The function f(x)=sin(3x)+cos(3x)f(x)=sin(3x)+cos(3x) is the result of series of transformations with the first one being a horizontal translation of the function sin(x)sin(x). Which of this describes the first transformation?
1 Answer
We can get the graph of
a horizontal translation of
pi/12π12 radians to the lefta stretch along
OxOx with a scale factor of1/313 units- a stretch along
OyOy with a scale factor ofsqrt(2)√2 units
Explanation:
Consider the function:
f(x) = sin(3x)+cos(3x) f(x)=sin(3x)+cos(3x)
Let us suppose we can write this linear combination of sine and cosine as a single phase shifted sine function, that is suppose we have:
f(x) -= Asin(3x+alpha) f(x)≡Asin(3x+α)
\ \ \ \ \ \ \ = A{sin3xcosalpha+cos3xsinalpha}
\ \ \ \ \ \ \ = Acosalpha sin3x + Asinalphacos3x
In which case by comparing coefficients of
Acos alpha = 1 \ \ \ and\ \ \ Asinalpha = 1
By squaring and adding we have:
A^2cos^2alpha+A^2sin^2alpha = 2 => A^2=2=> A=sqrt(2)
By dividing we have:
tan alpha => alpha=pi/4
Thus we can write,
f(x) -= sin(3x)+cos(3x)
\ \ \ \ \ \ \ = sqrt(2)sin(3x+pi/4)
\ \ \ \ \ \ \ = sqrt(2)sin(3(x+pi/12))
So we can get the graph of
- a horizontal translation of
pi/12 radians to the left- a stretch along
Ox with a scale factor of1/3 units- a stretch along
Oy with a scale factor ofsqrt(2) units
Which we can see graphically:
The graph of
graph{sinx [-10, 10, -2, 2]}
The graph of
graph{sin(x+pi/12) [-10, 10, -2, 2]}
The graph of
graph{sin(3x+pi/4) [-10, 10, -2, 2]}
The graph of
graph{sqrt(2)sin(3x+pi/4) [-10, 10, -2, 2]}
And finally, the graph of the original function for comparison:
graph{sin(3x)+cos(3x) [-10, 10, -2, 2]}