GCSE: Curved Graphs
Dr J Frost (jfrost@tiffin.kingston.sch.uk)
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 𝑦 = sin 𝑥 and 𝑦 = cos 𝑥,
within the range -360° to +360°
The graph shows 𝑦 =
cos 𝑥. Determine the
coordinate of point 𝐴.
q in
4 Find the values of p and
the function 𝑦 = 𝑝𝑞 𝑥 given
coordinates on the graph of
𝑦 = 𝑝𝑞 𝑥
“Given that 2,6 and 5,162 are
points on the curve 𝑦 = 𝑘𝑎 𝑥 , find
the value of 𝑘 and 𝑎.”
Skill #1: Recognising Graphs
Linear
𝒚 = 𝒂𝒙 + 𝒃
𝒚 = 𝒂𝒙 + 𝒃
When 𝑎 > 0
When 𝑎 < 0
?
?
?
The line is known as a straight line.
Skill #1: Recognising Graphs
Quadratic
𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
When 𝑎 > 0
When 𝑎 < 0
?
?
The line for a quadratic equation
is known as a parabola.
?
Skill #1: Recognising Graphs
Cubic
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
?
Skill #1: Recognising Graphs
Reciprocal
𝑎
𝑦=
𝑥
𝑎
𝑦=
𝑥
When 𝑎 > 0
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.
Skill #1: Recognising Graphs
Exponential
𝑦 = 𝑎 × 𝑏𝑥
y
?
𝑎
x
The y-intercept is 𝑎 because 𝑎 × 𝑏0 = 𝑎 × 1 =?𝑎.
(unless 𝑎 = 0, but let’s not go there!)
Skill #1: Recognising Graphs
Circle
The equation of this circle is:
𝑦
x2 + y2 =? 25
5
5
-5
-5
𝑥
! The equation of a circle with
centre at the origin and radius
r is:
𝑥2 + 𝑦2 = 𝑟2
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.
𝑥
0 8
90
𝑦
0
?
1
?
180
270
360
0?
-1?
0?
𝑦 = sin 𝑥
1
90
180
270
360
-1
Skill #2: Plotting and
recognising trig functions.
Click to
brosketch
Test Your Understanding
90
?1
180? 0
𝑥
0 8
90
𝑦
1
?
0
?
180
270
360
-1?
0?
1?
𝑦 = cos 𝑥
1
90
180
270
360
-1
Click to
brosketch
Quickfire Coordinates
𝑦 = sin 𝑥
𝑦 = sin 𝑥
𝑦 = cos 𝑥
𝐵
𝑦 = cos 𝑥
𝐶
𝐷
𝐴
𝐴 270,
? −1
𝑦 = sin 𝑥
𝐵 90,
?0
𝐶 360,
? 0
𝑦 = sin 𝑥
𝑦 = cos 𝑥
𝐺
𝐸
𝐷 0,
?1
𝑦 = cos 𝑥
𝐻
𝐹
𝐸 180,
? 0
𝐹 180,
? −1
𝐺 90,1
?
𝐻 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 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
Bro Tip for (b): Look at what
value has been substituted into
the equation in each case.
a) 𝑦 = −2
2
− 5 −2 − 3
?
= 11
b) i) When 𝑦 = 0, then using graph,
?
roughly 𝒙 = −𝟎. 𝟓 𝒐𝒓 𝒙 = 𝟓. 𝟓
ii) 𝒙 = −𝟏. 𝟒 𝒐𝒓 𝒙 = 𝟔. 𝟒
?
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
?
Using a Trig Graph
Suppose that sin 45 =
Q
1
1
Using the graph, find the other
1
solution to sin 𝑥 = 2
𝒙 = 𝟏𝟑𝟓°
?
2
𝟒𝟓
-1
1
2
90
𝟏𝟑𝟓
We can see by symmetry
that the difference
between 0 and 45 needs
to be the same as the
difference between 𝑥
and 180.
180
270
360
1
Q
Suppose that sin 210 = − 2
Using the graph, find the other
1
solution to sin 𝑥 = − 2
𝒙 = 𝟑𝟑𝟎°
?
Test Your Understanding
The graph shows the line with equation 𝑦 = cos 𝑥
1
1
a) Given that cos 60 = , find the other solution to cos 𝑥 =
2
2
𝒙 = 𝟑𝟎𝟎°
1
b) Given that cos 150° = −
90
-1
180
3
,
2
?
find the other solution to cos 𝑥 = −
𝒙 = 𝟐𝟏𝟎°
?
270
360
3
2
Exercise 1 (on provided sheet)
3 Match the graphs to their equations.
1 Identify the coordinates of the
indicated points.
𝐴
𝑦 = sin 𝑥
𝐶
𝐵
𝑥2 + 𝑦2 = 9
𝑦=
4
𝑥
𝐸
𝐷
1
𝑨 𝟗𝟎,
?𝟏
𝑪 𝟎,?𝟑
𝑬 𝟏,?𝟒
𝑩 𝟏𝟖𝟎,
?𝟎
𝑫 −𝟑,?𝟎
2 Which of these graphs could have the
equation 𝑦 = 𝑥 3 − 2𝑥 2 + 3?
a
b
c
c, because a is the wrong way up (given 𝒙𝟑 term has positive
coefficient) and b has the wrong y-intercept.
?
i. 𝑦 = 4 sin 𝑥
ii. 𝑦 = 4 cos 𝑥
iii. 𝑦 = 𝑥 2 − 4𝑥 + 5
iv. 𝑦 = 4 × 2𝑥
v. 𝑦 = 𝑥 3 + 4
4
vi. 𝑦 = 𝑥
E
B
F
C?
D
A
Exercise 1 (on provided sheet)
4
-15
?
-7
?
-6
?
?1
Reveal
Exercise 1
5
The graph shows 𝑦 = 𝑥 2 − 𝑥 − 2.
7 Using the cos graph below, and given
a that cos 45 = 12, find all solutions
to cos 𝑥 =
1
2
(other than 45).
𝒙 = 𝟑𝟏𝟓°
?
Use the graph to estimate the solution(s) to:
i) 𝑥 2 − 𝑥 − 2 = 4
𝒙 = −𝟐 𝒐𝒓 𝟑
2
ii) 𝑥 − 𝑥 − 2 = −1
𝒙 ≈ −𝟎. 𝟔 𝒐𝒓 𝟏. 𝟔
2
iii) 𝑥 − 𝑥 − 2 = 7
𝒙 ≈ −𝟐. 𝟓 𝒐𝒓 𝟑. 𝟓
?
?
?
6
The graph shows the line with equation
𝑦 = 6 + 2𝑥 − 𝑥 2
b
Given that cos 30 =
3
,
2
3
2
find all
solutions to cos 𝑥 =
𝒙 = 𝟑𝟑𝟎°
?
c
Use the graph to estimate the solution(s) to:
i) 6 + 2𝑥 − 𝑥 2 = 0
𝒙 ≈ −𝟏. 𝟔𝟓 𝒐𝒓 𝟑. 𝟐𝟓
2
ii) 6 + 2𝑥 − 𝑥 = 4
𝒙 ≈ −𝟎. 𝟕 𝒐𝒓 𝟐. 𝟕
iii) By drawing a suitable line onto the graph, estimate the
solutions to 6 + 2𝑥 − 𝑥 2 = 𝑥 + 2
𝒙 ≈ −𝟏. 𝟓𝟔 𝒐𝒓 𝟐. 𝟓𝟔
?
?
1
[Hard] Given cos 60 = , again
2
using the graph, find all solutions to
1
𝑐𝑜𝑠 𝑥 = −
2
𝒙 = 𝟏𝟐𝟎°, 𝟐𝟒𝟎°
?
Exercise 1
8
3
,
2
i)
Given sin 60 =
ii)
solutions to sin 𝑥 =
2
𝒙 = 𝟏𝟐𝟎 (, 𝟔𝟎)
1
Given sin 30 = , determine all
determine all
3
?
2
1
solutions to sin 𝑥 =
2
𝒙 = 𝟏𝟓𝟎 (, 𝟑𝟎)
iii) [Harder] Given sin 45 =
?
1
,
2
determine the two solutions to
1
sin 𝑥 = − (note the minus)
2
𝒙 = 𝟐𝟐𝟓°, 𝟑𝟏𝟓°
?
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 = 𝒌𝒂𝒙
(3,175)
(1,7)
Determine 𝑘 and 𝑎 (where 𝑘
and 𝑎 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?
𝟕 = 𝒌𝒂𝟏
𝟏𝟕𝟓 = 𝒌𝒂𝟑
𝟐𝟓 = 𝒂𝟐
𝒂 =?𝟓
Substituting back into 1st equation:
𝟕
𝒌=
𝟓
Test Your Understanding
Q
N
Given that 2,6 and 5,162 are points on the curve 𝑦 = 𝑘𝑎 𝑥 , find
the value of 𝑘 and 𝑎.
6 = 𝑘𝑎2
162 = 𝑘𝑎5
→ 27 = 𝑎3
?
𝒂=𝟑
𝟔
𝟐
𝒌= 𝟐=
𝟑
𝟑
9
Given that 3, 45 and 1, 5 are points on the curve 𝑦 = 𝑎2 𝑏 𝑥
where 𝑎 and 𝑏 are positive constants, find the value of 𝑎 and 𝑏.
45 = 𝑎2 𝑏3
9
= 𝑎2 𝑏
5
→ 25 = 𝑏2
?
𝒃=𝟓
𝒂=
𝟒𝟓
=
𝒃𝟑
𝟒𝟓
𝟗
=
𝟏𝟐𝟓 𝟐𝟓
Exercise 1 (continued)
9
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 = 𝒃𝒂𝟒
→ 𝟖 = 𝒂𝟑
𝒂=𝟐
𝒃=𝟑
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 =
?
?
2
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
?
4
𝟏
𝟐
Given that the points (3,
1
1
) and
72
7,
lie on the exponential
1152
curve with equation 𝑦 = 𝑏 2 𝑎 𝑥 ,
determine 𝑎 and 𝑏.
𝟏
𝟏
𝒂 = ,𝒃 =
𝟐
𝟑
?