lfg skething functions

8/9/2019 Lfg Skething Functions

1/13

BI SKETCHING GRAPHS FROM TABLESTry to make up tables of numbers which will correspond to thefollowing six graphs: (They do not need to rep resent realsituations).

In this booklet, you will he asked to explore several tables of data.and attempt to discover any patterns or t rends that they contain.How far can you see?

4

Look carefully at the table shown above.

Balloons Distance toheight the horizon(m) (km)

5 8l() II20 1630 2040 2350 25100 36500 801000 11 2

Without accurately plotting the points, try to sketch a roughgraph to descr ibe the relationship between the balloon\ height.and the distance to the horizon.Distanceto thehoriLon

Explain your method [or doing this.

0 xy

0y

() x 0 x

Now make up some tables of your own, and sketch thecorresponding graphs on a separate sheet of paper. (Againthey do not need to represent real situations). Pass onlythe tables to your neighbour.She must now try to sketch graphs from you r tables.Compare her solutions with yours. Balloons height

Shell Centre for Ma ema ica l Education. University of Nottingham. 1985


2/13

Time (minutes) 0 5 10 IS 20 25 30jj~iperaIure (C) 9(1 79 74) 62 55 49 .14

I. Cooling Coffee

2. Cooking Times for TurkeyWeight(lh) I 6181)0112 I~ 116118 121)1Time(hours) 22 3 3~ 41 , ~ 5~ 6

3. How a Baby Grew Before BirthI I ~I ~ 161 7J8J9J9ImnI24I30i~4I~I42I

4. After Three Pints of Beer...Time(hours) Jl 1213141516171Alcoholinthe I F I I I Iblood (mg/lOOmI) I I I ~ IS

5. Number of Bird Species on a Volcanic Island( Year 1881) 894) I 194H)I 1910 11920 11931)1194(11

Age (sears) Number o1 Age (years ) Number olSurvivors Survivors

0 1000 50 9135 979 60 808

10 978 70 57920 972 80 24830 963 90 3240 950 100 I

Without plotting, choose the best sketch graph (from theselection on page 3) to fit each of the tables shown below.Particular graphs may fit more than one tab le . Copy themost suitable graph, name the axes clearly, and explain yourchoice. If you cannot f ind the graph you want, draw yourown vers ion.

Age (months) 2~,, Length (cm) 4

(a ) (d)_ _ J7v~Numberol - I S 17 30 IIISpecies

6. Life Expectancy

Without plotting, try and sketch a g raph to illustrate the followingtable:

How daylight Nsummer temperaturevaries as yougo higher inthe atmosphere~2

Alitlude Temperature Altitude Temperature4km) (C) (km) ( C I

I) 211 N) 12II) 48 74)21) 511 HI) $1134) 38 441 MI411 IS lIE 75SI) 6 I III 211

C)I Shell Centre or Mathematical I ducation. Untversit~ ol Nottingham. 1985,


3/13

Bi. (contd) SOME HINTS ON SKETCHING GRAPHS FROM TABLESLook again at the balloon problem, How far can you see?

The following discussion should help you to see how you can go about sketchingquick graphs from tables without having.to spend a long time plotting points.* As the balloons height increases by equal amounts, what happens to the distance

to the horizon? Does it increase or decrease?nnfl ~s

Balloons height (m) 5 10 20 30 40 50 100 500 1000Distance to horizon (km) 8 11 16 20 23 25 36 80 11 2~4

CCDoes this distanceincrease by equal amounts?...

ill 20 30 40Balloons height

.or increase by greaterand greater amounts?.. .

10 2(1 3(1 40Balloons heightor increase by smaller andsmaller amounts?

U


4/13

B2 FINDING FUNCTIONS IN SITUATIONSFor each of the two situations which follow,

(i) Describe your answer by sketching a rough graph.(ii) Explain the shape of )our graph in words.(iii) Check your graph by constructing a table of values, andredraw it if necessary.(iv) Try t o f ind an algebraic formula.

The OutingA coach hire firm otters lo loanluxury coach for I 2(1 per da~ Theorganiser ot the irip decides to chareeevery member of the p.irts an equala moun I br t he ride -I-fo~ ~ ill the i/c of each personconirihution depend U[)Lifl ihe siteob the parR

Developing Photographs~ Happy Snaps photographic service offerio develop your film for LI (a fixedprice for processing) plus lop for eachprini - How does ihe cost of developinga film vary with the number of prinisS ou ~ ant des eloped

Think carefully about this situation, and discuss it w ith you rneighbour.* Describe, in writing, how the enclosed area will change as thelength increases through all possible values.* Illustrate your answer using a sketch graph:

The Rabbit Run

c-iaU

as

4 LengthA rectangular rabbit run is to he made from 22 metres of wirefencing. The owner is interested in knowing how the areaenclosed by the fence will depend upon ihe length of the run

I-

4

Enclosedarealength of the rabbit run

Shell Centre for Mathematical Lducai,on. Universicy of Nottingham. 1985.


5/13

Th e pupils shown below have all attempted thisproblem. Comment on their answerc. and try toexplain their mistakes.

~ AreaThe longer the rabbit run.then ihe bigger the

The amount ol ire is fixed.so as the run gets longer itgels narrower by the Santeamount.so t he a rea stays the same

II there is no length. thenthere is no area. and(the length is II metres.again there is no a reaso the graph turns round

In order to see how good your sketch is. construct a table ofvalues:i.englli ol run (metre.)Area (square metres)

Do you notice any patterns in this table?Write down what they are and try to explain why they occur.* Now, redraw your sketch using the patterns youobserved. (This does not need to be done accurately).

* Using your sketch and your table of values, f ind out what thedimensions of the boundary should he to obtain the greatestpossible space for the rabbit to move around in.

* Finally. iry to find an algebraic formula which tits thissituat ion.

Are a0 Length

I Linger runs are narrower.t he a rea drops

2

.irea0 Length

Area~0 Length

I ]Arej/\Length 11

Shell Centre for Mathematical Educat ion. University of Nottingham. 1985.


6/13

83 LOOKING AT EXPONENTIAL FUNCTIONS

I-)

* Check your sketch graphs by plotting afew points accuratelyon graph paper. Share this work out with your neighbour sothat i t doesnt take too long.

* Do just one of the two investigations shown below:

4

Hypnoflc Drugs

Sometimes, doctors prescribe hypnotic drugs (e.g.sleeping pills) to patients who, either through physicalpain or emotional tension, find that they cannot sleep.(Others are used as mild sedatives or for anaestheticsduring operations). There are many different kinds ofdrugs which can be prescribe d. One importantrequirement is that the effect of the drug should wear offby the following morning, otherwise the patient will findhimself drowsy al l through the next day. This cou ld bedangerous if, for example, he has to drive to work! Ofcourse, for someone confined to a hospital bed thiswouldnt matter so much.

Draw an accurate graph to show how the effect ofTriazolam wears off.After how many hours has the amount of drug in theblood halved?How does this Half life depend on the size of theinitial dose?Write down and explain your findings.

Investigate the effect of taking a 4~ig dose ofMethohexitone every hour.Draw an accura te graph and write about itsimplications.

I)I~tI%I ~ t~lniiiiigIi,iiii. I)H5


7/13

Drug name (and Brand name) Approximate formulaTriazolam (Halcion~) y = A x (0.84)Nitrazepam (Mogadon) y = A x (0.97)Pentohombitone (Sonitan) y = A x (1.15)Methohexitone (Brietal) y = A x (0.5)KEY A = size of the initial dose in the blood

y = amount of drug in the bloodx time in hours after the drug reaches the blood.

y = 4 x (0.84)* Please note that in this worksheet, doses and bloodconcentrations are not the same as those used in clinicalpractice, and the formulae may vary considerably owingto physiological differences between patients.

2

(on I inue the table below, using a calculakw. lo shc n~ In ~wdrug wears off during the first 10 hours.You do not need to plot a graph.Time (hours) Amount of drug left in the blood

x y0 4I 3.36( 4 x 0.84)2 2.82(=3.36x0.84)

~s y~ y~* On the same pair of axes, sketch four graphs to compare how a

4~sg dose* of each of the drugs will wear off.(Guess the graphsdo not draw them accurately)* Only three of the drugs are real. The other was intended as ajoke! Which is it? Explain how you can tell.What would happen if you took this drug?

3

Imagine that a doctor prescribed adrug called Triazolam. (Halcion9.After taking some pills, the drugeventually reaches a level* of 4p~g/t inthe blood plasma.How quickly will the drug wear off?Look at the table shown below:

(.3

N)C

For Triazolam, the formula is y = A x (0.84)In ou r problem the initial dose is 4 ~sg/ l, so this becomes

* Which of the following graphs best describes your data?Explain how you can tell without plotting

SheIl Centre for Mathemat ica l Education , n lvers it y 0 0 ing am.


8/13

At the moment, we have 3 variables; length, breadth, andthickness. If we keep two of these variables fixed, then we maybe able to discover a relationship between the third variable andthe weight the plank will support.So...* Collect together all the data which relates to a plank withbreadth 30cm and thickness 2cm, and make a table:Length of plank (1 metres)Maximum weight supported (w kg wt)Describe any patterns or rules that you spot. (Can you predict,for example, the value of w when I = 6?)Does your sketch graph agree with this table?Try to write down a formula to fit this data.

* Now look at all bridges with a fixed length and breadth, and tryC to !ind a connection between the thickness and the maximum~ weight it will support.3 Describe what you discover, as before.

* Now look at all planks with a fixed length and thickness.For geniuses only! Can you combine all your results toobtain a formula which can be used to predict the strength ofa bridge with any dimensions?

* Finally, what will happen in this situation?

70kg

4

B4 A FUNCTION WITH SEVERAL VARIABLESIn this booklet you will be considering the following problem:

Imagine the distance between the bridge supports (1) beingslowly changed. How will this affect the maximum weight (w)that can safely go across?

Sketch a graph to showhow w will vary with I.

I4)

I

Bridges

I

4m

How can you predict whether a plank bridge willcollapse under the weight of the person crossing it?

breadththickness = 70cm3cm

Shtii ( critre or 1;,IIiciui:iia:ii I IiIpt,liiI,r) I ipnrsiu~ i,i NIItIiI,gI,;IIII. 985


9/13

* Now imagine that, in turn, the thickness (1) and the breadth (1,)of the bridge are changed. Sketch two graphs to show the effecton w.

WI , vvVThkeep land b~:~~h1t in this

>1

constant in~~andgraph

The table on the next page shows the maximum weights that cancross bridges with different dimensions. The results are written inorder, from the strongest bridge to the weakest.* Try to discover patterns or rules by which the strength of abridge can be predicted from its dimensions.

Some Hints: Try reorganising this table, so that 1, bandvary in a systematic way.Try keeping b and t fixed, and look at howw depends on 1...

Distance Breadth Thickness Maximumbetween b(cm) ((cm) supportablesupports weight1(m) w(kg wt)2 40 5 250I 20 5 2502 50 4 2002 40 4 160I 20 4 1602 20 5 1252 30 4 120I 20 3 902 20 4 80I 30 2 604 40 3 451 20 2 402 10 4 402 30 2 303 30 2 203 10 3 154 30 2 155 30 2 121 20 1 104 40 1 5

Compare your graphs with those drawn by your neighbour.Try to convince her that your graphs are correct. It does notmatter too much if you cannot agree at this stage.Write down an explanation for the shape of each of yourgraphs.

If you are still stuck, then there are more hints on page 4.2

ShcII Centre for Mathematical Educat ion. University of Nottingham. 1985.


10/13

FINDING FUNCTIONS IN SITUATIONSRegular PolygonsHOCquare Regular Regularpentagon hexagonHow does the size of one of the interior angles depend upon thenumber of sides of the polygon?* Describe your answer in words and by means of a rough sketchgraph.

* Draw up a table of values, and check your sketch.(If you find this difficult, it may help if you first calculate thetotal sum of all the angles inside each polygon by subdividing itinto triangles, for example: sum of angles= 4 x 180= 7200

* Explain, in words, how you would calculate the size of aninterior angle for a regular n sided polygon.Can you write this as a formula?

4

For each of the four situations which follow,(i) Describe your answer by sketching a rough graph.Explain the shape of your graph in words.

Check your graph by constructing a table of values, andredraw it if necessary.(iv) If you can, try to find an algebraic formula, hut do not worry

too much if this proves difficult.I Renting a Television

A TV rental company charge 10per mon th fo r a colour set. Anintroductory offer allows you tohave the set rent-free for the firstm onth. H ow will the total costchange as the renta l periodincreases?

AquilateraltriangleCegularheptagon

(ii)(iii)

0egularoctagon 0egularnonagon 0egulardecagon2 The DepreciatingCar

so each angle is.. .)

When it was flew, my car cosi me 3JXHI. Its value isdepreciating at a rate of20~ per year. This means that alterone year its value was

[3.000 x (lJ< 2,400and after iwo> ears, its alue %%as

[2.4(H) x (1.8 LI .921) and sC) cmI lc~ does its ~alue conilnue to change

Shell Cenire k,r Mathematical I:dueaiion. University 01 Niiitingharn. l9H5.


11/13

The instructions on what to do for these two questions are atthe top of page 1 The Twelve Days of Christmas

On the first day of Christmas my true love sent to me:A partridge in a pear tree.On the second day of Christmas my true love sent to me:Two turtle doves and a partridge in a pear tree.On the third...4 The Film Show On the twelfth day of Christmas my true love sent to me:12 drummers drumming, 11 pipers piping, 10 lords a-leaping, 9 ladies dancing, 8 maids a-milking, 7 swansa-swimming, 6 geese a-laying, 5 gold rings, 4 callingbirds, 3 french hens, 2 turtle doves, and a partridge in apear tree.

After twelve days, the lady counts up all her gifts.* How many turtle doves did she receive altogether?(No, not two).

* If we call a partridge in a pear tree the first kind ofgift, a turtle dove the second kind of gift. . .etc, thenhow many gifts of the n th kind were received duringthe twelve days? Draw up a table to show your results.* Sketch a rough graph to illustrate your data. (You donot need to do this accurately).

* Which gift did she receive the most of?* Try to find a formula to fit your data

3

3 StaircasesThe normal pace length is 60cm.This must be decreased by 2cm fo revery 1 cm that the foot is raisedwhen climbing stairs.If stairs are designed according to this principal, how shouldthe tread length (see diagram) depend upon the height ofeach riser?

risertread-4 *

When a square colour slide is projected onto a screen,the area of the picture depends upon the distance of theprojector from the screen as illustrated below.(When the screen is 1 metre fromthe projector, the picture is20 cm x 20 cm). How does thearea of the picture vary as thescreen is moved away fromthe projector?3m

2mim

Om2

SheII Centre for Mathematical Education, University of Nottingham, 1985.


12/13

I. Speed conversion chartMiles jwr hour II) 21) ~ ~ ~) (~) 70Kilometre,, ~r hour 16.1 32.2 4S.3 ~ L %,6 112.7 l21&7

2. Radio frequencies and wavelengthsRadii, 4

Frequency (Kilt) Hfl 2(R) .Afl .141) 5U) (II) 7WWavelength (m) Ufl IS4( uRn 791 Hi) 9*) 429 ~75

FINDING FUNCTIONS IN TABLES OF DATATry the following problem. When you have finished, or when youget stuck, read on.

Dropping a stone

ci.)004. Temperature conversion

Sketch a rough graph to illustratethis data.Can you see any rules or patterns inthis table? Describe them in wordsand, if possible, by formulae.

*

Length of Time for 1(X)pendulum (cm) swings (seconds)0 05 4510 6315 7720 89

25 1(X)30 11035 11844) 1265)~jp4

CeIsw,00959085607570656055Sc4540.5

10

III

A stone is dropped from anaircraft. How far will it fall in 10seconds?

6

Fahrenheit21220394851675814940131

954~68~190dl

4

Tables of data often conceal a simple mathematical rule or functionwhich, when known, can b e u se d to predict unknown values.This function can be very difficult to find, especially if the table containsrounded numbers or experimental errors.It helps a great deal if you can recognise a function from the shape of itsgraph. On the next page is a rogues gallery of some of the mostimportant functions.* Which graph looks most like your sketch for the dropping a stoneproblem?

Shell C ent re for Mat hem;,i teal ltducai itt,. t niversi ty of Noninghani. I9~S


13/13

Quadratic

Reciprocal>~ ~Exponential

V

A and Bare numbersgreater than 0

LinearRogues Gallery

y=Ax y=~4s+8 y=Ax-8 y=m4x+8yv~ Yy= Air

I

Fitting a fonnula to the dataBy now, you have probably realised that the graph labelled y =Ax2 ~the only one which fits the dropping a stone data.In our case

y = distance fallen (metres)x = time (seconds)

and A is fixed positive number.* Try to find the value of A that makes the function fit the dataeither by trial and error or by substituting for values of x andyand solving the resulting equation.

* Use your resulting formula to find out how far the stone will fallin ten seconds.

y=AJ? +8 y= Ax1+B

y=!L+B y=-a+B

Vy = At (8> I)

Ay ABI (8< I) Extend this

collectionwhenever youmeet a newfunction...I x

Now look at the tables on the next page* Sketch a rough graph to illustrate the type of function shown ineach table. (You do not need to plot points accurately).

* Try to find patterns or rules in the tables and write about them.* Use the Rogues gallery to try to fit a function to the data ineach table.

* Some of the entries in the tables have been hidden by ink blots.Try to f ind out what these entries should be .

2

SheIl Centre for Mathematical Education. University of Nottingham. t985

lfg skething functions

Documents