A Resource for Free-standing Mathematics Qualifications

Transcription

1 Ope.ls The first sheet is show elow. It is set up to show graphs with equatios of the form = m + c At preset the values of m ad c are oth zero. You ca chage these values usig the scroll ars. Leave the value of c equal to zero, ad use the scroll ar to chage the value of m. Tr usig differet parts of the scroll ar to see what happes ad look at the tales ad graphs ou get. scroll ars You have ee lookig at some graphs with equatios of the form I each case is proportioal to. Aswer these questios aout graphs with equatios of this tpe (i.e. = m (sice c is zero). = m ) 1 What poit o the graph does the lie alwas pass through?. What happes to the lie as the value of m icreases from to 5? 3 What happes to the lie as the value of m decreases from to 5? The Nuffield Foudatio 1

2 Now keep m = ad use the scroll ar to chage the value of c. Aswer these questios aout such graphs i.e. graphs with equatios of the form 4 What happes to the lie as the value of c icreases from 4 to 4? = c Aswer these questios aout graphs with equatios = m + c Keep m = 1 ad use the scroll ar to chage the value of c. 5 What happes to the lie as the value of c icreases from 4 to 4? Keep m = 1 ad chage the value of c. 6 What happes to the lie as the value of c icreases from 4 to 4? Tr other values of m ad c. 7 What ca ou sa aout the lik etwee the lie ad the value of c? 8 What ca ou sa aout the lik etwee the lie ad the value of m? The Nuffield Foudatio

3 Quadratic Graphs Use the secod worksheet ( = a^ + + c) of.ls It is set up to show graphs with equatios of the form = a + + c Questios 1 to 4 are aout curves with equatios of the form = a Keep ad c at zero ad eperimet with differet values of a. 1 O this set of aes sketch the graph whe a is positive. Descrie what happes to the curve as the value of a icreases from to 5. 3 O this set of aes sketch the graph whe a is egative. 4 Descrie what happes to the curve as the value of a decreases from to 5. Questios 5 to 7 are aout curves with equatios of the form = a + c Keep a = 1 ad =. Use the scroll ar to chage the value of c. 5 What happes to the curve as the value of c icreases from 4 to 4? Keep a = 1 ad =. Chage the value of c. 6 What happes to the curve as the value of c icreases from 4 to 4? Tr other values of a ad c, keepig =. 7 What ca ou sa aout the lik etwee the curve ad the values of a ad c? The Nuffield Foudatio 3

4 Questios 8 to 11 are aout curves with equatios of the form = a + Keep a = 1 ad c = ad chage the value of. 8 Descrie what happes to the curve as the value of icreases from to 4. Write dow athig ou otice. 9 Descrie what happes to the curve as the value of decreases from to 4. Now keep a = 1 ad c = ad chage the value of. 1 Descrie what happes to the curve as the value of icreases from to 4? Write dow athig ou otice. 11 Descrie what happes to the curve as the value of decreases from to 4? Eperimet with differet values of a, ad c. Check what effect chagig each costat has o curves with equatios of the form Write a summar of our fidigs elow. = a + + c. The Nuffield Foudatio 4

5 Power Graphs Use the third worksheet ( = k^) of.ls It is set up to show graphs with equatios of the form = k where is a positive power. 1 Keepig k = 1 click o the scroll ar to the right of the cetral divider to icrease the value of from to 1, the, 3, 4,.1. a Eplai wh the graph is a straight lie whe = ad = O this set of aes sketch the geeral shape of the graph = k whe is a positive eve iteger. Descrie what happes to the curve as icreases. c O this set of aes sketch the geeral shape of the graph = k whe is a odd iteger greater tha 1. Descrie what happes to the curve as icreases. a Keepig =, icrease the value of k startig from 1. Chage to 3 ad agai icrease k from 1. Descrie what happes to the curves as k icreases... Keepig =, chage k from 1 to 1. Check if the same thig happes with other values of ad k whe ou chage k from positive to egative. Descrie what happes to the curve whe ou chage k from positive to egative. The Nuffield Foudatio 5

6 3 Keepig k = 1, start with =.5 the click the scroll ar to the right of the divider to icrease to 1.5 the.5, 3.5. Wh is there o curve whe <? Use the fourth worksheet ( = k^-) of.ls It is set up to show graphs with equatios of the form = k i.e for egative powers of. 4 Keepig k = 1 click o the scroll ar for to the right of the cetral divider so that the power of chages from to 1, the, 3,. 1. a O this set of aes sketch the geeral shape of the graph = k whe the power is a egative odd iteger. O this set of aes sketch the geeral shape of the graph = k whe the power is a egative eve iteger. 5 a Keep the power as (i.e. keep = ). Icrease the value of k startig from 1. Chage the power to 3 ad agai icrease k from 1. Descrie what happes to the curves as k icreases... Keepig the same (a value), chage k from 1 to 1. Check if the same thig happes with other values of ad k whe ou chage k from positive to egative. Descrie what happes to the curve whe ou chage k from positive to egative. The Nuffield Foudatio 6

7 Teacher Notes Uit Advaced Level, Workig with algeraic ad graphical techiques Skills used i this activit: usig Ecel to ivestigate the shape of graphs Preparatio Each studet will eed to have a cop of pages 1 to 6 ad access to Ecel. Chage the ame of the Ecel file to.ls ad protect it if ou wish efore allowig studets to use it. Notes o Activit This activit ca e used to itroduce the shape ad mai features of liear, quadratic ad power graphs. It ca e split ito three separate activities to e doe at differet poits i the course if ou wish. (Note that a alterative versio of the first part of the activit dealig with proportioal ad liear graphs ca e foud i Liear Graphs. This resource ca e foud i the skills activities sectio for Makig coectios i mathematics o the Nuffield wesite.) Studets should e told to alwas use the scroll ars to chage the values of the costats. Note that o the liear ad quadratic worksheets, clickig o a arrow at the ed of a scroll ar icreases or reduces the costat.1 whilst clickig o other parts of the ar icreases or reduces the costat 1. O the power graph worksheets clickig o a arrow at the ed of the scroll ar icreases or reduces the costat.5 whilst clickig o other parts of the ar icreases or reduces the costat 1. Alterativel a costat ca e varied draggig the cetral ar alog the scroll ar. If ou have the equipmet eeded to project the spreadsheet oto a scree, this will aid class discussio. Discussio could iclude patters i the values of ad i the tales as well as features of the graphs. Aswers/Poits for discussio Liear Graphs 1 Lies with equatios of the form = m alwas pass through (, ) i.e. the origi. As m icreases from to 5, the gradiet of the lie icreases i.e. the lie ecomes steeper. 3 The gradiet ow ecomes egative, with the lie gettig steeper as m decreases from to 5. 4 Lies with equatios of the form = c have zero gradiet. As c icreases from 4 to 4 the lie moves up the page from = 4 to = 4. 5 For lies with equatios of the form = + c, as c icreases from 4 to 4 the poit of itersectio of the lie with the ais moves from = 4 to = 4 (ad the gradiet is 1). 6 For lies with equatios of the form = + c, as c icreases from 4 to 4 the poit of itersectio of the lie with the ais moves from = 4 to = 4 (ad the gradiet is 1). 7 For all lies with equatios of the form = m + c, c gives the itercept o the ais. 8 For all lies with equatios of the form = m + c, m gives the gradiet. The Nuffield Foudatio 7

8 Quadratic Graphs 1 3 = a whe a is positive = a whe a is egative As a icreases from to 5 the curve arrows ad gets steeper. 4 As a decreases from to 5 the curve arrows ad gets steeper. 5 As c icreases from 4 to 4 the poit of itersectio of the curve with the ais moves from 4 to 4. 6 As c icreases from 4 to 4 the poit of itersectio of the curve with the ais moves from 4 to 4. 7 For quadratic curves with equatios of the form = a + c, c gives the itercept o the ais. The value of a affects the gradiet ad orietatio of the curve. 8 As icreases from to 4 the curve moves to the left ad dowwards. I fact quadratic curves with equatios of the form = + alwas cross the ais at the poits (, ) ad (, ). The miimum poit is,. 4 9 As decreases from to 4 the curve moves to the right ad dowwards. 1 As icreases from to 4 the curve moves to the right ad upwards. Quadratic curves with equatios of the form (, ). The maimum poit is,. 4 = + alwas cross the ais at the poits (, ) ad 11 As decreases from to 4 the curve moves to the left ad upwards. Summar for quadratic curves Fidigs could iclude: The itercept o the ais is alwas equal to c. I geeral chagig either a or chages the gradiet as well as the positio of the turig poit of the curve. I the special case whe a = the graph is a straight lie (uless ad c are also ). Whe a is positive, icreasig alwas moves the turig poit of the curve to the left. Whe a is egative, icreasig alwas moves the turig poit of the curve to the right. Notes The turig poit of a quadratic curve is give, c a 4a The gradiet of a quadratic curve is give a + The Nuffield Foudatio 8

9 Power Graphs 1 a Whe = the equatio reduces to = k ad whe = 1 it is equivalet to = k. c = k = k a positive eve iteger a odd iteger greater tha 1 I oth cases as icreases the curve ecomes flatter for 1 < < 1 ad steeper for < 1 ad > 1 a As k icreases the curve ecomes steeper. (The value of each poit o the curve = k is equal to the value o = Whe k chages from positive to egative the curve is reflected i the ais. multiplied k.) 3 Whe is.5, 1.5,.5, the equatios ivolve the square root of. 4 a = k odd = k eve 5 a The value of each poit o the curve = k is equal to the value o = multiplied k. Whe k chages from positive to egative the curve is reflected i the ais. The Nuffield Foudatio 9