The transformation needed to change the parent cosine function to y = 3cos( 10(x-3)) are: vertical stretch of 3, horizontal compression to a period of [tex]\frac{\pi}{5}[/tex] and a phase shift of 3 units to the right
The correct answer is option (B)
What is translation?
"It is a geometric transformation which is used to describe when object moves a certain distance, without change in shape and size."
What is scaling?
"It is a geometric transformation that enlarges or shrinks objects by a scale factor that is the same in all directions. "
What is phase shift?
"It is how far the function is shifted horizontally from the usual position."
What is general formula for cosine function?
"y = A cos(B(x + C)) + D"
where, A = amplitude
2π/B = period
C = phase shift (it is positive to the left)
D = vertical shift
For given question,
We have been given a function y = 3cos(10 (x - 3))
Comparing above function with general formula of cosine function,
A = 3, B = 10, C = -3
Since C = -3, the parent function y = cos(x) has a phase shift of 3 units to the right.
Also, the period is,
[tex]=\frac{2\pi}{B}\\\\=\frac{2\pi}{10}\\\\ =\frac{\pi}{5}[/tex]
We know, y = f(n x) is a stretch horizontally by scale factor 1/n
So, y = cos(10(x - 3)) means, the parent function y = cos(x) has horizontally compression to a period of [tex]\frac{\pi}{5}[/tex]
We know, y = n f(x) is a stretch vertically by scale factor of n
y = 3 cos(10(x - 3)) represents a function stretched on the y-axis by scale factor of 3.
Therefore, the transformation needed to change the parent cosine function to y = 3cos( 10(x-3)) are: vertical stretch of 3, horizontal compression to a period of [tex]\frac{\pi}{5}[/tex] and a phase shift of 3 units to the right
The correct answer is option (B)
Learn more about geometric transformation here:
https://brainly.com/question/5833423
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