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Plotting a Graph

Plotting a Graph. y=2x+3. The value of y depends on x For example: If x=1 then y=2(1)+3 =5. (0,3). 2(0)+3. 3. 5. 2(1)+3. (1,5). 2(2)+3. (2,7). 7. 2(3)+3. (3,9). 9. 2(4)+3. (4,11). 11. Gradients. Gradient=slope =rate of change

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Plotting a Graph

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  1. Plotting a Graph y=2x+3 The value of y depends on x For example: If x=1 then y=2(1)+3 =5 (0,3) 2(0)+3 3 5 2(1)+3 (1,5) 2(2)+3 (2,7) 7 2(3)+3 (3,9) 9 2(4)+3 (4,11) 11

  2. Gradients Gradient=slope =rate of change =how steep the line is =rise run m= 5 2

  3. To read a gradient: we start on the line and go up or down first and then back to the line m= 2/3 How’s this graph different from the other 2 graphs? m= 3/4 ?? m= - 3/4 m= 3/1 m= 3

  4. Draw lines with the following gradients 2/3 3 -1/2 -2 1) Decide which way the line goes by the + , – sign 2) Count rise and run

  5. Jonathan Climbs a hill. The gradient is 3/2 when he goes uphill, the gradient is -2/5 when he goes downhill. Draw the hill Jonathan climbs. (Assume that it’s a very pointy hill)

  6. y= 3x-2 y= -x+5 y=1/2x+2 y= 4-x

  7. Gradient Intercept Method The equation is in the form of y=mx+c gradient y intercept 2 points determine a line!!!!! (you need 2 points to draw a line)

  8. y=3x-2 y=-x+5 y=3 x + 6 5 y=4 - 1x 2

  9. Application of linear graphs

  10. Cost $ Cost $ Time hours Weight tonnes Applications of Graphs Gradient Water Level litres Time miniutes

  11. Interpreting Gradient Cooking time y Water level of a reservoir meter 5 hours 4 4 days x Gradient=m/day How many meters the water level drops per day 3 5 kg Gradient=hour/kg How many hours does it take to cook 1 kg of food. If the equation of the line is y= -x+5 How long would it take before the reservoir is empty? 7 Temperature of an item in the freezer 5 temp 2 minutes Gradient=temp/min How much the temp drops per minute 6.5 5 2

  12. The pizzeria Italiano specializes in selling large size Pizzas • The relationship between x ( the number of pizzas they sell) and • y (their daily costs) is given by the equation y=10x+50. • Draw a graph showing the number of pizzas on the x-axis, • and the daily costs on the y x-axis. • b) What are their costs if they sell 8 pizzas? • c) If their costs are $100, how many pizzas did they sell. • d) What’s the y-intercept, What does it represent? • e) Give the gradient. What does it represent?

  13. . The time it takes for a block of ice to melt can be represented by a straight-line graph. x=time (hours) y=volume of the block of ice (litres) There are two blocks of ice, A and B. The equation for block A is y=6-1/2x. The equation of block B is y=8-x • Draw the lines for the two blocks of ice • How large is block A to start with? • Explain what the y-intercept for block B graph represents? • When are the two blocks the same volume? • What do the x-intercepts for the 2 lines tell you?

  14. Linear graph revision

  15. Parabola factorised form

  16. Draw the following parabola y=3x2 y=(x+2)2 – 4 y=-x2 + 5 y= -(x+3)2 + 2

  17. y=(x-2)(x-3)Is it a parabola???

  18. y=(x-1)(x-5) Step a Find the two x intercepts Put y=0 (x-1)(x-5)=0 x=1 and x=5 Step b Find the y intercept Put x=0 y=(0-1)(0-5) y= 5 Step c Find the line of symmetry Mid-way between 1 and 5 x=(1+5)÷2=3 Step d Find the turning point Substitute x=3 into y=(x-1)(x-5) y=(3-1)(3-5) y=2×-2 y=-4 Turning point (3,-4)

  19. Step (a) Find the x- and y-intercepts by putting y = 0 and x = 0. Step (b) Find the axis of symmetry midway between the two x-intercepts. Step (c) Find the coordinates of turning point. Substitute the mid-point of two X-intercepts in to the equation to get y

  20. Parabolas in factorised form To find the symmetry

  21. Step (a) Find the x-intercepts by putting y = 0 Step Step (b) Find the y-intercept by putting x=0 Step (c) Find the axis of symmetry midway between the two x-intercepts. Step (d) Find the coordinates of turning point.( , ) Substitute the mid-point of two X-intercepts in to the equation to get y y=(x-4)(x+2)

  22. y=(x-4)(x+2) Step a Find the two x intercepts Put y=0 (x-4)(x+2)=0 x=4 and x=-2 Step b Find the y intercept Put x=0 y=(0-4)(0+2) y= -8 Step c Find the line of symmetry Mid-way between 2 and -3 x=(4+-2)÷2=1 Step d Find the turning point Substitute x=1 into y=(x-4)(x+2) y=(1-4)(1+2) y=-3×3 y=-9 Turning point (1,-9)

  23. 3) Line of symmetry is at y=-3 4) Turning point is at (-3,0)

  24. P259 14.3 Q1 and Q5

  25. The sketch shows the function y = x(x - 2) (a) What are the coordinates of A? (b) What are the coordinates of B? (c) What is the equation of m? (d) What are the coordinates of the turning point of the curve? (e) What is the minimum value of the function? The sketch shows the function y = (x-2)(x+3) (a) What are the coordinates of A? (b) What are the coordinates of B? (c) What is the equation of D? (d) What are the coordinates of the turning point of the curve? (e) What is the minimum value of the function?

  26. Match up each of the graphs with the following functions

  27. 3) Line of symmetry is at y=-3 4) Turning point is at (-3,0)

  28. Parabola applications

  29. x-intercept: (0,0) Y-intercept: None Vertex: (0,0) x-intercept: (-4,0) (0,0) y-intercept: (0,0) Vertex: (-2, -4) X-intercpet: (-3,0) (3,0) Y-intercpet: (0,5) Vertex: (5,0) Find 1) X intercepts 2) Y intercept 3) Vertex (turning point) Alternatively Draw parabola: y= x2 + 2x - 8

  30. For the helicopter to fly above the rainbow parabola, how high must the helicopter fly? (In other words what is the maximum value of the parabola)

  31. Problem 2) Nick threw a ball out of a window that is 4 units high. The position of the ball is determined by the parabola y = -x² + 4. At how many feet from the building does the ball hit the ground? You need to draw a parabola There are two solutions. 2 and − 2. This picture assumes that Nick threw the ball to the right so that the balls lands at 2 feet away from the building.

  32. Problem 4) A ball is dropped from a height of 36 feet. The quadratic equation d = -t² + 36 provides the distance, d, of the ball, after t seconds. After how many seconds, does the ball hit the ground? d When the ball hits the ground d=0 So we are looking for X intercpt 0= -t² + 36 t= ±6 t=-6 is not a sensible answer So t=6 After 6 seconds d = -t² + 36 t ?

  33. The stream of water from a fountain can be modelled by a parabola with equation: H=(2 -d)( 1+d ) H is the height of the water stream above the ground and d is the distance from the wall (both in meters) Calculate the height of the fountain’s spout. How far from the wall does the water stream hit the ground? Blake stands 1.3m from the wall. He is 1.7m tall. Will he get wet ? H Wall 1.3 d ground

  34. Equation for parabola

  35. y=(x-3)(x-4) y=(x-2)2 + 3 In an instantWhat do you know about the following three parabola? y=x2

  36. y=(x+a)(x+b) y=(x+1)(x-3) When you know x-intercepts y=(x-a)2 + b y=(x-4)2 + 1 When you know the turning point Write an equation for a parabolaThere are 2 forms of parabola -1 3 1 (4,1) 4

  37. a b c -3 1 -3 3 f e d 3 1 1 -3 -3 g i h 3 -1 3 -3 -1

  38. j k l 1 3 -3 -1 3 n m -3 3

  39. End of the year revision

  40. 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 = x y = 2x y = -2/3x+4 y = 7 x=-7

  41. Intercept Method 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 3y-2x=12 2y+4x=8

  42. Straight Line Equation y 10 All straight lines have the equation of the form 9 8 y = ax + b 7 Line are parallel same gradient 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 = -2x+2 y = x y = x+4 y = 4x+2

  43. Straight Line Equation General C Q1. Find the connection between cost (C) and electricity used (E) 50 40 30 Since line passes through 15 on the y-axis b =15 20 10 www.mathsrevision.com Gradient a = E 0 10 20 30 50 40 Equation is

  44. Straight Line Equation General W Q2. The graph shows the connection between water flowing out of a tank (W) and time (T) 50 40 Downward slope 30 Since line passes through 40 on the y-axis b = 40 20 10 www.mathsrevision.com T Gradient a = 0 10 20 30 50 40 Equation is

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