STRAIGHT LINE GRAPHS Learning Outcomes and Assessment Standards Lesson 15 Learning Outcome : Functions and Algebra The learner is able to investigate, analse, describe and represent a wide range o unctions and solve related problems. Assessment Standards: We know this when the learner is able to: Demonstrate the abilit to work with various tpes o unctions; Recognise relationships between variables in terms o numerical, graphical, verbal and smbolic representations and convert leibl between these representations (tables, graphs, words and ormulae) Overview In this lesson ou will: Revise straight line graphs Model straight line graphs. Lesson From Grade 9 = m + c Straight line graph Linear Function m is the gradient c is the intercept To draw the graph we simpl need two points DVD Eample Sketch = 1 Start with c c = 1 m = _ 1 Plot the gradient 4 3 1 4 3 1 0 1 1 3 4 3 4 Properties o straight line graphs a + b + c = 0 is the deining equation o a straight line graph. The deining equation is written in the orm = m + c where m is the gradient and c is the -intercept. 1 06 LC G10 MATHS LWB.indb 1 008/09/09 1:30:37 PM
The straight line graph is a linear unction Standard orm can be () = m + c or : m + c The domain o a unction The domain o unction a untion is the set o values can have. The range o a unction The range o a unction is the set o values can have. So, i () = m + c D = R R = R The zero ( intercept) This is the value o when = 0. The zero is the solution to the equation. m + c = 0 = c _ m This is also called the root. To ind the equation o a straight line ou need the gradient m and the - intercept c: = m + c Eample 1. Find the equation o and g For c = 3 m = 3 _ () = _ 3 + 3 For g the are perpendicular so m = _ c = 3 3 g() = _ 3 3. (i) Sketch : + 1 (ii) Show on our graph where ou will read o ( 1) with the letter A. (iii) What is the zero o? (iv) Find the equation o g i g is perpendicular to and passes through point (0; 1). g 3 3 06 LC G10 MATHS LWB.indb 008/09/09 1:30:38 PM
Solutions (i) Sketch : + 1 (ii) Show on our graph where ou will read o ( 1) with the letter A. (iii) What is the zero o? + 1 = 0 4 3 1 0 1 3 4 1 = 1 = _ 1 3 4 (iv) Find the equation o g i g is perpendicular to and passes through point (0; 1) g : _ 1 1 Another wa o sketching graphs: we use the dual intercept method Find the -intercept, = 0 Find the -intercept, = 0 5 4 A 3 1 Eample Sketch 3 = 6 Find the -intercept = 0 3 = 6 = (; 0) Find the - intercept = 6 = 3 (0; 3) Using graphs to solve real lie situations Conversion graphs Water boils at 100 C and this is 1 F. We also know that 0 C is 3 F. 1. Draw a graph that will enable ou to convert Centigrade to Fahrenheit. Put C on the horizontal ais and F on the vertical ais. 3 1 3 1 0 1 3 1 3 4 3 06 LC G10 MATHS LWB.indb 3 008/09/09 1:30:39 PM
Solution F 00 (100; 1) 150 100 50 3 50 100 150 00 C (a) Show on the graph with (A) where ou convert 100 F to C (b) Show on the graph with (B) where ou convert 70 C to F Solution F 00 B (100; 1) 150 100 50 3 A 50 100 150 00 C. Write a ormula to change C to F 4 Solution 1 3 m = 100 0 = _ 9 5 c = 3 F = 9 _ 5 c + 3 3. Write a ormula to change F to C Solution F 3 = _ 9 5 c 5F 160 = 9c c = _ 5 9 F _ 160 9 06 LC G10 MATHS LWB.indb 4 008/09/09 1:30:40 PM
Distance time graphs distance travelled Average speed = time taken Eample Mar walks rom her home to the shops and back. Her journe is represented b this distance time graph. 800 600 400 00 0 5 10 15 0 5 30 35 40 45 Time (mins) 1. How ar is it to the shops?. How long was Mar in the shops? 3. Calculate Mar s average speed in metres per hour or her journe to the shops and back. 4. Calculate Mar s average speed or her journe to the shops. 5. Calculate Mar s average speed or her journe rom the shops. Solutions 1. 800 m. 15 minutes 1 600 3. _ = 3 00 m per hour 1_ 800 4. _ = 4 800 m per hour 1_ 6 800 5. _ = 400 m per hour 1_ 3 For ou to do Put appropriate letters on aes like the ollowing and tell the stor o the graph. (Redraw our graph on graph paper) 5 06 LC G10 MATHS LWB.indb 5 008/09/09 1:30:40 PM
180 150 10 90 60 30 Cost problems 30 60 90 10 150 180 10 40 Time (mins) The cost o a journe with Foloza tais is R8 a kilometre. 1. Work out the cost o using Foloza tais or journes o i) 10, ii) 30, iii) 50, iv) 0 kilometres.. Plot this inormation on a graph and join the line (distance on the horizontal ais and cost on the vertical ais). 3. Suggest an equation or the graph 4. The cost o a journe with Sam s motors is worked out b halving the distance and adding R10 or insurance. 5. Plot this graph on the same aes. 6. Suggest an equation or this graph. 7. Use our graph to ind the distance or which the costs are the same. 8. When is it more economical to use Foloza Tais? 9. When is it more economical to use Sam s motors? Solutions: 1. (i) Distance 10 cost 80 (10; 80) (ii) Distance 30 cost 40 (30; 40) (iii) Distance 50 cost 400 (50; 400) (iv) Distance 0 cost 0 (0; 0) 6 06 LC G10 MATHS LWB.indb 6 008/09/09 1:30:41 PM
. c 500 = 8 400 00 100 0 10 0 30 40 50 60 70 80 90 3. Equation = m + c = 8 4. (0; 10) (0; 130) (40; 160) 5. c d 500 400 00 100 0 10 0 30 40 50 60 70 80 90 6. Suggest an equation or this graph = m + c = _ 10 0 + 10 = _ 1 + 10 7. Use our graph to ind the distance or which the costs are the same. 8 = _ 1 + 10 c 16 = + 40 = 8 500 15 = 40 400 = = 16 1_ + 10 00 100 d 0 10 0 30 40 50 60 70 80 90 d 7 06 LC G10 MATHS LWB.indb 7 008/09/09 1:30:4 PM
8. When < 16 km 9. When > 16 km Traic Volumes between 4:30 and 6:00 pm 1. Place the graphs given below in the correct chronological (time) order. (a) Traic Volume (b) 0 5 10 15 0 5 30 35 Distance rom cit centre (km) Traic Volume (c) 0 5 10 15 0 5 30 35 Distance rom cit centre (km) Traic Volume (d) 0 5 10 15 0 5 30 35 Distance rom cit centre (km) Traic Volume 0 5 10 15 0 5 30 35 Distance rom cit centre (km). Pick one o the 4 graphs and write a paragraph eplaining wh the graph has the shape it does. 8 06 LC G10 MATHS LWB.indb 8 008/09/09 1:30:4 PM
3. At what time o a 4 hour da do ou think the ollowing graph was made: Traic Volume Just or un 0 5 10 15 0 5 30 35 Distance rom cit centre (km) Just or un we are going to match strange looking bottles to the graphs that are ormed b the height o the water as the bottle is being illed. Which bottle matches with which graph? 1.. Height (cm) Height (cm) 3. Water (cups) Water (cups) Height (cm) Water (cups) A. B. C. 9 06 LC G10 MATHS LWB.indb 9 008/09/09 1:30:43 PM
10 INDIVIDUAL SUMMATIVE ASSESSMENT Activit 1 1. 5 miles is approimatel the same as 8 kilometres. (a) Use graph paper to draw the graph or converting kilometres to miles. (You will need a second point : perhaps ou can use the act that 0 miles is the same as 0 kilometres). Put miles on the horizontal ais and kilometres on the vertical ais. (b) Use our graph to convert the ollowing (write down the approimate answer and show where ou read o the answers with letters A; B; C; ). (i) 14 km s to miles (ii) 14 miles to km s (iii) 1 miles to km s (iv) 3 km s to miles (c) Write down a ormula or changing miles into kms (make k kilometres and M miles). 9 gallons is 50 litres. (a) Draw a graph or converting gallons to litres. (b) Use the graph to convert (i) 6 gallons to litres (ii) 44 litres to gallons (iii) 55 litres to gallons. (c) Write down a ormula or converting gallons to litres. (d) Write down a ormula or converting litres to gallons. 3. 50 kilograms 110 pounds. (a) Draw a conversion graph or converting kilograms to pounds (b) Use our graph to make the ollowing conversions (show with letters where ou make our readings) (i) 50 kg to pounds (ii) 80 kg to pounds (iii) 00 pounds to kg. (c) Devise a ormula or changing kilograms to pounds. (d) Devise a ormula or changing pounds to kilograms. 4. Katie travels b car rom her home in Johannesburg to Rustenburg and back. She leaves home at 7 am and travels 90 miles in hours. She stops to have a cup o coee or 30 minutes. She then drives a urther 50 miles 06 LC G10 MATHS LWB.indb 10 008/09/09 1:30:44 PM
and arrives in Rustenburg at 10.30 am. She leaves Rustenburg at pm and drives straight back, arriving home at 3.30 pm. (a) Draw a distance-time graph to illustrate her journe. (b) Calculate her average speed or each section o the journe. (c) Calculate her average speed or the whole journe. 5. Two car hire companies give the ollowing quotes or a das car hire. Aja : R150 + 80c per km Econo : R100 + R1 per km (a) Write down a ormula or the cost or 1 da or each quote (use C or cost in rands) (b) Draw each graph on the same set o aes. (c) Under what conditions is it more avourable to use Aja? (d) Under what conditions is it more avourable to use Econo? Activit 1. Find the gradients o the lines joining the ollowing points (a) ( 3; 1) and (6; ) (b) (7; 4) ( 3; 6) (c) (9; 8) and ( 1; ) (d) ( 3; 1) and ( 5; 7) (e) (4; 1) and (4; 5) () (; 3) and (6; 3) (g) (1; 3) and (8; 1) Activit 3 1. Use the gradient intercept method to draw sketches o the ollowing graphs (a) = 4 (b) = _ (c) : _ 1 + (d) = 3 (e) = 5 () 4 = 3 4. Sketch i : + 1. (i) Show on our graph where ou will read o ( 1) with the letter A. (ii) What is the zero o? 11 06 LC G10 MATHS LWB.indb 11 008/09/09 1:30:45 PM
(iii) On the same set o aes, sketch g i g is perpendicular to and passes through (0; 1) and write down the equation. Activit 4 1. Sketch the ollowing graphs on the same set o aes (ou an use an method) (a) + = (b) = 4 (c) = 3. Sketch : _ i ( 4 ; ] (a) Show on our graph with A where ou will read o ( 1). (b) Give the range o the unction. (c) What is the zero? 3. Sketch : { 3 i < { + 1 i < < 1 { i 1 (a) Name an zeros (b) Evaluate (3) (c) Find ( 4) (d) Give the range o the unction. 4. Sketch g : { 1 i < 1 { i 1 < < { + 3 i (a) Name an zeros (b) What is ( 3) ; ( 1) (c) What is (0) ; () (d) What is (5) (e) Show on our graph where ou ind (3) () Give the range o the unction 5. On the same set o aes sketch : _ and g: (a) Show on the graph where ou will solve the equation = g. 1 (b) Find when = g 6. On the same set o aes sketch (i) 4 + 5 = 0 06 LC G10 MATHS LWB.indb 1 008/09/09 1:30:46 PM
(ii) + = (iv) = 1 (v) = 1 Shade in the area that satisies the inequalities 4 + 5 0 + 1 1 Activit 5 1. Find the equations o the ollowing straight lines. (a) The gradient is 3 passing through pt ( 1; ) (b) The gradient is _ 1 passing through pt (1; ) (c) The gradient is 1 passing through pt (7; ) (d) Passing through A(1; 4) and B(3; ) (e) Passing through P(7; 1) and Q(6; 3) () Passing through R 5; 1) and T(1; ) (g) The gradient is O, passing through pt (7; 1) (h) Passing through ( 6; ) and (4; ) (i) The gradient is undeined, passing through pt (4; ) (j) Passing through A(; 1) and B(; 5) (k) Parallel to 4 = 1 passing through (; 3) (l) Parallel to the line joining ( ; 3) and ( 1; 1) passing through pt (4; ) (m) Perpendicular to 3 + = passing through ( 1; 5) (n) Perpendicular to the line AB i A(3; 1) and B( 1; ) and passing through pt (4; 4) 13 06 LC G10 MATHS LWB.indb 13 008/09/09 1:30:47 PM
. (5; 3) ( 1; 3) (a) Find the equation o and write it as : - - - - (b) Find the equation o g i g is parallel to passing through ( ; 3) and write down the and intercepts o g (c) Find the equation o h i h is perpendicular to passing through ( 1; 3). Write down the and intercepts o h (d) Find the points o intersection g and h (Draw these graphs on the same set o aes) 3. g 4 4 B A (a) Find the equation o and g (b) Find the co-ordinates o A and B 4. g D n A E ( 4; ; 1) (1; ) 14 C (a) Find the equations o ; g and h B 06 LC G10 MATHS LWB.indb 14 008/09/09 1:30:47 PM
(b) Find the co-ordinates o A; B; C; D and E 5. ( ; 3) (1; 1) (a) Find the equation o (b) Give the domain and range (c) What is the zero o? Activit 6 1. P A M g Q 0 T c D E N B R : + 3 g : _ 1 (a) Find the lengths o AB and CD (b) Find the co-ordinates o E (c) I OQ = units, ind the length o PR (d) I MN = units, ind the length o OT (e) Find the length o PC 15 06 LC G10 MATHS LWB.indb 15 008/09/09 1:30:48 PM
. B g C P K Q 0 4 A 4 R D (a) Find the equations o and g (b) Find the co-ordinates o A (c) I BD = 10 units, ind the length o OC (d) I OQ = 1 unit, ind the length o PR (e) Find the co-ordinates o PR and BD () Find the co-ordinates o P (g) Find the length o PK 3. n A D 0 E B C g : 1 g : + 4 h : + + 5 16 (a) Find the length o DE (b) Find the co-ordinates o A; B and C (c) Find the domain and range o the shaded area 06 LC G10 MATHS LWB.indb 16 008/09/09 1:30:49 PM
(d) Find the area o ΔABC 4. g 6 A 4 P R Q 4 0 T 3 (a) Find the equation o and g (b) Find the length o OT (c) I OR = units, ind the length o PQ 17 06 LC G10 MATHS LWB.indb 17 008/09/09 1:30:49 PM