### GCSE Curved Graphs

```GCSE: Curved Graphs
Dr J Frost ([email protected]/* <![CDATA[ */!function(t,e,r,n,c,a,p){try{t=document.currentScript||function(){for(t=document.getElementsByTagName('script'),e=t.length;e--;)if(t[e].getAttribute('data-cfhash'))return t[e]}();if(t&&(c=t.previousSibling)){p=t.parentNode;if(a=c.getAttribute('data-cfemail')){for(e='',r='0x'+a.substr(0,2)|0,n=2;a.length-n;n+=2)e+='%'+('0'+('0x'+a.substr(n,2)^r).toString(16)).slice(-2);p.replaceChild(document.createTextNode(decodeURIComponent(e)),c)}p.removeChild(t)}}catch(u){}}()/* ]]> */)
GCSE Revision Pack Reference: 94, 95, 96, 97, 98
GCSE Specification
1
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  = sin  and  = cos ,
within the range -360° to +360°
The graph shows  =
cos . Determine the
coordinate of point .
q in
4 Find the values of p and
the function  =   given
coordinates on the graph of
=
“Given that 2,6 and 5,162 are
points on the curve  =   , find
the value of  and .”
Skill #1: Recognising Graphs
Linear
=  +
=  +
When  > 0
When  < 0
?
?
?
The line is known as a straight line.
Skill #1: Recognising Graphs
=  2 +  +
=  2 +  +
When  > 0
When  < 0
?
?
The line for a quadratic equation
is known as a parabola.
?
Skill #1: Recognising Graphs
Cubic
=
=  3 +  2 +  +
3
When  > 0
When  > 0
y
?
=  3
When  < 0
?
x
=  3 +  2 +  +
When  < 0
y
?
x
?
Skill #1: Recognising Graphs
Reciprocal

=

=

When  > 0
When  < 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
?

x
The y-intercept is  because  × 0 =  × 1 =?.
(unless  = 0, but let’s not go there!)
Skill #1: Recognising Graphs
Circle
The equation of this circle is:

x2 + y2 =? 25
5
5
-5
-5

! The equation of a circle with
centre at the origin and radius
r is:
2 + 2 = 2
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
?
B: cubic ?
?
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

0 8
90

0
?
1
?
180
270
360
0?
-1?
0?
= sin
1
90
180
270
360
-1
Skill #2: Plotting and
recognising trig functions.
Click to
brosketch
90
?1
180? 0

0 8
90

1
?
0
?
180
270
360
-1?
0?
1?
= cos
1
90
180
270
360
-1
Click to
brosketch
Quickfire Coordinates
= sin
= sin
= cos

= cos

270,
? −1
= sin
90,
?0
360,
? 0
= sin
= cos

0,
?1
= cos

180,
? 0
180,
? −1
90,1
?
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  when  = −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)  = −2
2
− 5 −2 − 3
?
= 11
b) i) When  = 0, then using graph,
?
roughly  = −.    = .
ii)  = −.    = .
?
The graph shows the line with
equation  =  2 +  − 12
Find estimates for the
solutions of the following
equations:
i)  2 +  − 12 = 5
= −.
= .
?
ii)  2 +  − 12 = −7
= −.    = .
?
Using a Trig Graph
Suppose that sin 45 =
Q
1
1
Using the graph, find the other
1
solution to sin  = 2
= °
?
2

-1
1
2
90

We can see by symmetry
that the difference
between 0 and 45 needs
to be the same as the
difference between
and 180.
180
270
360
1
Q
Suppose that sin 210 = − 2
Using the graph, find the other
1
solution to sin  = − 2
= °
?
The graph shows the line with equation  = cos
1
1
a) Given that cos 60 = , find the other solution to cos  =
2
2
= °
1
b) Given that cos 150° = −
90
-1
180
3
,
2
?
find the other solution to cos  = −
= °
?
270
360
3
2
Exercise 1 (on provided sheet)
3 Match the graphs to their equations.
1 Identify the coordinates of the
indicated points.

= sin

2 + 2 = 9
=
4

1
,
?
,?
,?
,
?
−,?
2 Which of these graphs could have the
equation  =  3 − 2 2 + 3?
a
b
c
c, because a is the wrong way up (given  term has positive
coefficient) and b has the wrong y-intercept.
?
i.  = 4 sin
ii.  = 4 cos
iii.  =  2 − 4 + 5
iv.  = 4 × 2
v.  =  3 + 4
4
vi.  =
E
B
F
C?
D
A
Exercise 1 (on provided sheet)
4
-15
?
-7
?
-6
?
?1
Reveal
Exercise 1
5
The graph shows  =  2 −  − 2.
7 Using the cos graph below, and given
a that cos 45 = 12, find all solutions
to cos  =
1
2
(other than 45).
= °
?
Use the graph to estimate the solution(s) to:
i)  2 −  − 2 = 4
= −
2
ii)  −  − 2 = −1
≈ −.   .
2
iii)  −  − 2 = 7
≈ −.   .
?
?
?
6
The graph shows the line with equation
= 6 + 2 −  2
b
Given that cos 30 =
3
,
2
3
2
find all
solutions to cos  =
= °
?
c
Use the graph to estimate the solution(s) to:
i) 6 + 2 −  2 = 0
≈ −.   .
2
ii) 6 + 2 −  = 4
≈ −.   .
iii) By drawing a suitable line onto the graph, estimate the
solutions to 6 + 2 −  2 =  + 2
≈ −.   .
?
?
1
[Hard] Given cos 60 = , again
2
using the graph, find all solutions to
1
= −
2
= °, °
?
Exercise 1
8
3
,
2
i)
Given sin 60 =
ii)
solutions to sin  =
2
=  (, )
1
Given sin 30 = , determine all
determine all
3
?
2
1
solutions to sin  =
2
=  (, )
iii) [Harder] Given sin 45 =
?
1
,
2
determine the two solutions to
1
sin  = − (note the minus)
2
= °, °
?
SKILL #4: Finding constants of  =  ⋅
The graph shows two points
(1,7) and (3,175) on a line with
equation:
=
(3,175)
(1,7)
Determine  and  (where
and  are positive constants).
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?
=
=
=
=?
Substituting back into 1st equation:

=

Q
N
Given that 2,6 and 5,162 are points on the curve  =   , find
the value of  and .
6 = 2
162 = 5
→ 27 = 3
?
=

= =

9
Given that 3, 45 and 1, 5 are points on the curve  = 2
where  and  are positive constants, find the value of  and .
45 = 2 3
9
= 2
5
→ 25 = 2
?
=
=

=

=

Exercise 1 (continued)
9
Given that the points (1,6) and
4,48 lie on the exponential curve
with equation  =  ×   ,
determine  and .
=
=
→  =
=
=
3
Given that the points (1,3) and
3,108 lie on the exponential
curve with equation  =  ×   ,
determine  and .
= ,  =
?
?
2
Given that the points (2,48) and
5,3072 lie on the exponential
curve with equation  =  ×   ,
determine  and .
= ,  =
?
4

Given that the points (3,
1
1
) and
72
7,
lie on the exponential
1152
curve with equation  =  2   ,
determine  and .

= , =

?
```