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:
#1: Shapes of graphs
(quadratic, cubic, reciprocal)
and basic features (roots, yintercept, max/min points,
asymptotes)
C1
#2: Specific skills in sketching
(i) quadratics (ii) cubics and
(iii) reciprocals
C1
IGCSE FM
IGCSE FM
#4: Graph transformations
#3: Piecewise functions
IGCSE FM only
C1 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?
?
1
y=
+ 2(𝑥 + 2)2
𝑥+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
𝑥
e.g. 𝑦 = 𝑥 2 − 4𝑥 + 7
Parabola
(Quadratic Equation)
𝑦
𝑥
e.g. 𝑦 = 𝑥 3 − 𝑥 2 − 𝑥 + 1
Cubic
𝑥
1
e.g. 𝑦 = 𝑥 + 2
Reciprocal
At GCSE these were
previously centred at
the origin.
RECAP :: Sketching Quadratics
3 features needed in sketch?
y
Roots
?
x
General shape:
?
Smiley face or hill?
y-intercept
?
Example 1
1. Roots
2. y-intercept
3. Shape: smiley face or hill?
y
𝑦 = 𝑥2 − 𝑥 − 2
= (𝑥 + 1)(𝑥 − 2)
So if 𝑦 = 0, i.e.
𝑥 + 1 𝑥 − 2 = 0, then
𝑥 = −1 or 𝑥 = 2.
When 𝑥 = 0, clearly
𝑦 = −2.
?
-1
2
-2
x
Example 2
1. Roots
2. y-intercept
3. Shape: smiley face or hill?
y
𝑦 = −𝑥 2 + 5𝑥 − 4
= − 𝑥 2 − 5𝑥 + 4
= − 𝑥 − 1? 𝑥 − 4
= (𝑥 − 1)(4 − 𝑥)
Bro Tip: We can tidy up
by using the minus on the
front to swap the order in
one of the negations.
?
1
-4
4
x
Test Your Understanding So Far
𝑦 = 𝑥 2 + 3𝑥 + 2
𝑦 = −𝑥 2 + 2𝑥 + 8
x = -1,?-2
Roots?
y=2 ?
∩ or ∪ shape? ∪
?
y-Intercept?
Roots?
𝑥 = −2,
? 4
y-Intercept?
𝑦 = 8?
?
∩ or ∪ shape? ∩
y
y
8
2
-2
-1
?
x
?
-2
4
x
Graph → Equation
𝑦
−1
𝑦
1
2
Find an equation for this curve, in the
form 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 where 𝑎, 𝑏, 𝑐
are integers.
𝒚 = 𝟐𝒙 − 𝟏 𝒙 + 𝟏
𝒚 = 𝟐𝒙𝟐 +?𝒙 − 𝟏
𝑥
1
−
4
2
3
𝑥
Find an equation for this curve, in the
form 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 where 𝑎, 𝑏, 𝑐
are integers.
𝒚 = − 𝟑𝒙 − 𝟐 𝟒𝒙 + 𝟏
𝒚 = −𝟏𝟐𝒙𝟐?+ 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
? −3
tangent is 𝑦 =
RECAP :: Using completed square for min/max
2
𝑦 = 𝑥 − 6𝑥 + 10
2
?
= 𝑥−3 +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.
Write down
!
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 y at graph y-intercept Roots?
min
min
1 y = x2 + 2x + 5
y = (x + 1)2 + 4
-1
2 y = x2 – 4x + 7
y = (x –?2)2 + 3
2
3 y = x2 + 6x – 27
y = (x +?3)2 – 36
-3
1
?
?
4
5
3
7
?
-36 ?
2
-27
None
None?
?
?
x = 3?or -9
3
7
5
(-1,4)
?
-9
(2,3)
?
-27
(-3,-36)
3
Exercise 1
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
(c)
1
(d)
?
𝑦 = 𝑥 2 − 2𝑥 + 1
?
1
𝑦 = 3 − 𝑥2
2
3
?
(b)
(e)
𝑦 = 4 + 3𝑥 − 𝑥 2
?
−5
−5
3
− 3
𝑦 = 𝑥 2 + 4𝑥 − 5
4
1
−1
?
4
Exercise 1
2
(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
Find equations for the following
graphs, giving your answer in the
form
𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0
3
a
6
?
(−1,5)
2
−4
𝒚 = 𝒙𝟐 +
? 𝟐𝒙 − 𝟖
b
(b)
𝑦 = 𝑥 2 − 4𝑥 + 7
= 𝒙−𝟐 𝟐+𝟑
7
? (2,3)
5
2
−2
𝒚 = 𝟐𝒙 − 𝟓 𝒙 + 𝟐
= 𝟐𝒙𝟐 − 𝒙 − 𝟏𝟎
𝒚= 𝟓−𝒙 𝒙+𝟑
= −𝒙𝟐 + 𝟐𝒙 + 𝟏𝟓
?
c
?
−3
5
Exercise 1
(Exercises on provided sheet)
4 [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)
5 [AQA] The diagram shows a
quadratic graph that intersects the 𝑥1
axis when 𝑥 = 2 and 𝑥 = 5.
Work out the equation of the
quadratic graph, giving your answer
in the form 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 where
𝑎, 𝑏, 𝑐 are integers.
11
(−3,2)
?
(c) Find the value of the discriminant of
x2 + 6x + 11.
(2)
𝟔𝟐 − 𝟒 × 𝟏 × 𝟏𝟏 = −𝟖
?
𝟐𝒙 − 𝟏 𝒙 − 𝟓
= 𝟐𝒙𝟐 −?𝟏𝟏𝒙 + 𝟓
Exercise 1
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
𝟑
𝒙?=
𝟐
(Exercises on provided sheet)
7
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
= −𝒙𝟐 +
? 𝟒𝒙 − 𝟖
𝒂 = 𝟒, 𝒃 = −𝟖
(b) Determine the discriminant.
𝟏𝟔 − 𝟒 × −𝟏 × −𝟖
= 𝟏𝟔 − 𝟑𝟐
?
= −𝟏𝟔
#2b :: Sketching Cubics
A recap of their general shape from GCSE…
𝑦=
𝑦 = 𝑎𝑥 3 + 𝑏𝑥 2 + 𝑐𝑥 + 𝑑
𝑎𝑥3
When 𝑎 > 0
When 𝑎 > 0
y
?
𝑦 = 𝑎𝑥 3
When 𝑎 < 0
?
x
𝑦 = 𝑎𝑥 3 + 𝑏𝑥 2 + 𝑐𝑥 + 𝑑
When 𝑎 < 0
y
?
x
?
#2b :: Sketching Cubics
1. Is it uphill or downhill? Is the 𝑥 3 term + or -?
2. Consider the roots:
a) If (𝑥 − 𝑎) appears once, the line crosses at 𝑥 = 𝑎.
b) If 𝑥 − 𝑎 2 appears, the line touches at 𝑥 = 𝑎.
c) If 𝑥 − 𝑎 3 appears, we have a point of inflection at 𝑥 = 𝑎.
𝒚= 𝒙−𝟏
𝒚 = 𝒙(𝒙 − 𝟏)(𝒙 + 𝟏)
y
𝟐
𝒙+𝟐
y
2
?
-1
1
x
?
-2
1
x
More Examples
1. Is it uphill or downhill? Is the 𝑥 3 term + or -?
2. Consider the roots:
a) If (𝑥 − 𝑎) appears once, the line crosses at 𝑥 = 𝑎.
b) If 𝑥 − 𝑎 2 appears, the line touches at 𝑥 = 𝑎.
c) If 𝑥 − 𝑎 3 appears, we have a point of inflection at 𝑥 = 𝑎.
𝒚 = 𝒙𝟐 𝟐 − 𝒙
𝒚= 𝒙−𝟏
y
y
?
-1
𝟑
2
?
x
1
-1
x
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
𝑦
𝑥
-3
4
?
-1
2
𝑥
Sketch 𝑦 = 9𝑥 − 𝑥 3
(hint: factorise first!)
= 𝒙 𝟗 − 𝒙𝟐 = 𝒙 𝟑 + 𝒙 𝟑 − 𝒙
𝑦
?
-3
3
𝑥
Quickfire Questions!
Sketch the following, ensuring you indicate the values where the line intercepts the axes.
1
𝑦 = (𝑥 + 2)(𝑥 − 1)(𝑥 − 3)
4
𝑦 =𝑥 1−𝑥
2
6
𝑦 = 3−𝑥
27
6
-2
2
?
1
𝑦 = 𝑥(𝑥 − 1)(2 − 𝑥)
?
3
1
𝑦=𝑥 𝑥+1
?
3
2
2
5
?
3
1
7
𝑦 = −𝑥 3
3
𝑦 = 𝑥 + 2 2 (𝑥 − 1)
?
1
-2
?
-4
8
𝑦 = 1 − 𝑥 2 (3 − 𝑥)
3
-1
?
?
1
3
Exercise 2
(Exercises on provided sheet)
the curve
1 [Set 1 Paper 2] Sketch
𝑦 = 𝑥 3 − 12𝑥 2
2
a
𝑦 = 𝑥(2𝑥– 1)(𝑥 + 3)
?
-3
b
𝑦 = 𝑥2(𝑥 + 1)
-1
?
12
c
0.5
?
𝑦= 𝑥+2
3
8
?
-2
d
𝑦 = 2−𝑥 𝑥+3
18
-3
?
2
2
Exercise 2
3
[Set 4 Paper 2] A sketch of 𝑦 = 𝑓(𝑥),
where 𝑓(𝑥) is a cubic function, is
shown.
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
(Exercises on provided sheet)
4
[Set 2 Paper 2] Here is a sketch of
𝑥 3 + 𝑏𝑥 2 + 𝑐𝑥 + 𝑑
where 𝑏, 𝑐, 𝑑 are constants.
Work out the values of 𝑏, 𝑐, 𝑑.
𝒚= 𝒙+𝟏 𝒙−𝟐 𝒙−𝟓
= 𝒙 + 𝟏 𝒙𝟐 − 𝟕𝒙 + 𝟏𝟎
= 𝒙𝟑 − 𝟔𝒙𝟐 ?
+ 𝟑𝒙 + 𝟏𝟎
𝒃 = −𝟔, 𝒄 = 𝟑, 𝒅 = 𝟏𝟎
Exercise 2
(Exercises on provided sheet)
5
?
Exercise 2
6
(Exercises on provided sheet)
Suggest equations for the following cubic graphs.
(You need not expand out any brackets)
a
b
-4
-2
𝑦 = 𝑥 2?(𝑥 + 4)
c
3
𝑦 = 𝑥 +?
2 2 (𝑥 − 3)
d
-1
-3
𝑦 = 𝑥?+ 1
3
𝑦 = −𝑥?𝑥 + 3
2
Exercise 2
(Exercises on provided sheet)
7
?
#2c :: Reciprocal Graphs
𝑦
𝑥
At GCSE, you encountered
‘reciprocal graphs’, with
equations of the form:
𝒌
𝒚=
𝒙
where 𝑘 is a constant.
We’ll be able to sketch more complicated graphs of this form:
1
𝑦=
𝑥−3
4
𝑦 = −2
𝑥
1
𝑦=−
+1
𝑥+1
Example
Sketch
Is there a value of 𝑥 for which 𝑦 is not
defined?
𝑦
1
𝑦=
𝑥−3
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
𝑦
Because it’s a ‘negative
reciprocal graph’ (e.g.
1
like − 𝑥) the curve is the
other way up.
Sketch
1
𝑦=−
𝑥+1
?
𝑥 = −1
𝑥
−1
𝑦
Sketch
3
𝑦 = +2
𝑥
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
1
solving 0 = + 2
𝑥
𝑦=2
?
3
−
2
𝑥
Test Your Understanding
Sketch
Sketch
1
𝑦=
+2
𝑥+4
1
𝑦=−
+3
𝑥−2
7
2
9
4
𝑦=3
?
9
−
2
𝑥=2
𝑦
𝑥 = −4
𝑦
?
𝑦=2
𝑥
7
3
𝑥
Exercise 3
1 Sketch the following, ensuring you
indicate the equation of any
asymptotes and the coordinates of
any points where the graph crosses
the axes.
1
(a)
𝑦 = −2
𝑥
(c)
𝑦=
2
𝑥+1
𝑥 = −1 𝑦
2
?
𝑦
(d)
?
𝑥
1
2
𝑦
1
𝑦 = − 𝑥−2
𝑥=2
1
2
?
𝑦 = −2
(b)
𝑥
𝑥
3
𝑦 =𝑥+1
𝑦
(e)
𝑦=1
−3
?
2
𝑦 = 𝑥+3 − 1
𝑥 = −3 𝑦
𝑥
−1
−
?
𝑥
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
𝑥2
𝑓 𝑥 =
1
3−𝑥
0≤𝑥<1
1≤𝑥<2
2≤𝑥<3
(1, 1)
Sketch
Sketch
Sketch
This example
was used on the
specification
itself!
(2, 1)
(3, 0)
Exercise 4
(Exercises on provided sheet)
1 [Jan 2013 Paper 2] A function 𝑓(𝑥) is defined as:
4
𝑓 𝑥 =
𝑥2
12 − 4𝑥
𝑥 < −2
−2 ≤ 𝑥 ≤ 2
𝑥>2
𝑥+3
𝑓 𝑥 =
3
5 − 2𝑥
(a) Draw the graph of 𝑦 = 𝑓(𝑥) for
−4 ≤ 𝑥 ≤ 4
(b) Use your graph to write down how many
solutions there are to 𝑓 𝑥 = 3
3 sols
b?
(c) Solve 𝑓 𝑥 = −10
c?
𝟏𝟐 − 𝟒𝒙 = −𝟏𝟎 → 𝒙 =
2 [June 2013 Paper 2] A function 𝑓(𝑥) is
defined as:
Draw the graph of 𝑦 = 𝑓(𝑥) for
−3 ≤ 𝑥 < 2
𝟏𝟏
𝟐
?
a?
−3 ≤ 𝑥 < 0
0≤𝑥<1
1≤𝑥≤2
Exercise 4
(Exercises on provided sheet)
3 [Set 1 Paper 1] A function 𝑓(𝑥) is defined as:
3
0≤𝑥<2
𝑓 𝑥 = 𝑥+1 2≤𝑥 <4
9−𝑥 4≤𝑥 ≤9
Draw the graph of 𝑦 = 𝑓(𝑥) for 0 ≤ 𝑥 ≤ 9.
4 [Specimen 1 Q4] A function 𝑓(𝑥) is
defined as:
3𝑥
𝑓 𝑥 =
3
12 − 3𝑥
0≤𝑥<1
1≤𝑥<3
3≤𝑥≤4
Calculate the area enclosed by the
graph of 𝑦 = 𝑓 𝑥 and the 𝑥 −axis.
?
Sketch ?
Area = 𝟗
?
Exercise 4
(Exercises on provided sheet)
6
5 [AQA Worksheet Q9]
−𝑥 2
𝑓 𝑥 =
−4
2𝑥 − 10
0≤𝑥<2
2≤𝑥<3
3≤𝑥≤5
Draw the graph of 𝑓(𝑥) from 0 ≤ 𝑥 ≤ 5.
[AQA Worksheet Q10]
2𝑥
3−𝑥
𝑓 𝑥 = 𝑥−7
3
0≤𝑥<1
1≤𝑥<4
4≤𝑥≤7
2
1
-1
-2
-3
-4
2
3
?
4
5
3
-1
7
Show that 𝑎𝑟𝑒𝑎 𝑜𝑓 𝐴: 𝑎𝑟𝑒𝑎 𝑜𝑓 𝐵 =
3: 2
𝟏
Area of 𝑨 = 𝟐 × 𝟑 × 𝟐 = 𝟑
𝟏
𝟐
?
Area of 𝑩 = × 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)
𝒙
𝒇
𝟑
𝒇 𝒙 +𝟒
 Stretch x by? factor of ½
𝟑𝒇 𝒙
?
↕ Stretch y by factor
of 3.
↔ Stretch x by factor
of 3
?
↑4
?
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)
1
y=1
x
?
-1
?
x
y = -1
(2, -3)
Change outside f
brackets, so times
y values by -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
Sketch 𝒚 = 𝒙 − 𝟏
𝟑
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.
−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
𝑥 = −1
-2
?
x
-0.5
𝑦 = −1
𝑓 𝑥 =
1
𝑥
1
𝑓 𝑥+2 =
𝑥+2
1
𝑓 𝑥+2 −1=
−1
𝑥+2
So translation 2 left 1 down.
Test Your Understanding
C1 June 2009 Q10
a)
b)
𝑦
𝑥 𝑥 2 − 6𝑥
a +2? 9
=𝑥 𝑥−3
c) If 𝑓 𝑥 = 𝑥 3 − 6𝑥 2 + 9𝑥
Then 𝑓 𝑥 − 2
= 𝑥−2 3−6 𝑥−2 2+9 𝑥−2
This is a translation right of 2.
b?
3
𝑥
c?
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
𝑥
𝑓 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)
?
?
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),
(b) y = 2f(x).
(3)
(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),
(b) y = −f(x),
(c) y = f(x + p), where 0 < p < 3.
?
(3)
(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)
(b)
(c)
y = f(x) + 2,
y = 4f(x),
y = f(x + 1).
(2)
(2)
(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.
?