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GCSE: Curved Graphs
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GCSE Specification Plot and recognise quadratic, cubic, reciprocal, exponential and circular functions. Plot and recognise trigonometric functions 𝑦= sin 𝑥 and 𝑦= cos 𝑥 , within the range -360° to +360° Use the graphs of these functions to find approximate solutions to equations, eg given x find y (and vice versa) Find the values of p and q in the function 𝑦=𝑝 𝑞 𝑥 given coordinates on the graph of 𝑦=𝑝 𝑞 𝑥 1 3 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 4 “Given that 2,6 and 5,162 are points on the curve 𝑦=𝑘 𝑎 𝑥 , find the value of 𝑘 and 𝑎.” The graph shows 𝑦= cos 𝑥 . Determine the coordinate of point 𝐴.
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Skill #1: Recognising Graphs
Linear 𝒚=𝒂𝒙+𝒃 When 𝑎>0 𝒚=𝒂𝒙+𝒃 When 𝑎<0 ? ? ? The line is known as a straight line.
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Skill #1: Recognising Graphs
Quadratic 𝑦=𝑎 𝑥 2 +𝑏𝑥+𝑐 When 𝑎>0 𝑦=𝑎 𝑥 2 +𝑏𝑥+𝑐 When 𝑎<0 ? ? The line for a quadratic equation is known as a parabola. ?
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Skill #1: Recognising Graphs
Cubic 𝑦=𝑎 𝑥 3 +𝑏 𝑥 2 +𝑐𝑥+𝑑 When 𝑎>0 𝑦=𝑎𝑥3 When 𝑎>0 y ? ? x 𝑦=𝑎 𝑥 3 When 𝑎<0 𝑦=𝑎 𝑥 3 +𝑏 𝑥 2 +𝑐𝑥+𝑑 When 𝑎<0 y ? ? x
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Skill #1: Recognising Graphs
Reciprocal 𝑦= 𝑎 𝑥 When 𝑎>0 𝑦= 𝑎 𝑥 When 𝑎<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.
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Skill #1: Recognising Graphs
Exponential 𝑦=𝑎× 𝑏 𝑥 ? y 𝑎 x The y-intercept is 𝑎 because 𝑎× 𝑏 0 =𝑎×1=𝑎. (unless 𝑎 = 0, but let’s not go there!) ?
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Skill #1: Recognising Graphs
Circle The equation of this circle is: x2 + y2 = 25 𝑦 ? 5 The equation of a circle with centre at the origin and radius r is: 𝑥 2 + 𝑦 2 = 𝑟 2 𝑥 -5 5 -5
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Quickfire Circles ? ? ? ? ? ? x2 + y2 = 1 x2 + y2 = 9 x2 + y2 = 16
3 4 -1 1 -3 3 -4 4 -1 -3 -4 ? ? x2 + y2 = 1 x2 + y2 = 9 x2 + y2 = 16 ? ? 8 10 6 -8 8 -10 10 -6 6 -8 -10 -6 ? x2 + y2 = 64 x2 + y2 = 100 x2 + y2 = 36
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Matching Activity Match the graphs with the equations. A B C D E F G H
Equation types: A: quadratic B: cubic C: quadratic D: cubic E: cubic F: reciprocal G: cubic H: reciprocal I: exponential J: linear K: Trig Function ? ? ? ? ? E F G H ? ? ? ? ? ? I J K 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 ix) y = 2x3 xii) y = 2x – 3
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𝑦= sin 𝑥 ? ? ? ? ? Skill #2: Plotting and recognising trig functions.
8 1 -1 𝑦= sin 𝑥 𝑥 90 180 270 360 𝑦 1 -1 ? ? ? ? ? Get students to sketch axes and tables in their books. Skill #2: Plotting and recognising trig functions. Click to sketch
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Test Your Understanding
? ?
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8 1 -1 𝑦= cos 𝑥 𝑥 90 180 270 360 𝑦 1 -1 ? ? ? ? ? Get students to sketch axes and tables in their books. Click to brosketch
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Quickfire Coordinates
𝑦= sin 𝑥 𝑦= cos 𝑥 𝑦= sin 𝑥 𝑦= cos 𝑥 𝐷 𝐵 𝐶 𝐴 𝐴 270, −1 ? 𝐵 90, 0 ? 𝐶 360, 0 ? 𝐷 0, 1 ? 𝑦= sin 𝑥 𝑦= cos 𝑥 𝑦= sin 𝑥 𝑦= cos 𝑥 𝐺 𝐸 𝐻 𝐹 𝐸 180, 0 ? 𝐹 180,−1 ? 𝐺 90,1 ? 𝐻 270, 0 ?
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SKILL #3: Using graphs to estimate values
The diagram shows the graph of y = x2 – 5x – 3 a) Find the exact value of 𝑦 when 𝑥=−2. b) Use the graph to find estimates for the solutions of (i) x2 – 5x – 3 = 0 (ii) x2 – 5x – 3 = 6 Tip for (b): Look at what value has been substituted into the equation in each case. 𝑦= −2 2 −5 −2 −3 =11 i) When 𝑦=0, then using graph, roughly 𝒙=−𝟎.𝟓 𝒐𝒓 𝒙=𝟓.𝟓 ii) 𝒙=−𝟏.𝟒 𝒐𝒓 𝒙=𝟔.𝟒 ? ? ?
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Test Your Understanding
The graph shows the line with equation 𝑦= 𝑥 2 +𝑥−12 Find estimates for the solutions of the following equations: i) 𝑥 2 +𝑥−12=5 𝒙=−𝟒.𝟔 𝒐𝒓 𝒙=𝟑.𝟔 ii) 𝑥 2 +𝑥−12=−7 𝒙=−𝟐.𝟖 𝒐𝒓 𝒙=𝟏.𝟖 ? ?
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Using a Trig Graph Q ? Q ? 𝒙=𝟏𝟑𝟓° 𝒙=𝟑𝟑𝟎° Suppose that sin 45 = 1 2
Using the graph, find the other solution to sin 𝑥 = 1 2 1 1 2 𝒙=𝟏𝟑𝟓° ? 𝟒𝟓 𝟏𝟑𝟓 -1 We can see by symmetry that the difference between 0 and 45 needs to be the same as the difference between 𝑥 and 180. Q Suppose that sin 210 =− 1 2 Using the graph, find the other solution to sin 𝑥 =− 1 2 𝒙=𝟑𝟑𝟎° ?
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Test Your Understanding
1 -1 The graph shows the line with equation 𝑦= cos 𝑥 Given that cos 60 = 1 2 , find the other solution to cos 𝑥 = 1 2 𝒙=𝟑𝟎𝟎° Given that cos 150° =− , find the other solution to cos 𝑥 =− 𝒙=𝟐𝟏𝟎° ? ? Get students to sketch axes and tables in their books.
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Exercise 1 (on provided sheet)
Identify the coordinates of the indicated points. Match the graphs to their equations. 1 3 𝐴 𝑦=3× 2 𝑥 𝑦= sin 𝑥 𝐶 𝐵 𝑦= 4 𝑥 𝑥 2 + 𝑦 2 =9 𝐸 𝐷 1 𝑨 𝟗𝟎,𝟏 𝑩 𝟏𝟖𝟎,𝟎 𝑪 𝟎,𝟑 𝑫 −𝟑,𝟎 𝑬 𝟏,𝟒 ? ? ? ? ? Which of these graphs could have the equation 𝑦= 𝑥 3 −2 𝑥 2 +3? 2 i. 𝑦=4 sin 𝑥 E ii. 𝑦=4 cos 𝑥 B iii. 𝑦= 𝑥 2 −4𝑥 F iv. 𝑦=4× 2 𝑥 C v. 𝑦= 𝑥 D vi. 𝑦= 4 𝑥 A ? a b c c, because a is the wrong way up (given 𝒙 𝟑 term has positive coefficient) and b has the wrong y-intercept. ?
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Exercise 1 (on provided sheet)
4 ? ? ? ? Reveal
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Exercise 1 The graph shows 𝑦= 𝑥 2 −𝑥−2. Use the graph to estimate the solution(s) to: i) 𝑥 2 −𝑥−2= 𝒙=−𝟐 𝒐𝒓 𝟑 ii) 𝑥 2 −𝑥−2=− 𝒙≈−𝟎.𝟔 𝒐𝒓 𝟏.𝟔 iii) 𝑥 2 −𝑥−2= 𝒙≈−𝟐.𝟓 𝒐𝒓 𝟑.𝟓 5 7 Using the cos graph below, and given that cos 45 = , find all solutions to cos 𝑥 = (other than 45). 𝒙=𝟑𝟏𝟓° a ? ? ? ? Given that cos 30 = , find all solutions to cos 𝑥 = 𝒙=𝟑𝟑𝟎° [Hard] Given cos 60 = 1 2 , again using the graph, find all solutions to 𝑐𝑜𝑠 𝑥 =− 1 2 𝒙=𝟏𝟐𝟎°, 𝟐𝟒𝟎° 6 The graph shows the line with equation 𝑦=6+2𝑥− 𝑥 2 Use the graph to estimate the solution(s) to: i) 6+2𝑥− 𝑥 2 = 𝒙≈−𝟏.𝟔𝟓 𝒐𝒓 𝟑.𝟐𝟓 ii) 6+2𝑥− 𝑥 2 = 𝒙≈−𝟎.𝟕 𝒐𝒓 𝟐.𝟕 iii) By drawing a suitable line onto the graph, estimate the solutions to 6+2𝑥− 𝑥 2 =𝑥 𝒙≈−𝟏.𝟓𝟔 𝒐𝒓 𝟐.𝟓𝟔 b ? c ? ? ? ?
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Exercise 1 8 Given sin 60 = , determine all solutions to sin 𝑥 = 𝒙=𝟏𝟐𝟎 (,𝟔𝟎) Given sin 30 = 1 2 , determine all solutions to sin 𝑥 = 1 2 𝒙=𝟏𝟓𝟎 (,𝟑𝟎) [Harder] Given sin 45 = , determine the two solutions to sin 𝑥 =− (note the minus) 𝒙=𝟐𝟐𝟓°, 𝟑𝟏𝟓° ? ? ?
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SKILL #4: Finding constants of 𝑦=𝑎× 𝑏 𝑥
The graph shows two points (1,7) and (3,175) on a line with equation: 𝒚=𝒌 𝒂 𝒙 Determine 𝑘 and 𝑎 (where 𝑘 and 𝑎 are positive constants). (3,175) Answer: 𝟕=𝒌 𝒂 𝟏 𝟏𝟕𝟓=𝒌 𝒂 𝟑 Dividing: 𝟐𝟓= 𝒂 𝟐 𝒂=𝟓 Substituting back into 1st equation: 𝒌= 𝟕 𝟓 (1,7) 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?
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Test Your Understanding
Given that 2,6 and 5,162 are points on the curve 𝑦=𝑘 𝑎 𝑥 , find the value of 𝑘 and 𝑎. 6=𝑘 𝑎 =𝑘 𝑎 5 →27= 𝑎 3 𝒂=𝟑 𝒌= 𝟔 𝟑 𝟐 = 𝟐 𝟑 Given that 3, 45 and 1, are points on the curve 𝑦= 𝑎 2 𝑏 𝑥 where 𝑎 and 𝑏 are positive constants, find the value of 𝑎 and 𝑏. 45= 𝑎 2 𝑏 = 𝑎 2 𝑏 →25= 𝑏 2 𝒃=𝟓 𝒂= 𝟒𝟓 𝒃 𝟑 = 𝟒𝟓 𝟏𝟐𝟓 = 𝟗 𝟐𝟓 = 𝟑 𝟓 Q ? N ?
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Exercise 1 (continued) ? ? ? ? 1 3 4* 2
Given that the points (1,6) and 4,48 lie on the exponential curve with equation 𝑦=𝑏× 𝑎 𝑥 , determine 𝑎 and 𝑏. 𝟔=𝒃𝒂 𝟒𝟖=𝒃 𝒂 𝟒 →𝟖= 𝒂 𝟑 𝒂=𝟐 𝒃=𝟑 Given that the points (2,48) and 5,3072 lie on the exponential curve with equation 𝑦=𝑏× 𝑎 𝑥 , determine 𝑎 and 𝑏. 𝒂=𝟒, 𝒃=𝟑 Given that the points (1,3) and 3,108 lie on the exponential curve with equation 𝑦=𝑏× 𝑎 𝑥 , determine 𝑎 and 𝑏. 𝒂=𝟑, 𝒃= 𝟏 𝟐 Given that the points (3, 1 72 ) and 7, lie on the exponential curve with equation 𝑦= 𝑏 2 𝑎 𝑥 , determine 𝑎 and 𝑏. 𝒂= 𝟏 𝟐 ,𝒃= 𝟏 𝟑 1 3 ? ? 4* 2 ? ?
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