1 of 52. Section B Linear and step graphs

Transcription

1 1 of 5 Section B Linear and step graphs

2 of 5 Copy into your notes Words to know Y axis 5 (3, 5) Origin (0, 0) 3 (3, 5) X axis Notice the first number of the coordinate pair is found on the x axis x-coordinate y-coordinate Together, the x-coordinate and the y-coordinate are called a coordinate pair.

3 SLO How to draw a graph using a table of values 3 of 5 Drawing graphs using plotting tables

4 Copy into your notes Finding the coordinates for an equation Given an equation, we can find coordinate points by constructing a table of values. E.g. find the coordinates for the equation y = x + 3 Construct a table (each click adds a number to table): x y = x ( 3, 0) (, 1) ( 1, ) (0, 3) (1, 4) (, 5) (3, 6) 4 of 5 Convert table into points to plot

5 5 of 5 Copy into your notes Drawing graphs from equations E.g. Draw a graph of y = x : 1) Complete a table of values: y y = x x y = x ) Plot the points on a grid. x 3) Draw a line through the points, extend to edge of axes (extrapolate). 4) Label the line. 5) Check that other points on the line fit the rule.

6 6 of 5 Your Turn: Draw a graph of y = x + 1 Step 1) Table for y = x + 1 x y (calculated from y = x + 1) Co-ordinate -3 (-3) + 1 = - (-3, -) (-1) + 1 = 0 (0) + 1 = 1 (1) + 1 = () + 1 = 3 (-1, 0) (0, 1) (1, ) (, 3)

7 7 of 5 Step ) Plot the points Step 3) Draw the line (, 3) -5-4 (-1, 0) (1, ) (0, 1) (-3, -)

8 8 of 5 Your Turn: Draw the graph of y = x Step 1) Table for y = x x y (calculated from y = x) Co-ordinate - (-) = -4 (-, -4) (-1) = - (0) = 0 (1) = () = 4 (-1, -) (0, 0) (1, ) (, 4)

9 9 of 5 Step ) Plot the points Step 3) Draw the line -5 (0, 0) (-1, -) (-, -4) (, 4) (1, )

10 10 of 5 Your Turn: Draw the graph of y = x + 1 Step 1: Table for y = x + 1 x y (calculated from y = x + 1) Co-ordinate -3 (-3) + 1 = -5 (-3, -5) (-1) + 1 = -1 (0) + 1 = 1 (1) + 1 = 3 () + 1 = 5 (-1, -1) (0, 1) (1, 3) (, 5)

11 11 of 5 Step ) Plot the points Step 3) Draw the line (0, 1) (1, 3) (, 5) -5-4 (-3, -5) -3 - (-1, -1)

12 Drawing graphs from equations 1 of 5

13 13 of 5 Questions to do from the books Gamma P99 EX Achieve Merit Excellence EAS P5 Q9 1 P8 Q3-44

14 SLO To find the gradient and y intercept of a linear graph 14 of 5 Finding gradient & y-intercept from linear graph

15 15 of 5 Copy into your notes Gradients of linear graphs The gradient of a line is a measure of how steep the line is. an upwards slope y a horizontal line y a downwards slope y Positive gradient x Zero gradient x Negative gradient x If a line is vertical its gradient cannot by specified.

16 16 of 5 Your Turn State if the following gradients are positive, negative, zero or undefined Positive Negative Zero Undefined

17 17 of 5 Copy into your notes Finding the Gradient Gradient = Rise Run E.g. Find the gradient of the graph below Step 1: Draw a right angled triangle on line (any size, anywhere) Step : Measure rise (height of triangle) i.e. 4 Step 3: Measure run (base of triangle) i.e. Step 4: Divide rise by run i.e. 4 =

18 Exploring gradients (This gives rise and run) 18 of 5

19 Your Turn: Gradients Move line by dragging dots on line and then reveal gradient (last fraction) students need to find rise and run 19 of 5

20 0 of Your Turn: Find the gradients of the following lines. y y y x x x 4 6 m m m m m 3 1 3

21 Investigating parallel lines (move lines and see what equations for parallel lines have in common) 1 of 5

22 of 5 Matching parallel lines

23 3 of 5 Copy into your notes Y intercept The y intercept is where the graph crosses the y axis E.g. Find the y intercept of the graph below Answer: The graph crosses the y axis at 1, so the y intercept is 1 (It s that easy!)

24 Intercept: chose a value of m and then change values of c to show how this changes where graph crosses y axis 4 of 5

25 5 of 5 Your Turn: Find the gradient and intercept of the following graphs Gradient = 4 = 1 Gradient = 3 5 Intercept = 3 Intercept =

26 6 of 5 Your Turn: Find the gradient and intercept of the following graphs Gradient = Intercept = 5 = 1 1 Gradient = 5 0 Intercept = 4

27 7 of 5 Your Turn: Find the gradient and intercept of the following graphs Gradient = 1 5 Gradient = 3 4 Intercept = 3 Intercept = 4

28 8 of 5 Questions to do from the books Gamma Achieve Merit Excellence P100 EX. 8.0 Q-1 EAS P11 Q45 5

29 9 of 5 SLO General equation of a straight line

30 30 of 5 Copy into your notes Equation of a straight line The general equation of a straight line can be written as: y = mx + c Gradient is number in front of x Y intercept is number on its own. For example, the line y = 3x + 4 has a gradient of 3 and crosses the y-axis at the point (0, 4).

31 31 of 5 Y = mx + c (positive gradients only): Use to demonstrate how m and c values effect graph.

32 Your Turn: Write the gradient and intercept for the following equations Equation Gradient y Intercept y = 4x y = x + 7 y = 5 + x y = 7 3 x 5 y = -4x + 8 y = 6x y = 9 7x 7 + 3x = y of 5 Which of the above graphs is the steepest? y = 9-7x

33 33 of 5 Your Turn: Fill in the blanks in the table equation gradient y-intercept y = 3x + 4 x y = 5 y = 3x 3 (0, 4) 1 (0, 5) 3 (0, ) y = x y = x 7 1 (0, 0) (0, 7)

34 SLO Drawing graphs from equations (not tables) 34 of 5 Drawing graphs using y = mx+c (not from table]

35 Copy into your notes Another way to draw graphs A quick way to draw linear graphs is to use the equation and not draw a table of values e.g. Draw the graph of y = x 1 Step 1: Identify gradient and y intercept 35 of 5 Gradient = y intercept = -1 Step : Place your first point at the y intercept Step 3: Use the gradient of i.e. rise, run 1 to find the next point. Step 4: Repeat step 3 several times. Step 5: Join points with line (extend if needed)

36 36 of 5 Your Turn: Draw the following graphs y = x + 1 y = 1 x + 1

37 37 of 5 Questions to do from the books Achieve Merit Excellence Gamma P104 EX Q1,,4 EAS P13 Q53 64

38 SLO Write equation given graph 38 of 5 Given the graph, write down the equation [y = mx+c] (uses y = mx + B)

39 Copy into your notes Equations from graphs Remember y = mx + c Gradient is number in front of x Y intercept is number on its own. Step 1: Find the y intercept by from where graph crosses y axis i.e. 5 Step : Find gradient using rise over run i.e. 4 3 Step 3: Substitute values in y = mx + c 1 i.e. y = 4 x of 5

40 40 of 5 What is the equation of the line?

41 41 of 5 Your Turn: Find the equations of the following graphs y = 1 5 x + 4 y = 1 x +

42 4 of 5 Your Turn: Find the equations of the following graphs y = 1 5 x y = x

43 43 of 5 Your Turn: Find the equations of the following graphs y = x y = 3x 1

44 Your Turn: find the equations of the following graphs y = 3 4 x y 6 x = y = x 4 1 y = 3 x x y = 3x of

45 45 of 5 Questions to do from the books Achieve Merit Excellence Gamma P104 EX Q5,10 EAS P14 Q65 76

46 46 of 5 SLO Vertical graphs Graph y = k, x = k and find equation from horizontal and vertical graphs

47 47 of 5 Graphs parallel to the y-axis What do these coordinate pairs have in common? (, 3), (, 1), (, ), (, 4), (, 0) and (, 3)? The x-coordinate in each pair is equal to. Look what happens when these points are plotted on a graph. y All of the points lie on a straight line parallel to the y-axis. x This line is called x =. x =

48 Copy into your notes Vertical Graphs E.g. x = 3 is a vertical line crossing the x axis at 3 Graphs such as x = #, will be vertical of

49 49 of 5 A: x = -4 B: x = -1 C: x = 0 D: x = 4 Your Turn: Find the equations of graphs A - D A B C D

50 50 of 5 SLO Horizontal graphs Graph y = k, x = k and find equation from horizontal and vertical graphs

51 51 of 5 Graphs parallel to the x-axis What do these coordinate pairs have in common? (0, 1), (3, 1), (, 1), (, 1), (1, 1) and ( 3, 1)? The y-coordinate in each pair is equal to 1. Look what happens when these points are plotted on a graph. y All of the points lie on a straight line parallel to the x-axis. y = 1 x This line is called y = 1.

52 Copy into your notes E.g. y = 3 is a horizontal line crossing the y axis at 3 Horizontal Graphs Graphs such as y = #, will be horizontal of

53 Drag line up and down to produce different horizontal graphs 53 of 5

54 A: y = 4 B: y = 1 Your Turn: Find the equations of graphs A - D 5 A 4 3 B C: y = - C - D: y = -4 D of 5-5

55 55 of 5 Questions to do from the books Achieve Merit Excellence Gamma P105 EX Q3,6 EAS P13 Q64 P14 Q68, 74

56 56 of 5 SLO Each year in the examination students are expected to know all aspects of y = mx + c. The next few slides are activities to help revise this concept.

57 57 of 5 Write down the equations of these lines: y y = x y = x + y = -x + 1 y = -x + x y = 3x + 1 x = 4 y = -3

58 Matching statements 58 of 5

59 59 of 5 Match the equations to the graphs

60 60 of 5 What is the equation? Look at this diagram: y 10 A What is the equation of the line passing through the points a) A and E x = G F H 5 B b) A and F c) B and E y = x + 6 y = x -5 D 0 E 5 C x 10 d) C and D e) E and G y = y = x f) A and C? y = 10 x

61 Pairs 61 of 5

62 SLO Linear graphs of the form Ax + By = C 6 of 5 Graph the intercept form ax + by = c using cover up method

63 63 of 5 Graphing Ax + By = C There are two ways to draw a graphs of the form Ax + By = C Method 1: Use algebra skills to rearrange the equation back into the form y = mx + c (This, for some students, is too hard so we will use Method ) Method : Cover up method

64 Copy into your notes 64 of 5 Graphing Ax + By = C using the Cover up method E.g. Draw the graph Step 1: Cover up the x part of the equation and solve for y. This will tell you where the line crosses the y axis

65 Copy into your notes 65 of 5 Step : Cover up the y part of the equation and solve for x. This will tell you where the line crosses the x axis Step 3: Join the two points

66 66 of 5 Example : Find the two intercepts of x + 3y = 6 and hence plot the graph x + 3y = 6 x + 3(0) = 6 x = 6 x = 3 The x-intercept is 3. x + 3y = 6 (0) + 3y = 6 3y = 6 y = The y-intercept is

67 Your Turn: Find the intercepts of the following graphs Equation X intercept Y intercept x y = x + 3y = x + y = Merit y = 0 4x 4x = 1 + 3y of 5

68 68 of 5 Your Turn: Plot the following graphs Equation X intercept Y intercept x + 3y =

69 69 of 5 Your Turn: Plot the following graphs Equation X intercept Y intercept x + y =

70 Your Turn: Plot the following graphs Equation X intercept Y intercept x y = of

71 Your Turn: Plot the following graphs Equation X intercept Y intercept 3x + 4y = of

72 7 of 5 Questions to do from the books Achieve Merit Excellence Gamma P106 EX Q1 17 EAS P16 Q77 9

73 SLO Graphs and simultaneous equations 73 of 5 Solve simultaneous equations graphically

74 74 of 5 Simultaneous Equations A single equation with unknowns is impossible to solve i.e. x + y = 18 (there are many possible solutions) Simultaneous equations are equations with the same value for both letters e.g. x + y = 7 x y = 3 As we know x and y have the same value in both equations it is possible to solve. See if you can find the answer. x = 5 and y = One way to solve harder simultaneous equations is by using algebra, but it is far easier to use graphs.

75 75 of 5 Continued on next slide Simultaneous Equations There is one pair of values that solves both these equations: x + y = 3 y x = 1 We can find the pair of values by drawing the lines x + y = 3 and y x = 1 on the same graph. 3 y y x = 1 The point where the two lines intersect gives us the solution to both equations. 0 3 x x + y = 3 This is the point (1, ). At this point x = 1 and y =.

76 76 of 5 Continuation from previous slide The values of x and y that solve both equations are x = 1 and y =, as we found by drawing graphs. We can check this solution by substituting these values into the original equations. x + y = 3 y x = = 3 1 = 1 Both the equations work so the solution is correct.

77 77 of 5 Copy into your notes Simultaneous equations and graphs E.g. Solve the following graphically x + y = 7 x y = 3 Step 1: Draw both graphs Step : Crossing point is solution to equations i.e. (5, ) x = 5 and y = Step 3: Substitute values into both equations to confirm answer. i.e. 5 + = 7 (correct) 5 = 3 (correct)

78 E.g. Solve y + 6x = 1 and y = x + 1 graphically Step 1: Plot both graphs Find coordinates for y + 6x = 1 using cover up method: 6x = 1 x = y = 1 y = 6 Find co-ordinates for y = x + 1 x = 0 y = (x0) + 1 y = 1 (0, 1) x = 1 y = (x1) + 1 y = 3 (1, 3) x = y = (x) + 1 y = 5 (, 5) Step : The co-ordinate of the crossing point (1, 3). Therefore, the solutions to the simultaneous equations are: y y + 6x = 1 y = x x 78 of 5 x = 1 and y = 3 Step 3: Check Substitute into equations (both work)

79 Your Turn: Solve the pair of simultaneous equations x = y = of 5-5

80 Your Turn: Solve the pair of simultaneous equations x = y = 1 80 of

81 Your Turn: Solve the pair of simultaneous equations x = y = - 81 of

82 8 of 5 Solving simultaneous equations graphically

83 83 of 5 Questions to do from the books Gamma Achieve Merit Excellence P105 EX Q8,9 P84 Ex 6.07 P86 Ex 6.09 EAS P3 Q10 11 P6 Q

84 SLO Real life Graphs 84 of 5 Linear graphs: application problems

85 85 of 5 There are 4 parts of a real life graph we need to understand Gradient x-intercept Real life Graphs y-intercept Join points?

86 Boxes packed Copy into your notes Gradient The steeper the graph (bigger the gradient) the faster something is happening. The blue line is steeper so he works faster. 86 of 5 Time (hr)

87 Copy into your notes 87 of 5 kg side New Zealand dollars Boxes packed Gradient is read side unit per base unit i.e. base cm Kg per cm British pounds Dollar per pound Time (hr) Boxes per hour

88 88 of 5 The gradient is the rate of change of the graph. Find the meaning of the gradient in each of the following graphs. Cost $ Cost $ Distance km Time hours cost gradient = time $ = hr = pay rate = charge per hour Weight tonnes cost gradient = weight $ = tonne = charge per tonne Petrol Litres distance gradient = petrol usage km = litre = distance travelled per litre of petrol used

89 Push play once and then try and predict graph of next flask before pushing play again 89 of 5

90 distance Copy into your notes Gradient of distance-time graphs Speed is gradient of distance-time graph. Run: change in time time Rise: change in distance Gradient/speed = rise run The steeper the line, the faster the object is moving. Finding rates of change and distance time graphs (best I could find!!!!!) 90 of 5

91 Distance from home Copy into your notes Distance-Time Graph The steeper the graph the faster the object is going Horizontal = stopped Returning back to starting point 8 4 The constant slope = constant speed Time in hours 91 of 5 Interactive distance time graph:

92 9 of 5 Distance in km from home Your 0 Turn: 16 1 C D 1) In the first hour how far did she travel? ) How long did she stop for? 3) At which points do you think she ran? 8 4 4) At which point was she walking the slowest? A 5) What was her speed for B? B Time in hours 1km 3hr 10Km/hr B and E D E

93 93 of 5 Questions to do from the books Achieve Merit Excellence Gamma P189 Ex 15.O4 EAS P19 Q93 95 P0 Q96 101

94 Copy into your notes y-intercept The y intercept is the starting amount e.g. Cost $ Cost $ Distance km Time hours Weight tonnes Petrol Litres Cost per no time ie initial cost or cover charge or call out cost Cost per no weight ie delivery charge cartage cost Petrol usage for no kilometres travelled i.e. starting engine. 94 of 5 This must be interpreted in the context of the question

95 95 of 5 Copy into your notes x-intercept The x intercept is the amount of what is on the x axis when the y axis is zero. Profit Water Temperature Units sold time No Profit for some units sold Break even point Time to reach zero degrees C Time to start to thaw.

96 96 of 5 Money spent on music Copy into your notes Join points? The intermediate values on the real life graph might have no meaning. In such cases do not join the points to form a line. Points not joined as you can not have fractions of a song. Songs on ipod

97 97 of 5 Continued on next slide Example 1: Plotting real life Graph When we plot a real life graph we usually start with a table of values or an equation. For example, a Quad bike hire company charges $30 to hire a car and then $5 for each day that the bike is hired. This would give us the following table of values (fill in the numbers): Number of days, d Cost in $, c The cost of the bike hire depends on the number of days. The number of days must therefore go in the top row.

98 Cost ($) Plotting real life Graph example Use the table of values to plot the points on the graph. Number of days, d Cost in $, c of 5 0 Cost of bike hire Number of days Is it appropriate to join the points together? Yes: if you could hire the bike part days. NO: if you could only hire for full days. What does the gradient show? Gradient = Cost per day What does the y-intercept show? Intercept = standing charge

99 Cost ($) Plotting real life Graph example 99 of Cost of bike hire Number of days Which company would you use for less than three days? First/black as lower Find the equation of the line c = 5d + 30 Write the equation of another bike hire company who charges $0 per day and a $50 standing charge c = 0d + 50 Explain how this second graph would be different to the first. Less steep and higher intercept What does the crossing point of the two graphs show? Same cost for same number of days

100 Example Real life graphs A group of pupils are doing an experiment to explore the effect of friction on an object moving down a ramp. They attach weights of different mass to the object and time how long the object takes to reach the bottom of the ramp. They put their results in a table and use the table to plot a graph of their results. Mass of object moving down ramp (grams) Time taken for object to move down ramp (seconds) of 5 Continued on next slide

101 Time taken (seconds) Example Real life graphs Mass of object moving down ramp (grams) Time taken for object to move down ramp (seconds) Explain if we can join the points using a straight line. 101 of Mass of object (grams) YES: The intermediate points represent real weights If another ramp was used and its points plotted a steeper graph, describe this ramp. Steeper ramp

102 Example 3 Real life graphs The formula for the total number of circles below is C = 3n + 1 n = C of n Explain if we can join the points using a straight line. NO: You can only have whole pattern numbers so we do not join the points with a straight line.

103 Cost ($) of 5 Your Turn: The equation for a phone companies' cost of texting is c = n where c = cost and n = number of texts. Plot a suitable graph Number of texts 6 Do we join the points? NO: can t have part texts Explain what the 5 and the 0.5 in the equation has to do with the costing. $5 = standing charge $0.5 = cost of each text. Another phone company has no standing charge but higher texting rate. Explain the difference in graphs. Starts at origin (lower than other) Higher gradient

104 104 of 5 Questions to do from the books Achieve Merit Excellence Gamma P108 Ex 8.05 EAS P33 Q18-19

105 SLO Drawing and reading step graphs 105 of 5 Note: there are no questions on this topic in the books.

106 Copy into your notes 106 of 5 Step Graphs Step graphs display data which falls into separate intervals (groups) that do not overlap. An open circle is used to indicate no value at this point. A closed circle is used to show a value is valid at this point.

107 107 of 5 Copy into your notes Drawing a step graph A table of values in intervals/groups is an indication that a step graph is required. Age At this point the value is taken from the higher line as it is a closed circle Cost 0 A < A < 6 6 A < A < A < 15 5 Interval values on x-axis At this point the value is taken from the lower line as it is a closed circle

108 108 of 5 Copy into your notes Example of step graph Draw the graph for the starting times of a school triathlon. Age Start time 7 and 8 8:00 am 9 and 10 9:00 am 11 and 1 10:00 am 13 and 14 11:00 am 15 and 16 1:00 pm

109 Your Turn: step graph 109 of 5 Weight (oz) Cost (c) 0 to 1 39c Greater than 1 and up to Greater than and up to 3 Greater than 3 and up to 4 Greater than 4 and up to 5 Greater than 5 and up to 6 41c 43c 45c 47c 49c Draw the graph for the postage costs

110 Your Turn: step graph Valves Number of employees Draw the graph of the number of staff a factory employs and the number of valves it produces. 110 of 5 Circles at ether end of the step are not always required

111 Answer 011 Exam paper of 5 ONE (a)(i) Graph drawn from table. $ fare n Table is linked to graph for stages and costs, (rather than stations and costs) as a straight line, $ = 3n, but this simplifies the situation for Q1aii. Accept sloping lines connecting the steps. Graph drawn as horizontal sets of points for stages according to multiples of three. Accept ambiguity at boundaries between fares, eg when bars touch. Graph is drawn as discrete step function. Accept continuous steps and accept bars, so long as the fares are correct for each number of stations and none are ambiguous.

112 11 of 5 01 Exam paper

113 113 of 5 01 Answers (b)(i) Graph drawn from table. If graph is drawn from the WORDS of the question, or a mixture of the information from words and table, fixed amount could be $7 or $11, and a part-bundle delivery could count! Linking table to graph for ONE section as a straight line. OR Set of points. (From either step end.) OR Step function with no change 300. Graph shows change after 300 leaflets with: TWO sets of points. OR TWO straight lines. Graph drawn as step function with distinction made after the 300 leaflets have been delivered.

114 114 of 5 The examination papers from previous years appear to concentrate on 3 main aspects: Writing an equation given a linear graph. In application questions understanding the relationship between the gradient/intercept of the graph and the equation i.e. if a taxi firm charges more per km, how will the affect the graph or if their call out fee is higher but their rate per km is the same, how will this affect the graph. This understanding can also be related to understanding the link between a matchstick problem and the resulting linear equation. Step graphs

115 115 of 5 Random Revision questions

116 116 of 5 3) Find the equation 160 F W 40 F 4W 160

117 117 of 5 y 10 4) Draw 3 y x 5 4 x y 5 3x x

118 Use the cover up method to plot x + y = 10 4) Use the intercept intercept method to plot 5x y 10 y Cut x axis y 0 5x (0) 10 5x 10 x (,0) Cut y axis x 0 5(0) y 10 y 10 y 5 (0,5) x 118 of 5

119 5) Find the equation M Q M 1 Q of 5

120 Use the cover up method to plot 3x 4y 8 = 0 3) Use the intercept intercept method to plot 3x 4y 8 0 Cuts x axis y 0 y 3x 4(0) x 8 8 x 3 Cuts y axis x 0 3(0) 4y x 4y 8 y y of 5 8

121 11 of 5 4) Find the equation of the line K m K 3m 90

122 4) Find the equation of the line 70 K 0 m 7 K m 70 1 of 5

123 13 of 5 R ) Find the equation 30 g R g 10

124 Find the equations of these graphs Write the equations of these lines C 350 F R 30 W 70 H 30 R 30 C 5W 350 F 4 3 H 80 Q 70 Q 7 W 5 14 of 5 50 W

125 15 of 5 Write the equations of these lines a) H 1t 0 5 b) H t 3 H b a Time (t) When do these have the same H value? Lines intersect at t 30 K a) K 3s 35 b) K 3s 5 10 s What can you say a b about the rate? Why? Rate is the same Lines are parallel

126 16 of 5 Reading the equation of a graph R S 70 F C 500 C S 30 a 100 t 7 a 70 F 3t 00 3 R 10 A 300 W R 30 h 40 4h 10 A 7.5b b W R 70 R 40

127 Write the equations of these lines C 50 F R 40 C 0 W 5 W 50 F 3 P H 30 H 55 R P 4 5 B 50 B 17 of 5

128 18 of 5 Write the equations of these lines H a) H t 7 b) H 5t When do these have the same H value? Lines intersect at t b a Time (t) K 6 a) K s b) K s What can you say b s about the rate? Why? Rate is the same Lines are parallel a

129 Jacqui is investigating the cost to lay different types of pavers. Depending on the type of paver to be laid the rates differ. All the rates comprise a fixed daily charge plus a cost per paver to lay. The graph shows the cost to lay 3 types y Cost $ Flagstone Bevelstone Mountain Sandstone Work out the points for Mountain Pavers C.5P of No of Pavers (P) x

130 130 of 5 70 Cost $ ) Which is the cheapest option? 7) How does the graph show this? y No of Pavers (P) 3) For what number of pavers does it cost the same for Flagstone and Bevelstone pavers? Flagstone Bevelstone Mountain x Sandstone 1) Which paver costs the most to lay per paver? ) How does the graph show this? 4) How does the graph show this? 5) Write the equation for the other graphs

131 131 of 5 Find the equation of the following graphs 10 y 4 y = 4 5 x 7 8 y = x 6 4 y = 1 3 x 1 x = x