Question

Which transformations are needed to change the parent cosine function to y = 0. 35 cosine (8 (x minus StartFraction pi Over 4 EndFraction))? vertical stretch of 0. 35, horizontal stretch to a period of 16 pi, phase shift of StartFraction pi Over 4 EndFraction units to the right vertical compression of 0. 35, horizontal compression to a period of 4 pi, phase shift of StartFraction pi Over 4 EndFraction units to the left vertical compression of 0. 35, horizontal compression to a period of StartFraction pi Over 4 EndFraction, phase shift of StartFraction pi Over 4 EndFraction units to the right vertical stretch of 0. 35, horizontal stretch to a period of StartFraction pi Over 4 EndFraction, phase shift of StartFraction pi Over 4 EndFraction units to the right.

Answer 1

The transformations that are needed to change the parent cosine function to y = 0.35×cos(8(x-π/4)) are:

• vertical stretch of 0.35
• horizontal compression of period of $\pi/4$
• phase shift of $\pi/4$ to right

How does transformation of a function happens?

The transformation of a function may involve any change.

Usually, these can be shift horizontally (by transforming inputs) or vertically (by transforming output), stretching (multiplying outputs or inputs) etc.

If the original function is $y = f(x)$, assuming horizontal axis is input axis and vertical is for outputs, then:

• Horizontal shift (also called phase shift):

1. Left shift by c units: $y = f(x+c)$earlier)
2. Right shift by c units: $y = f(x-c)$output, but c units late)

• Vertical shift:

1. Up by d units: $y = f(x) + d$
2. Down by d units: $y = f(x) - d$

• Stretching:

1. Vertical stretch by a factor k: $y = k \times f(x)$
2. Horizontal stretch by a factor k: $y = f(\dfrac{x}{k})$

For this case, we're specified that:

y = cos(x) (the parent cosine function) was transformed to $y = 0.35\cos(8(x-\pi/4))$

We can see its vertical stretch by 0.35, right shift by $\pi/4$horizontal stretch by 1/8

Period of cos(x) is of $2\pi$ length. But 1.8 stretching makes its period shrink to $2\pi/8 = \pi/4$

Thus, the transformations that are needed to change the parent cosine function to y = 0.35×cos(8(x-π/4)) are:

• vertical stretch of 0.35
• horizontal compression to period of $\pi/4$ (which means period of cosine is shrunk to $\pi/4$ which originally was $2\pi$ )
• phase shift of $\pi/4$ to right

Learn more about transformation of functions here:

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