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Straight Line Graphs

S4 Credit. Straight Line Graphs. The gradient Vertical ÷ Horizontal. The gradient from Coordinates. Drawing Straight line graphs. General Equation y = mx + c. www.mathsrevision.com. Equation from 2 Points. Modelling Situations. Best – fitting straight line. Exam Type Questions.

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Straight Line Graphs

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  1. S4 Credit Straight Line Graphs The gradient Vertical ÷ Horizontal The gradient from Coordinates Drawing Straight line graphs General Equation y = mx + c www.mathsrevision.com Equation from 2 Points Modelling Situations Best – fitting straight line Exam Type Questions

  2. S4 Credit Starter Questions Q1. Calculate 7 – 5 x 2 Q2. Calculate www.mathsrevision.com Q3. Is this triangle right angled ? Explain 9 8 5 Created by Mr.Lafferty Maths Dept

  3. S4 Credit The Gradient of a Line Learning Intention Success Criteria • To show how to get the equations for horizontal and vertical lines. • Understand that vertical lines have equations of the form x = a • 2. Understand that horizontal lines have equations of the form y = a www.mathsrevision.com Created by Mr.Lafferty Maths Dept

  4. y 10 9 8 7 6 5 4 3 2 1 x -1 0 -10 -7 1 2 3 4 5 6 7 9 10 -9 -8 -6 -4 -3 -2 8 -5 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 Equations for Vertical and Horizontal lines x = -6 Mark the points on your grid below and then join them together Vertical lines have equations of the form x = 8 Horizontal lines have equations of the form y = 2 y = -9 Created by Mr. Lafferty Maths Dept

  5. The Gradient S4 Credit Now Try MIA Exercise 2.1 Ch7 (page 136) www.mathsrevision.com Created by Mr.Lafferty Maths Dept

  6. S4 Credit Starter Questions Q1. Write this number in full to 1 sig. figs Q2. Calculate the volume of the triangular prism. 20cm www.mathsrevision.com 12cm 8cm Created by Mr.Lafferty Maths Dept

  7. S4 Credit The Gradient of a Line Learning Intention Success Criteria • To explain how to calculate the gradient using a right angle triangle • Gradient is : • change in vertical height divided by • change in horizontal distance www.mathsrevision.com • 2. Calculate simple gradients. Created by Mr.Lafferty Maths Dept

  8. Change in vertical height Change in horizontal distance Difference in y -coordinates S4 Credit The Gradient The gradient is the measure of steepness of a line www.mathsrevision.com Difference in x -coordinates The steeper a line the bigger the gradient Created by Mr.Lafferty Maths Dept

  9. S4 Credit 3 The Gradient 4 3 2 www.mathsrevision.com 3 5 2 6 Created by Mr.Lafferty Maths Dept

  10. The Gradient S4 Credit Now Try MIA Exercise 3.1 Ch7 (page 139) www.mathsrevision.com Created by Mr.Lafferty Maths Dept

  11. S4 Credit Starter Questions Q1. A house 4 years ago is valued at £50 000. Calculate it’s value if it has increased by 5%. Q2. Calculate 3.36 x 70 to 2 significant figures. www.mathsrevision.com Q3. Write down the 3 ways of factorising. Created by Mr.Lafferty Maths Dept

  12. S4 Credit The Gradient of a Line Learning Intention Success Criteria • To explain positive and negative gradients using coordinates. • Know gradient formula. • 2. Calculate gradients given two coordinates. www.mathsrevision.com Created by Mr.Lafferty Maths Dept

  13. The gradient using coordinates S4 Credit m = gradient y-axis y2 www.mathsrevision.com y1 x-axis O x1 x2 www.mathsrevision.com

  14. The gradient using coordinates S4 Credit Find the gradient of the line. m = gradient y-axis www.mathsrevision.com x-axis O www.mathsrevision.com

  15. The gradient using coordinates S4 Credit Find the gradient of the two lines. y-axis x-axis O www.mathsrevision.com

  16. The gradient using coordinates S4 Credit The gradient formula is : www.mathsrevision.com It is a measure of how steep a line is A line sloping up from left to right is a positive gradient A line sloping down from left to right is a negative gradient Created by Mr.Lafferty Maths Dept

  17. The gradient using coordinates S4 Credit Now try MIA Ex 4.2 Ch7 (page 143) www.mathsrevision.com Created by Mr.Lafferty Maths Dept

  18. S4 Credit Straight Line Graphs Learning Intention Success Criteria • To draw graphs by using a coordinate table • Understand the keypoints of drawing a straight line graph • Be able to plot a straight line graph www.mathsrevision.com Created by Mr.Lafferty Maths Dept

  19. y 10 9 8 7 6 5 4 3 2 1 x 0 -10 1 2 3 4 5 6 7 9 10 -9 -8 -6 -4 -3 -2 8 -7 -5 -1 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 Drawing Straight Line Graphs y = ax + b y = x x y 0 3 -4 y = 3x+1 0 3 -4 x y 2 -2 0 -5 1 7 y = 2x y = x - 3 x y x y 8 0 4 0 3 -4 -3 1 5 0 6 -8 Created by Mr. Lafferty Maths Dept

  20. S4 Credit Straight Line Graphs • Key Points • Make a table • Calculate and plot 3 coordinates • 3. Draw a line through points www.mathsrevision.com Created by Mr.Lafferty Maths Dept

  21. Straight Line Graphs S4 Credit Now try MIA Ex5.1 Q5 Only Ch7 (page 145) www.mathsrevision.com Created by Mr.Lafferty Maths Dept

  22. S4 Credit Starter Questions Q1. Write out in full to 2 sig. figs. Q2. A superstore make 20% profit on each can of soup they sell. If they buy in a can for 50p. What is the selling price. www.mathsrevision.com Q3. Explain why 4x2 – 16 = 4(x + 2)( x – 2) Created by Mr.Lafferty Maths Dept

  23. S4 Credit The Straight Line Equation Learning Intention Success Criteria • To explain the connection between the straight line equation and the gradient. • Understand the term standard form. • 2. Identity the gradient m from the standard form. www.mathsrevision.com Created by Mr.Lafferty Maths Dept

  24. y 10 9 8 7 6 5 4 3 2 1 x 0 -10 1 2 3 4 5 6 7 9 10 -9 -8 -6 -4 -3 -2 8 -7 -5 -1 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 Straight line equation and the gradient connection y = -x - 5 y = 2x + 1 x y 3 0 1 x y 0 1 3 -5 -6 -8 1 3 7 m = -1 m = 2 Created by Mr. Lafferty Maths Dept

  25. Straight Line Equation S4 Credit All straight lines have the equation of the form y = mx + c Let’s investigate properties www.mathsrevision.com (You need GeoGebra to run link) Created by Mr.Lafferty Maths Dept

  26. Straight Line Equation y S4 Credit 10 All straight lines have the equation of the form lines are parallel if they have the same gradient 9 8 y = mx + c 7 6 5 4 3 www.mathsrevision.com Where line meets y-axis 2 Gradient 1 x 0 10 1 2 3 4 5 6 7 8 9 Find the equations of the following lines y = -0.5x+2 y = x y = x+4 y = 4x+2 Created by Mr.Lafferty Maths Dept

  27. The Gradient of a Line S4 Credit Now try MIA Ex 6.1 Ch7 (page 146) www.mathsrevision.com Created by Mr.Lafferty Maths Dept

  28. S4 Credit Starter Questions Q1. Write out in full to 1 significant fig. Q2. A computer store buys in a laptop for £500. They want to make a 40% profit. How much do they sell it for. www.mathsrevision.com Q3. A line is parallel to y = 2x. Write down its equation Created by Mr.Lafferty Maths Dept

  29. lines are parallel if same gradient Straight Line Equation S4 Credit All straight lines have the equation of the form Slope left to right upwards positive gradient y = mx + c www.mathsrevision.com y - intercept Gradient y intercept is were line cuts y axis Slope left to right downwards negative gradient Created by Mr.Lafferty Maths Dept

  30. Equation of a Straight Line y = mx+c S4 Credit To find the equation of a straight line we need to know Two points that lie on the line ( x1, y1) and ( x2, y2) Or The gradient and a point on the line m and (a,b) www.mathsrevision.com www.mathsrevision.com

  31. Equation of a Straight Line y = mx+c S4 Credit Find the equation of the straight line passing through the points (4, 4) and (8,24). Solution www.mathsrevision.com Using the point (4,4) and the gradient m = 5 sub into straight line equation y = mx + c  4 = 5 x 4 + c  c = 4 - 20 = -16 Equation : y = 5x - 16

  32. Equation of a Straight Line y = mx+c S4 Credit Find the equation of the straight line passing through the points (3, -5) and (6,4). Solution www.mathsrevision.com Using the point (6,4) and the gradient m = 3 sub into straight line equation y = mx + c  4 = 3 x 6 + c  c = 4 - 18 = -14 Equation : y = 3x - 14

  33. Straight Line Graphs S4 Credit Now try MIA Ex 6.2 Ch7 (page 147 ) www.mathsrevision.com Created by Mr.Lafferty Maths Dept

  34. S4 Credit Starter Questions Q1. The points ( 1, 4) and (3, 11) lie on a line. Find the gradient of the line. Q2. Complete the table given : y = 3x + 1 www.mathsrevision.com Q3. Are the two lines parallel. Explain your answer y = x + 2 and y = 2x + 2 Created by Mr.Lafferty Maths Dept

  35. S4 Credit Modelling Using Straight Line Equation Learning Intention Success Criteria • How to model real life situations using straight line theory • Be able to work out gradient and y intercept using a graph. www.mathsrevision.com • Form an equation for any straight line graph. Created by Mr.Lafferty Maths Dept

  36. Pick any 2 points Modelling Real – life C The cost for hiring a plumber per hour is shown below 100 90 80 10 70 Cost £ (7,100) 60 (a) Calculate the gradient. 50 40 30 • What is the value of C • when T = 0 20 30 30 10 T 0 1 2 3 4 5 6 7 9 10 8 • Write down an equation • connecting C and T. Time (hours) (0,30) C = T + (d) Find the cost for a plumber for 10 hours? T = 10 C = 10x10 + 30 = £130

  37. Pick any 2 points Modelling Real – life V The graph shows how a the volume of water tank drains over time. 100 90 80 -5 70 Volume 60 (a) Calculate the gradient. 50 (0,80) 40 30 • What is the value of V • when T = 0 20 80 80 10 T (8,40) 0 1 2 3 4 5 6 7 9 10 8 • Write down an equation • connecting V and T. Time (mins) V = T + (d) How long before the tank is empty? V = 0 0 = -5xT + 80 T = (-80) ÷(–5) = 16mins

  38. Straight Line Equation S4 Credit Now try Ex 7.1 MIA (page 149) www.mathsrevision.com Created by Mr.Lafferty Maths Dept

  39. S4 Credit Starter Questions Q1. The points ( 5, 7) and (7, 21) lie on the same line. Find the gradient of the line. Q2. Complete the table given : y = 2x + 10 www.mathsrevision.com Q3. Are the two lines parallel. Explain answer y = -2x + 1 and y = 2x + 1 Created by Mr.Lafferty Maths Dept

  40. S4 Credit Best Fit Straight Line Equation Learning Intention Success Criteria • How to model real life situations using straight line theory • Be able to work out gradient and y intercept using a graph. www.mathsrevision.com • Form an equation for any straight line graph. Created by Mr.Lafferty Maths Dept

  41. Best-Fitting Straight line Data was collected on pupils weight and height. Data is plotted below. Pick any 2 points (a) Is there correlation between height and weight. Yes, a positive correlation h 200 • Draw in the best-fit line and • find an equation relating • height and weight. x (70,180) 180 160 140 120 Height (cms) 100 2.57 80 60 c = 0 0 40 20 (0,0) x w 0 10 20 30 40 50 60 70 90 100 80 h = w + Weight (Kgs) (d) What height is a pupils who weights 40 kgs? w = 40 h = 2.57x40 = 102.8 cms

  42. Best-Fitting Straight line A survey was carried out on the value of cars depending on their age The data is plotted below. Pick any 2 points (a) Is there correlation between value and age. v Yes, a negative correlation 20 18 (0,18) • Draw in the best-fit line and • find an equation relating • value and year. x 16 14 12 value £ ‘ 000 10 8 (8,8) x -1.25 6 4 c = 18 18 2 y 0 1 2 3 4 5 6 7 9 10 8 Years v = y + (d) What is the cost of a car after 4years? y = 4 v = -1.25x 4 + 18 = 13 Value = £13 000

  43. Best Fit Straight Line Equation S4 Credit Now try Ex 8.1 MIA (page 151) www.mathsrevision.com Created by Mr.Lafferty Maths Dept

  44. Exam Type Straight Line Questions S4 Credit Example The graph shows an electrician’s charging system. He has a call-out charge plus an hourly rate. www.mathsrevision.com a) Work out the equation of the line. b) Calculate the cost of a 10 hour job.

  45. Exam Type Straight Line Questions = 10 S4 Credit a) Pick two convenient points on the line, say (0, 30) and (5, 80). Thinking of the form y = mx + c The y-intercept gives c = 30 The equation is: y = 10x + 30 The label on the y-axis is C and the label on the x-axis is T So the equation is C = 10 T + 30 www.mathsrevision.com So the equation is C = 10 T + 30 C = 10 x 10 + 30 = 130 b) Find C when T = 10 The cost of a 10 hr job is £130.

  46. Exam Type Straight Line Questions S4 Credit The tank of a car contains 5 litres of petrol. 1. The graph below shows how the volume of petrol in this tank changes as a further 45 litres of petrol is pumped in at a steady rate for 60 seconds. www.mathsrevision.com Find the equation of the straight line in terms of V and t.

  47. Exam Type Straight Line Questions S4 Credit In the diagram below. A is the point (-1, -7) and B is the point (4, 3). 2. • (a) Find the gradient of the line AB. • AB cuts the y-axis at the point (0, -5). • Write down the equation of the line AB. • The point (3k, k) lies on the line AB. • Find the value of k.

  48. Exam Type Straight Line Questions S4 Credit 3. A water pipe runs between two buildings. These are represented by the points A and B in the diagram below. www.mathsrevision.com (a) Using the information in the diagram, show that the equation of the line AB is 3y – x = 6.

  49. Exam Type Straight Line Questions S4 Credit • An emergency outlet pipe has to be built across the main pipe. The line representing this outlet pipe has equation • 4y + 5x = 46. www.mathsrevision.com Calculate the coordinates of the point on the diagram at which the outlet pipe will cut across the main water pipe.

  50. Exam Type Straight Line Questions S4 Credit www.mathsrevision.com

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