Describing a Transformation of Trigonometric Functions

[nycmathguy]

Section 4.5

I need help with 58 parts (a - c). Seeking a detailed reply with steps.

 

[MathLover1]

58.

g(x)=4-sin(2x+pi/2)

a.
descriebe transformations

Compare it to g(x)=a*sin(bx+c)+d

.a=-1, b=2, c=pi/2, d=4

Amplitude =| a|=1

Period=2pi/b=2pi/2=pi

vertical shift: 4 units up

b.
sketch the graph

c.
use function notation to write g(x) in terms of f(x)

g(x) =4-f(2x+pi/2)

 

[nycmathguy]

58.

g(x)=4-sin(2x+pi/2)

a.
descriebe transformations

Compare it to g(x)=a*sin(bx+c)+d

.a=-1, b=2, c=pi/2, d=4

Amplitude =| a|=1

Period=2pi/b=2pi/2=pi

vertical shift: 4 units up

b.
sketch the graph

c.
use function notation to write g(x) in terms of f(x)

g(x) =4-f(2x+pi/2)

Thanks. Nice-done.

 

[nycmathguy]

Question 54

g(x) =sin(2x + pi)

Part (a)

Compare given function to g(x) = a•sin(bx + c) + d.

Note: a = 1, b = 2, c = pi, d = 0

Amplitude = A = | 1 | = 1

Period = P = 2pi/b = 2pi/2 = pi

No vertical shift as d = 0 indicates.

Part (b)

Part (c)

g(x) = 4 - f(2x + pi)

 

[nycmathguy]

Question 56

We can write the given function like this:

g(x) = 1 + cos(x + pi)

We compare the given function with
g(x) = d + a•cos(bx + pi).

Note: a = 1, b = 1, c = pi, d = 1

There's a vertical shift upward as indicated by d = 1.

Part (b)

Part (c)

g(x) = 1 + f(x + pi)