IGCSE FM/C1 Sketching Graphs Dr J Frost (jfrost@tiffin.kingston.sch.uk) Objectives: (from the IGCSE FM specification) Last modified: 9th October 2015

Overview Over the next 5 lessons: #2: Specific skills in sketching (i) quadratics (ii) cubics and (iii) reciprocals #1: Shapes of graphs (quadratic, cubic, reciprocal) and basic features (roots, y-intercept, max/min points, asymptotes) C1 IGCSE FM C1 IGCSE FM #4: Graph transformations C1 only #3: Piecewise functions IGCSE FM only

#1 :: Features of graphs There are many features of a graph that we might want to identify when sketching. y-intercept? ? y as 𝑥→∞? ? ? y as 𝑥→−∞? ? Turning Points? ? Roots? ? Asymptotes? y= 1 𝑥+2 +2( 𝑥+2) 2 ! An asymptote is a straight line that a curve approaches at infinity (indicated by dotted line).

#1 :: Types of graphs There are three types of graphs you need to be able to deal with in C1 and/or IGCSE FM: 𝑦 𝑦 𝑦 𝑥 𝑥 𝑥 e.g. 𝑦= 𝑥 2 −4𝑥+7 e.g. 𝑦= 𝑥 3 − 𝑥 2 −𝑥+1 e.g. 𝑦= 1 𝑥 +2 Parabola (Quadratic Equation) Cubic Reciprocal At GCSE these were previously centred at the origin.

General shape: Smiley face or hill? RECAP :: Sketching Quadratics y 3 features needed in sketch? ? Roots x General shape: Smiley face or hill? ? y-intercept ?

Example 1 ? 𝑦= 𝑥 2 −𝑥−2 =(𝑥+1)(𝑥−2) Roots y-intercept Shape: smiley face or hill? ? y 𝑦= 𝑥 2 −𝑥−2 =(𝑥+1)(𝑥−2) x -1 2 So if 𝑦=0, i.e. 𝑥+1 𝑥−2 =0, then 𝑥=−1 or 𝑥=2. -2 When 𝑥=0, clearly 𝑦=−2.

Example 2 ? ? 𝑦=− 𝑥 2 +5𝑥−4 =− 𝑥 2 −5𝑥+4 =− 𝑥−1 𝑥−4 = (𝑥−1)(4−𝑥) Roots y-intercept Shape: smiley face or hill? y ? 𝑦=− 𝑥 2 +5𝑥−4 =− 𝑥 2 −5𝑥+4 =− 𝑥−1 𝑥−4 = (𝑥−1)(4−𝑥) ? x 1 4 Bro Tip: We can tidy up by using the minus on the front to swap the order in one of the negations. -4

Test Your Understanding So Far 𝑦= 𝑥 2 +3𝑥+2 𝑦=− 𝑥 2 +2𝑥+8 Roots? x = -1, -2 ? Roots? 𝑥=−2, 4 ? y-Intercept? y = 2 ? y-Intercept? 𝑦=8 ? ∩ or ∪ shape? ∪ ? ∩ or ∪ shape? ? ∩ ? y ? y 8 2 x x -2 -1 -2 4

Graph → Equation ? ? 𝑦 𝑦 𝑥 𝑥 1 2 −1 − 1 4 2 3 Find an equation for this curve, in the form 𝑎 𝑥 2 +𝑏𝑥+𝑐=0 where 𝑎,𝑏,𝑐 are integers. 𝒚= 𝟐𝒙−𝟏 𝒙+𝟏 𝒚=𝟐 𝒙 𝟐 +𝒙−𝟏 Find an equation for this curve, in the form 𝑎 𝑥 2 +𝑏𝑥+𝑐=0 where 𝑎,𝑏,𝑐 are integers. 𝒚=− 𝟑𝒙−𝟐 𝟒𝒙+𝟏 𝒚=−𝟏𝟐 𝒙 𝟐 +𝟓𝒙+𝟐 ? ?

Understanding features of a quadratic IGCSE FM June 2012 Paper 2 Q4 ? Positive (to give ∪ shape) ? Negative (this is 𝑦-intercept) ? Since 𝑦=0 the solutions are the roots. Thus (using the graph) one is positive, one negative. From graph, 𝑦 never drops below -3, so 0 solutions. ? ? Line is horizontal. So tangent is 𝑦=−3

𝑦= 𝑥 2 −6𝑥+10 = 𝑥−3 2 +1 RECAP :: Using completed square for min/max ? = 𝑥−3 2 +1 ? How could we use this completed square to find the minimum point of the graph? (Hint: how do you make 𝑦 as small as possible in this equation?) ? Anything squared must be at least 0. So to make the RHS as small as possible, we want 𝑥−3 2 to be 0. This happens when 𝑥=3. When 𝑥=3, 𝑦=1.

𝑦= 𝑥+𝑎 2 +𝑏 ! Write down The minimum point is −𝑎, 𝑏 . When we have a quadratic in the form: 𝑦= 𝑥+𝑎 2 +𝑏 The minimum point is −𝑎, 𝑏 .

Complete the table, and hence sketch the graphs Equation Completed Square x at graph min y at graph min y-intercept Roots? 1 y = x2 + 2x + 5 y = (x + 1)2 + 4 -1 4 5 None 2 y = x2 – 4x + 7 ? y = (x – 2)2 + 3 ? 2 ? 3 ? 7 None ? ? ? ? ? ? 3 y = x2 + 6x – 27 y = (x + 3)2 – 36 -3 -36 -27 x = 3 or -9 1 2 ? 3 ? 7 3 -9 5 (2,3) (-1,4) -27 (-3,-36)

Exercise 1 ? ? ? ? ? (Exercises on provided sheet) Sketch the following parabolas, ensuring you indicate any intersections with the coordinate axes. If the graph has no roots, indicate the minimum/maximum point. (a) 𝑦= 𝑥 2 −2𝑥 (b) 𝑦= 𝑥 2 +4𝑥−5 (c) 𝑦= 𝑥 2 −2𝑥+1 (d) 𝑦=3− 𝑥 2 (e) 𝑦=4+3𝑥− 𝑥 2 1 ? 1 1 ? ? 2 3 − 3 3 ? ? 4 −5 1 −5 −1 4

Exercise 1 ? ? ? ? ? (Exercises on provided sheet) Sketch the following parabolas. These have no roots, so complete the square to identify the minimum/maximum point. (a) 𝑦= 𝑥 2 +2𝑥+6 = 𝒙+𝟏 𝟐 +𝟓 (b) 𝑦= 𝑥 2 −4𝑥+7 = 𝒙−𝟐 𝟐 +𝟑 Find equations for the following graphs, giving your answer in the form 𝑎 𝑥 2 +𝑏𝑥+𝑐=0 2 3 ? a 6 (−1,5) −4 2 𝒚= 𝒙 𝟐 +𝟐𝒙−𝟖 ? b ? −2 5 2 𝒚= 𝟐𝒙−𝟓 𝒙+𝟐 =𝟐 𝒙 𝟐 −𝒙−𝟏𝟎 ? 7 (2,3) c 𝒚= 𝟓−𝒙 𝒙+𝟑 =− 𝒙 𝟐 +𝟐𝒙+𝟏𝟓 ? −3 5

Exercise 1 ? ? ? ? (Exercises on provided sheet) [C1 May 2010 Q4] (a) Show that x2 + 6x + 11 can be written as (x + p)2 + q, where p and q are integers to be found. (2) 𝒙+𝟑 𝟐 +𝟐 (b) Sketch the curve with equation y = x2 + 6x + 11, showing clearly any intersections with the coordinate axes. (2) (c) Find the value of the discriminant of x2 + 6x + 11. (2) 𝟔 𝟐 − 𝟒×𝟏×𝟏𝟏 =−𝟖 [AQA] The diagram shows a quadratic graph that intersects the 𝑥-axis when 𝑥= 1 2 and 𝑥=5. Work out the equation of the quadratic graph, giving your answer in the form 𝑦=𝑎 𝑥 2 +𝑏𝑥+𝑐 where 𝑎,𝑏,𝑐 are integers. 4 5 ? ? 11 (−3,2) ? 𝟐𝒙−𝟏 𝒙−𝟓 =𝟐 𝒙 𝟐 −𝟏𝟏𝒙+𝟓 ?

Exercise 1 ? ? ? ? (Exercises on provided sheet) A parabola has a maximum point of 2,−4 . (a) Given the quadratic equation is of the form y= −𝑥 2 +𝑎𝑥+𝑏, determine 𝑎 and 𝑏. −𝟒− 𝒙−𝟐 𝟐 =− 𝒙 𝟐 +𝟒𝒙−𝟖 𝒂=𝟒, 𝒃=−𝟖 (b) Determine the discriminant. 𝟏𝟔− 𝟒×−𝟏×−𝟖 =𝟏𝟔−𝟑𝟐 =−𝟏𝟔 6 [Set 2 Paper 2] Here is a sketch of 𝑦=10+3𝑥− 𝑥 2 (a) Write down the two solutions of 10+3𝑥− 𝑥 2 =0 𝒙=−𝟐, 𝟓 (b) Write down the equation of the line of symmetry of 𝑦=10+3𝑥− 𝑥 2 𝒙= 𝟑 𝟐 7 ? ? ? ?

#2b :: Sketching Cubics ? ? ? ? 𝑦=𝑎 𝑥 3 +𝑏 𝑥 2 +𝑐𝑥+𝑑 𝑦=𝑎𝑥3 𝑦=𝑎 𝑥 3 A recap of their general shape from GCSE… 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

#2b :: Sketching Cubics ? ? 𝒚=𝒙(𝒙−𝟏)(𝒙+𝟏) 𝒚= 𝒙−𝟏 𝟐 𝒙+𝟐 Is it uphill or downhill? Is the 𝑥 3 term + or -? Consider the roots: If (𝑥−𝑎) appears once, the line crosses at 𝑥=𝑎. If 𝑥−𝑎 2 appears, the line touches at 𝑥=𝑎. If 𝑥−𝑎 3 appears, we have a point of inflection at 𝑥=𝑎. 𝒚=𝒙(𝒙−𝟏)(𝒙+𝟏) 𝒚= 𝒙−𝟏 𝟐 𝒙+𝟐 y y ? ? 2 x x -1 1 -2 1

More Examples ? ? 𝒚= 𝒙 𝟐 𝟐−𝒙 𝒚= 𝒙−𝟏 𝟑 Is it uphill or downhill? Is the 𝑥 3 term + or -? Consider the roots: If (𝑥−𝑎) appears once, the line crosses at 𝑥=𝑎. If 𝑥−𝑎 2 appears, the line touches at 𝑥=𝑎. If 𝑥−𝑎 3 appears, we have a point of inflection at 𝑥=𝑎. 𝒚= 𝒙 𝟐 𝟐−𝒙 𝒚= 𝒙−𝟏 𝟑 ? y ? y x x -1 2 1 -1 A point of inflection is where the curve changes from concave to convex (or vice versa). Think of it as a ‘plateau’ when ascending or descending a hill.

Test Your Understanding 𝑦 Sketch 𝑦=(𝑥+1) 𝑥−2 2 , ensuring you indicate where the graph cuts/touches either axes. Suggest an equation for this graph. 𝒚=𝒙 𝒙+𝟑 𝟐 ? 𝑥 ? 𝑦 -3 4 Sketch 𝑦=9𝑥− 𝑥 3 (hint: factorise first!) =𝒙 𝟗− 𝒙 𝟐 =𝒙 𝟑+𝒙 𝟑−𝒙 𝑥 ? -1 2 𝑦 𝑥 -3 3

Quickfire Questions! Sketch the following, ensuring you indicate the values where the line intercepts the axes. 𝑦=(𝑥+2)(𝑥−1)(𝑥−3) 𝑦= 3−𝑥 3 1 4 𝑦=𝑥 1−𝑥 2 6 ? ? 27 ? 6 3 -2 1 3 1 2 𝑦=𝑥(𝑥−1)(2−𝑥) 7 𝑦= 𝑥+2 2 (𝑥−1) 5 𝑦=− 𝑥 3 ? ? ? 1 2 -2 1 -4 𝑦=𝑥 𝑥+1 2 3 8 𝑦= 1−𝑥 2 (3−𝑥) ? ? 3 -1 1 3

Exercise 2 ? ? ? ? ? (Exercises on provided sheet) [Set 1 Paper 2] Sketch the curve 𝑦= 𝑥 3 −12 𝑥 2 1 2 a 𝑦=𝑥(2𝑥–1)(𝑥+3) ? ? -3 0.5 b 𝑦=𝑥2(𝑥+1) ? -1 12 c 𝑦= 𝑥+2 3 ? 8 -2 d 𝑦= 2−𝑥 𝑥+3 2 ? 18 -3 2

Exercise 2 ? ? ? (Exercises on provided sheet) [Set 4 Paper 2] A sketch of 𝑦=𝑓(𝑥), where 𝑓(𝑥) is a cubic function, is shown. [Set 2 Paper 2] Here is a sketch of 𝑥 3 +𝑏 𝑥 2 +𝑐𝑥+𝑑 where 𝑏,𝑐,𝑑 are constants. 3 4 There is a maximum point at 𝐴 2,10 . (a) Write down the equation of the tangent to the curve at 𝐴. 𝒚=𝟏𝟎 (b) Write down the equation of the normal to the curve at 𝐴. 𝒙=𝟐 Work out the values of 𝑏,𝑐,𝑑. 𝒚= 𝒙+𝟏 𝒙−𝟐 𝒙−𝟓 = 𝒙+𝟏 𝒙 𝟐 −𝟕𝒙+𝟏𝟎 = 𝒙 𝟑 −𝟔 𝒙 𝟐 +𝟑𝒙+𝟏𝟎 𝒃=−𝟔, 𝒄=𝟑, 𝒅=𝟏𝟎 ? ? ?

Exercise 2 (Exercises on provided sheet) 5 ?

Exercise 2 ? ? ? ? (Exercises on provided sheet) 6 Suggest equations for the following cubic graphs. (You need not expand out any brackets) a b -4 3 -2 ? 𝑦= 𝑥 2 (𝑥+4) ? 𝑦= 𝑥+2 2 (𝑥−3) c d -1 -3 ? 𝑦= 𝑥+1 3 ? 𝑦=−𝑥 𝑥+3 2

Exercise 2 (Exercises on provided sheet) 7 ?

We’ll be able to sketch more complicated graphs of this form: #2c :: Reciprocal Graphs 𝑦 At GCSE, you encountered ‘reciprocal graphs’, with equations of the form: 𝒚= 𝒌 𝒙 where 𝑘 is a constant. 𝑥 We’ll be able to sketch more complicated graphs of this form: 𝑦= 1 𝑥−3 𝑦= 4 𝑥 −2 𝑦=− 1 𝑥+1 +1

Example Sketch 𝑦= 1 𝑥−3 Is there a value of 𝑥 for which 𝑦 is not defined? 𝑦 We can’t divide by 0. This occurs when 𝑥=3. We draw a dotted line (known as an asymptote) and MUST give its equation. 𝑥 ? − 1 3 𝑥=3 The rest of the curve will be the same as before (consider for example what happens when 𝑥→∞ or 𝑥→−∞). YOU MUST WORK OUT THE INTERCEPTS.

Example Sketch 𝑦=− 1 𝑥+1 Sketch 𝑦= 3 𝑥 +2 ? ? 𝑦 𝑥 𝑦 𝑥 Because it’s a ‘negative reciprocal graph’ (e.g. like − 1 𝑥 ) the curve is the other way up. Sketch 𝑦=− 1 𝑥+1 𝑥 −1 𝑥=−1 𝑦 ? Increasing the 𝑦 values by 2 shifts the graph up. We now have a horizontal asymptote! Note we now also have a root, which we work out in the usual way by solving 0= 1 𝑥 +2 Sketch 𝑦= 3 𝑥 +2 𝑦=2 𝑥 − 3 2

Test Your Understanding Sketch 𝑦= 1 𝑥+4 +2 Sketch 𝑦=− 1 𝑥−2 +3 ? 𝑦 ? 𝑦 𝑥=−4 𝑥=2 7 2 9 4 𝑦=3 𝑦=2 𝑥 𝑥 − 9 2 7 3

Exercise 3 𝑥=−1 𝑦 1 Sketch the following, ensuring you indicate the equation of any asymptotes and the coordinates of any points where the graph crosses the axes. (a) 𝑦= 1 𝑥 −2 (b) 𝑦= 3 𝑥 +1 (c) 𝑦= 2 𝑥+1 (d) 𝑦=− 1 𝑥−2 (e) 𝑦= 2 𝑥+3 −1 ? 2 𝑥 𝑦 ? 𝑦 ? 𝑥=2 𝑥 1 2 1 2 𝑦=−2 𝑥 𝑦 ? 𝑥=−3 𝑦 ? 𝑦=1 𝑥 −3 𝑥 −1 − 1 3 𝑦=−1

Exercise 3 2 ?

Exercise 3 3 ?

Exercise 3 4 ?

#3 :: Piecewise Functions Sometimes functions are defined in ‘pieces’, with a different function for different ranges of 𝑥 values. Sketch > Sketch > Sketch > (2, 9) (0, 5) (-1, 0) (5, 0)

Test Your Understanding 𝑓 𝑥 = 𝑥 2 0≤𝑥<1 1 1≤𝑥<2 3−𝑥 2≤𝑥<3 Sketch Sketch Sketch This example was used on the specification itself! (2, 1) (1, 1) (3, 0)

Exercise 4 b ? c ? ? a ? (Exercises on provided sheet) [Jan 2013 Paper 2] A function 𝑓(𝑥) is defined as: 𝑓 𝑥 = 4 𝑥<−2 𝑥 2 −2≤𝑥≤2 12−4𝑥 𝑥>2 Draw the graph of 𝑦=𝑓(𝑥) for −4≤𝑥≤4 Use your graph to write down how many solutions there are to 𝑓 𝑥 =3 3 sols Solve 𝑓 𝑥 =−10 𝟏𝟐−𝟒𝒙=−𝟏𝟎 →𝒙= 𝟏𝟏 𝟐 [June 2013 Paper 2] A function 𝑓(𝑥) is defined as: 𝑓 𝑥 = 𝑥+3 −3≤𝑥<0 3 0≤𝑥<1 5−2𝑥 1≤𝑥≤2 Draw the graph of 𝑦=𝑓(𝑥) for −3≤𝑥<2 1 2 b ? ? c ? a ?

Exercise 4 ? Sketch ? ? (Exercises on provided sheet) [Specimen 1 Q4] A function 𝑓(𝑥) is defined as: 𝑓 𝑥 = 3𝑥 0≤𝑥<1 3 1≤𝑥<3 12−3𝑥 3≤𝑥≤4 Calculate the area enclosed by the graph of 𝑦=𝑓 𝑥 and the 𝑥−axis. 3 [Set 1 Paper 1] A function 𝑓(𝑥) is defined as: 𝑓 𝑥 = 3 0≤𝑥<2 𝑥+1 2≤𝑥<4 9−𝑥 4≤𝑥≤9 Draw the graph of 𝑦=𝑓(𝑥) for 0≤𝑥≤9. 4 ? Sketch ? Area = 𝟗 ?

Exercise 4 ? ? (Exercises on provided sheet) [AQA Worksheet Q9] 6 𝑓 𝑥 = − 𝑥 2 0≤𝑥<2 −4 2≤𝑥<3 2𝑥−10 3≤𝑥≤5 Draw the graph of 𝑓(𝑥) from 0≤𝑥≤5. 6 [AQA Worksheet Q10] 𝑓 𝑥 = 2𝑥 0≤𝑥<1 3−𝑥 1≤𝑥<4 𝑥−7 3 4≤𝑥≤7 5 ? 2 -1 -2 -3 -4 1 2 3 4 5 3 7 -1 Show that 𝑎𝑟𝑒𝑎 𝑜𝑓 𝐴:𝑎𝑟𝑒𝑎 𝑜𝑓 𝐵= 3:2 Area of 𝑨= 𝟏 𝟐 ×𝟑×𝟐=𝟑 Area of 𝑩= 𝟏 𝟐 ×𝟒×𝟏=𝟐 ?

#4 :: Graph Transformations – GCSE Recap Suppose we sketch the function y = f(x). What happens when we sketch each of the following? 𝒇(𝒙+𝟑)  3 ? 𝒇 𝒙−𝟐  2 ? 𝒇(𝟐𝒙)  Stretch x by factor of ½ ? 𝒇 𝒙 𝟑 ↔ Stretch x by factor of 3 ? ↑4 𝒇 𝒙 +𝟒 ? 𝟑𝒇 𝒙 ↕ Stretch y by factor of 3. ? If inside f(..), affects x-axis, change is opposite. If outside f(..), affects y-axis, change is as expected.

RECAP :: 𝑓(−𝑥) vs −𝑓 𝑥 We don’t have to reason about these any differently! y = f(x) y Bro Tip: Ensure you also reflect any min/max points, intercepts and asymptotes. (2, 3) 1 x y = -1 𝑦=𝑓 −𝑥 𝑦=−𝑓 𝑥 ? y ? y Change inside f brackets, so times 𝑥 values by -1 (-2, 3) y = 1 x 1 -1 Change outside f brackets, so times y values by -1 x (2, -3) y = -1

Test Your Understanding C1 Jan 2009 Q5 Figure 1 shows a sketch of the curve C with equation y = f(x). There is a maximum at (0, 0), a minimum at (2, –1) and C passes through (3, 0).   On separate diagrams, sketch the curve with equation (a) y = f(x + 3), (3) (b) y = f(–x). (3) On each diagram show clearly the coordinates of the maximum point, the minimum point and any points of intersection with the x-axis. a ? b ?

Drawing transformed graphs Bro Tip: To sketch many functions, it’s best to start with a similar simpler function (in this case 𝑓 𝑥 = 𝑥 3 ), then consider how it’s been transformed. Sketch 𝒚= 𝒙−𝟏 𝟑 −𝟖 ? y 7 -1 x If 𝑓 𝑥 = 𝑥 3 𝑓 𝑥−1 = 𝑥−1 3 𝑓 𝑥−1 −8= 𝑥−1 3 −8 𝑓 𝑥−1 −8 gives you a translation right by 1 unit and 8 down.

Drawing transformed graphs Sketch 𝒚= 𝟏 𝒙+𝟐 −𝟏 (Hint: If 𝑓 𝑥 =1/𝑥, then what is the above function?) ? y -2 𝑦=−1 𝑥=−1 -0.5 x 𝑓 𝑥 = 1 𝑥 𝑓 𝑥+2 = 1 𝑥+2 𝑓 𝑥+2 −1= 1 𝑥+2 −1 So translation 2 left 1 down.

Test Your Understanding C1 June 2009 Q10 a) 𝑥 𝑥 2 −6𝑥+9 =𝑥 𝑥−3 2 a ? 𝑦 b ? b) c) If 𝑓 𝑥 = 𝑥 3 −6 𝑥 2 +9𝑥 Then 𝑓 𝑥−2 = 𝑥−2 3 −6 𝑥−2 2 +9 𝑥−2 This is a translation right of 2. c ? 𝑥 3 2 5

Exercise 5 ? ? (Exercises on provided sheet) 1 [C1 Jan 2011 Q5] Figure 1 shows a sketch of the curve with equation 𝑦=𝑓(𝑥) where 𝑓 𝑥 = 𝑥 𝑥−2 , 𝑥≠2 The curve passes through the origin and has two asymptotes, with equations y = 1 and x = 2, as shown in Figure 1.   (a) Sketch the curve with equation y = f(x − 1) and state the equations of the asymptotes of this curve. (3) (b) Find the coordinates of the points where the curve with equation y = f(x − 1) crosses the coordinate axes. (4) 1 ? ?

Exercise 5 ? ? ? (Exercises on provided sheet) 2 [C1 May 2010 Q6] Figure 1 shows a sketch of the curve with equation y = f(x). The curve has a maximum point A at (–2, 3) and a minimum point B at (3, – 5). On separate diagrams sketch the curve with equation (a) y = f (x + 3), (3) (b) y = 2f(x). (3)   On each diagram show clearly the coordinates of the maximum and minimum points. The graph of y = f(x) + a has a minimum at (3, 0), where a is a constant. (c) Write down the value of a. (1) ? ?

Exercise 5 ? ? ? (Exercises on provided sheet) 3 [C1 May 2011 Q8] Figure 1 shows a sketch of the curve C with equation y = f(x). The curve C passes through the origin and through (6, 0). The curve C has a minimum at the point (3, –1). On separate diagrams, sketch the curve with equation   (a) y = f(2x), (3) (b) y = −f(x), (3) (c) y = f(x + p), where 0 < p < 3. (4) On each diagram show the coordinates of any points where the curve intersects the x-axis and of any minimum or maximum points. ? ?

Exercise 5 ? ? ? ? (Exercises on provided sheet) 4 [C1 May 2012 Q10] Figure 1 shows a sketch of the curve C with equation y = f(x), where f(x) = x2(9 – 2x) There is a minimum at the origin, a maximum at the point (3, 27) and C cuts the x-axis at the point A.   (a) Write down the coordinates of the point A. (b) On separate diagrams sketch the curve with equation (i) y = f(x + 3), (ii) y = f(3x). On each sketch you should indicate clearly the coordinates of the maximum point and any points where the curves cross or meet the coordinate axes. The curve with equation y = f(x) + k, where k is a constant, has a maximum point at (3, 10). (c) Write down the value of k. ? ? ?

Exercise 5 ? (Exercises on provided sheet) 5 [Jan 2010 Q8] The curve 𝑦=𝑓 𝑥 has a maximum point (–2, 5) and an asymptote y = 1, as shown in Figure 1. On separate diagrams, sketch the curve with equation   (a) y = f(x) + 2, (2) (b) y = 4f(x), (2) (c) y = f(x + 1). (3) On each diagram, show clearly the coordinates of the maximum point and the equation of the asymptote.

Exercise 5 ? ? (Exercises on provided sheet) 6 [C1 June 2008 Q3] Figure 1 shows a sketch of the curve with equation y = f(x). The curve passes through the point (0, 7) and has a minimum point at (7, 0). On separate diagrams, sketch the curve with equation (a) y = f(x) + 3, (3) (b) y = f(2x). (2) On each diagram, show clearly the coordinates of the minimum point and the coordinates of the point at which the curve crosses the y-axis. ?