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GCSE: Curved Graphs Dr J Frost ([email protected]) GCSE Revision Pack Reference: 94, 95, 96, 97, 98 Last modified: 31st December 2014 GCSE Specification 1 Plot and recognise quadratic, cubic, reciprocal, exponential and circular functions. 3 Use the graphs of these functions to find approximate solutions to equations, eg given x find y (and vice versa) The diagram shows the graph of y = x2 – 5x – 3 (a) Use the graph to find estimates for the solutions of (i) x2 – 5x – 3 = 0 (ii) x2 – 5x – 3 = 6 2 Plot and recognise trigonometric functions = sin and = cos , within the range -360° to +360° The graph shows = cos . Determine the coordinate of point . q in 4 Find the values of p and the function = given coordinates on the graph of = “Given that 2,6 and 5,162 are points on the curve = , find the value of and .” Skill #1: Recognising Graphs Linear = + = + When > 0 When < 0 ? ? ? The line is known as a straight line. Skill #1: Recognising Graphs Quadratic = 2 + + = 2 + + When > 0 When < 0 ? ? The line for a quadratic equation is known as a parabola. ? Skill #1: Recognising Graphs Cubic = = 3 + 2 + + 3 When > 0 When > 0 y ? = 3 When < 0 ? x = 3 + 2 + + When < 0 y ? x ? Skill #1: Recognising Graphs Reciprocal = = When > 0 When < 0 ? ? The lines x = 0 and y = 0 are called asymptotes. ! An asymptote is a straight line which the ? curve approaches at infinity. You don’t need to know this until A Level. Skill #1: Recognising Graphs Exponential = × y ? x The y-intercept is because × 0 = × 1 =?. (unless = 0, but let’s not go there!) Skill #1: Recognising Graphs Circle The equation of this circle is: x2 + y2 =? 25 5 5 -5 -5 ! The equation of a circle with centre at the origin and radius r is: 2 + 2 = 2 Quickfire Circles 1 3 1 -1 -1 2 = 16 x2 + y? 6 10 8 -8 x2 + y2 = 64 10 -10 4 -4 -4 x2 + y 2 = 9 8 ? 3 -3 x2 + y?2 = 1 -8 ? -3 4 -10 2 = 100 x2 + y? ? -6 6 -6 x2 + y2 = 36 Card Sort A Match the graphs with the equations. B E C F I G J K D Equation types: H A: quadratic ? B: cubic ? C: quadratic ? D: cubic ? E: cubic ? F: reciprocal ? G: cubic ? H: reciprocal ? ? I: exponential J: linear ? ? K: sinusoidal ? L: fictional L i) y = 5 - 2x2 iv) y = 3/x vii) y=-2x3 + x2 + 6x x) y = x2 + x - 2 ii) y = 4x v) y = x3 – 7x + 6 viii) y = -2/x xi) y = sin (x) iii) y = -3x3 vi) ix) y = 2x3 xii) y = 2x – 3 Click to reveal answers. 0 8 90 0 ? 1 ? 180 270 360 0? -1? 0? = sin 1 90 180 270 360 -1 Skill #2: Plotting and recognising trig functions. Click to brosketch Test Your Understanding 90 ?1 180? 0 0 8 90 1 ? 0 ? 180 270 360 -1? 0? 1? = cos 1 90 180 270 360 -1 Click to brosketch Quickfire Coordinates = sin = sin = cos = cos 270, ? −1 = sin 90, ?0 360, ? 0 = sin = cos 0, ?1 = cos 180, ? 0 180, ? −1 90,1 ? 270, ? 0 SKILL #3: Using graphs to estimate values The diagram shows the graph of y = x2 – 5x – 3 a) Find the exact value of when = −2. b) Use the graph to find estimates for the solutions of (i) x2 – 5x – 3 = 0 (ii) x2 – 5x – 3 = 6 Bro Tip for (b): Look at what value has been substituted into the equation in each case. a) = −2 2 − 5 −2 − 3 ? = 11 b) i) When = 0, then using graph, ? roughly = −. = . ii) = −. = . ? Test Your Understanding The graph shows the line with equation = 2 + − 12 Find estimates for the solutions of the following equations: i) 2 + − 12 = 5 = −. = . ? ii) 2 + − 12 = −7 = −. = . ? Using a Trig Graph Suppose that sin 45 = Q 1 1 Using the graph, find the other 1 solution to sin = 2 = ° ? 2 -1 1 2 90 We can see by symmetry that the difference between 0 and 45 needs to be the same as the difference between and 180. 180 270 360 1 Q Suppose that sin 210 = − 2 Using the graph, find the other 1 solution to sin = − 2 = ° ? Test Your Understanding The graph shows the line with equation = cos 1 1 a) Given that cos 60 = , find the other solution to cos = 2 2 = ° 1 b) Given that cos 150° = − 90 -1 180 3 , 2 ? find the other solution to cos = − = ° ? 270 360 3 2 Exercise 1 (on provided sheet) 3 Match the graphs to their equations. 1 Identify the coordinates of the indicated points. = sin 2 + 2 = 9 = 4 1 , ? ,? ,? , ? −,? 2 Which of these graphs could have the equation = 3 − 2 2 + 3? a b c c, because a is the wrong way up (given term has positive coefficient) and b has the wrong y-intercept. ? i. = 4 sin ii. = 4 cos iii. = 2 − 4 + 5 iv. = 4 × 2 v. = 3 + 4 4 vi. = E B F C? D A Exercise 1 (on provided sheet) 4 -15 ? -7 ? -6 ? ?1 Reveal Exercise 1 5 The graph shows = 2 − − 2. 7 Using the cos graph below, and given a that cos 45 = 12, find all solutions to cos = 1 2 (other than 45). = ° ? Use the graph to estimate the solution(s) to: i) 2 − − 2 = 4 = − 2 ii) − − 2 = −1 ≈ −. . 2 iii) − − 2 = 7 ≈ −. . ? ? ? 6 The graph shows the line with equation = 6 + 2 − 2 b Given that cos 30 = 3 , 2 3 2 find all solutions to cos = = ° ? c Use the graph to estimate the solution(s) to: i) 6 + 2 − 2 = 0 ≈ −. . 2 ii) 6 + 2 − = 4 ≈ −. . iii) By drawing a suitable line onto the graph, estimate the solutions to 6 + 2 − 2 = + 2 ≈ −. . ? ? 1 [Hard] Given cos 60 = , again 2 using the graph, find all solutions to 1 = − 2 = °, ° ? Exercise 1 8 3 , 2 i) Given sin 60 = ii) solutions to sin = 2 = (, ) 1 Given sin 30 = , determine all determine all 3 ? 2 1 solutions to sin = 2 = (, ) iii) [Harder] Given sin 45 = ? 1 , 2 determine the two solutions to 1 sin = − (note the minus) 2 = °, ° ? SKILL #4: Finding constants of = ⋅ The graph shows two points (1,7) and (3,175) on a line with equation: = (3,175) (1,7) Determine and (where and are positive constants). Answer: Dividing: Bro Hint: Substitute the values of the coordinates in to form two equations. You’re used to solving simultaneous equations by elimination – either adding or subtracting. Is there another arithmetic operation? = = = =? Substituting back into 1st equation: = Test Your Understanding Q N Given that 2,6 and 5,162 are points on the curve = , find the value of and . 6 = 2 162 = 5 → 27 = 3 ? = = = 9 Given that 3, 45 and 1, 5 are points on the curve = 2 where and are positive constants, find the value of and . 45 = 2 3 9 = 2 5 → 25 = 2 ? = = = = Exercise 1 (continued) 9 Given that the points (1,6) and 4,48 lie on the exponential curve with equation = × , determine and . = = → = = = 3 Given that the points (1,3) and 3,108 lie on the exponential curve with equation = × , determine and . = , = ? ? 2 Given that the points (2,48) and 5,3072 lie on the exponential curve with equation = × , determine and . = , = ? 4 Given that the points (3, 1 1 ) and 72 7, lie on the exponential 1152 curve with equation = 2 , determine and . = , = ?