds. = the height of the ball in meters. L. Film projector A film is shown on a screen using a small projector. = the distance from the projector to the screen in feet. = the area of the picture in square feet. G G #!" '#" '!" &#" &!" %#" %!" $#" $!" #"!" 0" 00" 80" 60" 0" 0" 0"!" $" %" &" '" #" (" 0" " " 6" 8" 0" " " 6" 8" 0" A9 = 0 The ball hits the ground when: (6 ) = 0; = 0 or = 6. Ball hits ground after 6 seconds. A8 = When projector is 0 feet awa, area of picture is square feet.
Identif the domain and range of a function from its graph. Identifing Domain and Range from a Function s Graph GREAT QUESTION! I peeked below and saw that ou are using interval notation. What should I alread know about this notation? Recall that square brackets indicate endpoints that are included in an interval. Parentheses indicate endpoints that are not included in an interval. Parentheses are alwas used with or -. For more detail on interval notation, see Section P.9, pages 0. Figure. illustrates how the graph of a function is used to determine the function s domain and its range. Domain: set of inputs Range = f() Found on the -ais Range: set of outputs Found on the -ais Domain FIGURE. Domain and range of f
= f() Range: Outputs on -ais etend from to, inclusive. Let s appl these ideas to the graph of the function shown in Figure.6. To find the domain, look for all the inputs on the @ais that correspond to points on the graph. Can ou see that the etend from - to, inclusive? The function s domain can be represented as follows: Domain: Inputs on -ais etend from to, inclusive. The set of all Using Set-Builder Notation { } [, ]. such that is greater than or equal to and less than or equal to. Using Interval Notation The square brackets indicate and are included. Note the square brackets on the -ais in Figure.6. FIGURE.6 Domain and range of f To find the range, look for all the outputs on the @ais that correspond to points on the graph. The etend from to, inclusive. The function s range can be represented as follows: Using Set-Builder Notation { } Using Interval Notation [, ]. The set of all such that is greater than or equal to and less than or equal to. The square brackets indicate and are included. Note the square brackets on the -ais in Figure.6.
EXAMPLE 8 Identifing the Domain and Range of a Function from Its Graph Use the graph of each function to identif its domain and its range. a. = f() b. = f() c. = f() d. e. = f() = f() SOLUTION
SOLUTION For the graph of each function, the domain is highlighted in purple on the @ais and the range is highlighted in green on the @ais. a. Range: Outputs on -ais etend from 0 to, inclusive. = f() Domain: Inputs on -ais etend from to, inclusive. b. = f() Range: Outputs on -ais etend from, ecluding, to, including. Domain: Inputs on -ais etend from, ecluding, to, including. c. = f() Range: Outputs on -ais etend from to, inclusive. Domain: Inputs on -ais etend from, including, to, ecluding. Do main = { - } or [-, ] Ran ge = { 0 } or [0, ] GREAT QUESTION! The range in Eample 8(e) was identified as {,, }. Wh didn t ou also use interval notation like ou did in the other parts of Eample 8? Interval notation is not appropriate for describing a set of distinct numbers such as {,, }. Interval notation, [, ], would mean that numbers such as. and.99 are in the range, but the are not. That s wh we onl used set-builder notation. Do main = { - 6 } or (-, ] Ran ge = { 6 } or (, ] d. Range: Outputs on -ais include real numbers greater than or equal to O. = f() Domain: Inputs on -ais include real numbers less than or equal to. Do main = { } or (-, ] Ran ge = { Ú 0} or [0, ) Do main = { - 6 } or [-, ) Ran ge = { } or [, ] e. Range: Outputs on -ais "step" from to to. = f() Domain: Inputs on -ais etend from, including, to, ecluding. Do main = { 6 } or [, ) Ran ge = { =,, }
Check Point 8 Use the graph of each function to identif its domain and its range. a. b. c. = f() = f() = f()
Understand and use piecewise functions. Piecewise Functions A cellphone compan offers the following plan: $0 per month bus 60 minutes. Additional time costs $0.0 per minute. We can represent this plan mathematicall b writing the total monthl cost, C, as a function of the number of calling minutes, t. C(t) 80 60 0 0 C(t) = 0 if 0 t 60 0 FIGURE. C(t) = 0 + 0.0 (t 60) if t > 60 80 0 60 00 t 0 if 0 t 60 C(t)= e 0+0.0(t-60) if t>60 $0 for first 60 minutes $0.0 per minute times the number of calling minutes eceeding 60 The cost is $0 for up to and including 60 calling minutes. The cost is $0 plus $0.0 per minute for additional time for more than 60 calling minutes. A function that is defined b two (or more) equations over a specified domain is called a piecewise function. Man cellphone plans can be represented with piecewise functions. The graph of the piecewise function described above is shown in Figure..
EXAMPLE Evaluating a Piecewise Function Use the function that describes the cellphone plan C(t) = b to find and interpret each of the following: a. C(0) b. C(00). 0 if 0 t 60 0 + 0.0(t - 60) if t 7 60
SOLUTION a. To find C(0), we let t = 0. Because 0 lies between 0 and 60, we use the first line of the piecewise function. C(t) = 0 This is the function s equation for 0 t 60. C(0) = 0 Replace t with 0. Regardless of this function s input, the constant output is 0. This means that with 0 calling minutes, the monthl cost is $0. This can be visuall represented b the point (0, 0) on the first piece of the graph in Figure.. b. To find C(00), we let t = 00. Because 00 is greater than 60, we use the second line of the piecewise function. C(t) = 0 + 0.0(t - 60) This is the function s equation for t 7 60. C(00) = 0 + 0.0(00-60) Replace t with 00. = 0 + 0.0(0) Subtract within parentheses: 00-60 = 0. = 0 + 6 Multipl: 0.0(0) = 6. = 6 Add: 0 + 6 = 6. Thus, C(00) = 6. This means that with 00 calling minutes, the monthl cost is $6. This can be visuall represented b the point (00, 6) on the second piece of the graph in Figure..
Check Point Use the function in Eample to find and interpret each of the following: a. C(0) b. C(80). Identif our solutions b points on the graph in Figure.. b
EXAMPLE Graphing a Piecewise Function Graph the piecewise function defined b SOLUTION f() = b + if if 7.
b GREAT QUESTION! When I graphed the function in Eample, here s what I got: SOLUTION We graph f in two parts, using a partial table of coordinates to create each piece. The tables of coordinates and the completed graph are shown in Figure.. Is m graph ok? No. You incorrectl ignored the domain for each piece of the function and graphed each equation as if its domain was (-, ). Graph f() = + for. Domain = (, ] ( ) 0 f() f()=+= f(0)=0+= f( )= += f( )= +=0 f( )= += f( )= += (, f()) (, ) (0, ) (, ) (, 0) (, ) (, ) FIGURE. Graphing a piecewise function Graph f() = for >. Domain = (, ) ( 7 ). f() f(.)= f()= f()= f()= f()= (, f()) (., ) (, ) (, ) (, ) (, )
FIGURE. (repeated) We can use the graph of the piecewise function in Figure. to find the range of f. The range of the blue piece on the left is { }. The range of the red horizontal piece on the right is { = }. Thus, the range of f is { } { = }, or (-, ] {}. Check Point Graph the piecewise function defined b if - f() = b - if 7-. Some piecewise functions are called step functions because their graphs form discontinuous steps. One such function is called the greatest integer function, smbolized b int() or Œœ, where For eample, int() = the greatest integer that is less than or equal to.
Œ œ For eample, int()=, int(.)=, int(.)=, int(.9)=. Here are some additional eamples: is the greatest integer that is less than or equal to,.,., and.9. f() = int() FIGURE. The graph of the greatest integer function int()=, int(.)=, int(.)=, int(.9)=. is the greatest integer that is less than or equal to,.,., and.9. Notice how we jumped from to in the function values for int(). In particular, If 6, then int() =. If 6, then int() =. The graph of f() = int() is shown in Figure.. The graph of the greatest integer function jumps verticall one unit at each integer. However, the graph is constant between each pair of consecutive integers. The rightmost horizontal step shown in the graph illustrates that If 6 6, then int() =. In general, If n 6 n +, where n is an integer, then int() = n.
Due Dates Four Situa9ons (/7) Unit 7 Project Report (/9- printed) Presenta9ons (/)