GCSE Curved Graphs

Report
GCSE: Curved Graphs
Dr J Frost ([email protected])
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  = 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
Quadratic
 =  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
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.

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
Test Your Understanding
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)  = −.    = . 
?
Test Your Understanding
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
 = °
?
Test Your Understanding
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).
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?
 = 
 = 
 = 
 =?
Substituting back into 1st equation:

=

Test Your Understanding
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 .


 = , =


?

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