- Level: AS and A Level
- Subject: Maths
- Word count: 1446
Investigation into combined transoformations of 6 trigonometric functions
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Introduction
Keval Chauhan
Maths Core Three
Investigation of combined transformations
A line has the general formula of, the part in brackets must always be there the others are variables, and may not be there in some equations for a line.
Y= at(bx+c)+d
T, can be any trigonometric function (Sine, cosine, tangent, secant, cosecant or cotangent.)
Below is a picture that shows each of the 6 trigonometric functions.
A is a stretch in the y-axis.
B can give a stretch in the x-axis which can change the frequency of a graph. If b is greater than 1 there will be an increased frequency, but if b is less than 1 there will be a lower frequency.
C translates f(x) by c if c is +ve(positive) the graph will shift to the right by c, if c was –ve(negative) the graph will shift to the left by c.
D is a translation in the y-axis, if d is +ve the graph will shift up, but if d is -ve the graph will shift down.
In this investigation I will be looking at combined transformations with 2 transformations max per equation. There are 6 possible combinations by 4C2(nCr).
Middle
Graph 3 Graph 4
Graph 5
As you can see from all the results my predictions were correct. Graphs 2 and 3 are identical as well are graphs 4 and 5.
The third transformation combination im going to be investigating will be the combination of ad the combination of stretching in the y-direction and translating in the y-axis. This time round I’ll be investigating these changes on the trigonometric function of tangent shown by a red dotted line on figure 1.0
A tan graph has a period of every π, it has 2 vertical asymptotes -∞at x=− π/2 and ∞ at x= π/2, a tan graph has a rotational symmetry of 180 o about the origin (0,0) I’ll be looking at the graph of
y=atan(x)+d
I’ll be using the following values for a and d.
Graph 1 | Graph 2 | Graph 3 | Graph 4 | Graph 5 | |
a | 1 | 2 | 2 | -2 | -2 |
d | 1 | 2 | -2 | 2 | -2 |
Predictions
Graph 1 – Will cross the x-axis at (0,1)
Graph 2 – Will cross the x-axis at (0,2)
Graph 3 – Will cross the x-axis at (0,-2)
Graph 4 – Will cross the x-axis at (0,2)
Graph 5 – Will cross the x-axis at (0,-2)
Results
Graph 1 Graph 2 Graph 3
Graph 4 Graph 5
As you can see from all the results my predictions were correct.
Conclusion
A cosecant graph has 4 asymptotes at 0π and π, and has a period every 2π with a range of -∞<y<∞ it also has a rotational symmetry of 180 o about the origin (0,0). I’ll be looking at the graph of
y=csc(bx)+d
I’ll be changing b and d to the following values.
Graph 1 | Graph 2 | Graph 3 | Graph 4 | Graph 5 | |
b | 1 | 2 | 2 | -2 | -2 |
c | 1 | 2 | -2 | 2 | -2 |
Predictions
Graph 1 –
Graph 2 –
Graph 3 –
Graph 4 –
Graph 5 –
Results
Graph 1 Graph 2
Graph 3 Graph 4
Graph 5
The last combined transformation I’m going to be investigating is what happens when you combine c(translating in x-direction) and d(translating in the y-direction). I will be studying this combination with the trigonometric function of cotangent which is 1/tan(x), it is shown by a aqua dotted line on figure 1.0.
A cotangent graph has a period every π and has 2 vertical asymptotes at one at 0 and the other at π. It also has a rotational symmetry of 180 o about the origin (0,0). It has a range of -∞<y<∞ .
). I’ll be looking at the graph of
y=cot (x+c)+d
I’ll be changing b and d to the following values.
Graph 1 | Graph 2 | Graph 3 | Graph 4 | Graph 5 | |
c | 1 | 2 | 2 | -2 | -2 |
d | 1 | 2 | -2 | 2 | -2 |
Predictions
Graph 1 –
Graph 2 –
Graph 3 –
Graph 4 –
Graph 5 –
Results
Graph 1 Graph 2
Graph 3 Graph 4
Graph 5
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