1. IGCSE FM/C1 Sketching Graphs
Dr J Frost
Objectives: (from the IGCSE FM specification)
Last modified: 9th October 2015

2. 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

3. #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).

4. #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.

5. 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 ?

6. 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
So if 𝑦=0, i.e. 𝑥+1 𝑥−2 =0, then 𝑥=−1 or 𝑥=2. -2
When 𝑥=0, clearly
𝑦=−2.

7. 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 =− 𝑥−1 𝑥− = (𝑥−1)(4−𝑥) ? x
Bro Tip: We can tidy up by using the minus on the front to swap the order in one of the negations. -4

8. 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

9. 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 ? ?

10. 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

11. 𝑦= 𝑥 2 −6𝑥+10 = 𝑥−3 2 +1 RECAP :: Using completed square for min/max ?
= 𝑥− ?
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.

12. 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 .

13. 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)

14. 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

15. 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

16. 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 (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 ?

17. 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 ? ? ? ?

18. #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

19. #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

20. 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.

21. Test Your Understanding𝑦
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

22. 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

23. 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

24. 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 ? ? ?

25. Exercise 2
(Exercises on provided sheet) 5 ?

26. 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

27. Exercise 2
(Exercises on provided sheet) 7 ?

28. 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

29. 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.

30. 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

31. 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

32. 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

33. Exercise 3 2 ?

34. Exercise 3 3 ?

35. Exercise 3 4 ?

36. #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)

37. 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)

38. 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 = sols
Solve 𝑓 𝑥 =− 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 ?

39. 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 ?

40. 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 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 ?

41. #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.

42. 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

43. 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 ?

44. 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− 𝑓 𝑥−1 −8= 𝑥−1 3 −8
𝑓 𝑥−1 −8 gives you a translation right by 1 unit and 8 down.

45. 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.

46. 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− 𝑥−2
This is a translation right of 2.
c ? 𝑥 3 2 5

47. 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 ? ?

48. 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) ? ?

49. 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. ? ?

50. 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. ? ? ?

51. 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.

52. 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. ?