Answer :
Let's address each part of the problem step-by-step for both given functions.
### Function i) [tex]\( y = 4 \sin \left[\frac{1}{3}(x + 30^\circ)\right] - 1 \)[/tex]
#### a) Amplitude
- The amplitude of a function in the form [tex]\( y = A \sin(Bx + C) + D \)[/tex] is given by [tex]\(|A|\)[/tex].
- For the function [tex]\( y = 4 \sin \left[\frac{1}{3}(x + 30^\circ)\right] - 1 \)[/tex], the amplitude [tex]\( A \)[/tex] is [tex]\( 4 \)[/tex].
- Amplitude: 4
#### b) Period
- The period of a sine function [tex]\( y = A \sin(Bx + C) + D \)[/tex] is given by [tex]\( \frac{360^\circ}{|B|} \)[/tex] when using degrees.
- For [tex]\( y = 4 \sin \left[\frac{1}{3}(x + 30^\circ)\right] - 1 \)[/tex], the coefficient [tex]\( B = \frac{1}{3} \)[/tex].
- The period is [tex]\( \frac{360^\circ}{\frac{1}{3}} = 360^\circ \times 3 = 1080^\circ \)[/tex].
- Period: [tex]\( 1080^\circ \)[/tex]
#### c) Phase Shift
- The phase shift of the function [tex]\( y = A \sin(Bx + C) + D \)[/tex] is [tex]\(-\frac{C}{B}\)[/tex].
- In this case, [tex]\( C = \frac{1}{3} \cdot 30^\circ = 10^\circ \)[/tex].
- Hence, phase shift [tex]\( = -\frac{10^\circ}{\frac{1}{3}} = -30^\circ \)[/tex].
- Phase Shift: [tex]\( -30^\circ \)[/tex] (to the left)
#### d) Vertical Shift
- The vertical shift of the function [tex]\( y = A \sin(Bx + C) + D \)[/tex] is [tex]\( D \)[/tex].
- Here, [tex]\( D = -1 \)[/tex].
- Vertical Shift: -1
### Function ii) [tex]\( y = -\frac{1}{2} \cos \left[4(x + 135^\circ)\right] + 2 \)[/tex]
#### a) Amplitude
- The amplitude of the function [tex]\( y = A \cos(Bx + C) + D \)[/tex] is given by [tex]\(|A|\)[/tex].
- For the function [tex]\( y = -\frac{1}{2} \cos \left[4(x + 135^\circ)\right] + 2 \)[/tex], the amplitude [tex]\( A \)[/tex] is [tex]\( \left| -\frac{1}{2} \right| = \frac{1}{2} \)[/tex].
- Amplitude: [tex]\(\frac{1}{2}\)[/tex]
#### b) Period
- The period of a cosine function [tex]\( y = A \cos(Bx + C) + D \)[/tex] is given by [tex]\( \frac{360^\circ}{|B|} \)[/tex] when using degrees.
- For [tex]\( y = -\frac{1}{2} \cos \left[4(x + 135^\circ)\right] + 2 \)[/tex], the coefficient [tex]\( B = 4 \)[/tex].
- The period is [tex]\( \frac{360^\circ}{4} = 90^\circ \)[/tex].
- Period: [tex]\( 90^\circ \)[/tex]
#### c) Phase Shift
- The phase shift of the function [tex]\( y = A \cos(Bx + C) + D \)[/tex] is [tex]\(-\frac{C}{B}\)[/tex].
- In this case, [tex]\( C = 4 \cdot 135^\circ = 540^\circ \)[/tex].
- Hence, phase shift [tex]\( = -\frac{540^\circ}{4} = -135^\circ \)[/tex].
- Phase Shift: [tex]\( -135^\circ \)[/tex] (to the left)
#### d) Vertical Shift
- The vertical shift of the function [tex]\( y = A \cos(Bx + C) + D \)[/tex] is [tex]\( D \)[/tex].
- Here, [tex]\( D = 2 \)[/tex].
- Vertical Shift: 2
### e) Graph and Compare
To compare the given characteristics of each function with their graphs, you can use a graphing tool like Desmos or a graphing calculator.
Here's what you should expect:
- For [tex]\( y = 4 \sin \left[\frac{1}{3}(x + 30^\circ)\right] - 1 \)[/tex]:
- A wavy sine curve oscillating between 3 and -5 (due to amplitude of 4, vertically shifted down by 1).
- The wave completes one full cycle over an interval of [tex]\( 1080^\circ \)[/tex].
- The curve starts shifted 30 degrees to the left.
- For [tex]\( y = -\frac{1}{2} \cos \left[4(x + 135^\circ)\right] + 2 \)[/tex]:
- A wavy cosine curve oscillating between [tex]\( 2.5 \)[/tex] and [tex]\( 1.5 \)[/tex] (due to amplitude of [tex]\( \frac{1}{2} \)[/tex], vertically shifted up by 2).
- The wave completes one full cycle over an interval of [tex]\( 90^\circ \)[/tex].
- The curve starts shifted 135 degrees to the left.
When you graph these, make sure to analyze if the graphs match these properties in terms of amplitude, period, phase shift, and vertical shift.
### Function i) [tex]\( y = 4 \sin \left[\frac{1}{3}(x + 30^\circ)\right] - 1 \)[/tex]
#### a) Amplitude
- The amplitude of a function in the form [tex]\( y = A \sin(Bx + C) + D \)[/tex] is given by [tex]\(|A|\)[/tex].
- For the function [tex]\( y = 4 \sin \left[\frac{1}{3}(x + 30^\circ)\right] - 1 \)[/tex], the amplitude [tex]\( A \)[/tex] is [tex]\( 4 \)[/tex].
- Amplitude: 4
#### b) Period
- The period of a sine function [tex]\( y = A \sin(Bx + C) + D \)[/tex] is given by [tex]\( \frac{360^\circ}{|B|} \)[/tex] when using degrees.
- For [tex]\( y = 4 \sin \left[\frac{1}{3}(x + 30^\circ)\right] - 1 \)[/tex], the coefficient [tex]\( B = \frac{1}{3} \)[/tex].
- The period is [tex]\( \frac{360^\circ}{\frac{1}{3}} = 360^\circ \times 3 = 1080^\circ \)[/tex].
- Period: [tex]\( 1080^\circ \)[/tex]
#### c) Phase Shift
- The phase shift of the function [tex]\( y = A \sin(Bx + C) + D \)[/tex] is [tex]\(-\frac{C}{B}\)[/tex].
- In this case, [tex]\( C = \frac{1}{3} \cdot 30^\circ = 10^\circ \)[/tex].
- Hence, phase shift [tex]\( = -\frac{10^\circ}{\frac{1}{3}} = -30^\circ \)[/tex].
- Phase Shift: [tex]\( -30^\circ \)[/tex] (to the left)
#### d) Vertical Shift
- The vertical shift of the function [tex]\( y = A \sin(Bx + C) + D \)[/tex] is [tex]\( D \)[/tex].
- Here, [tex]\( D = -1 \)[/tex].
- Vertical Shift: -1
### Function ii) [tex]\( y = -\frac{1}{2} \cos \left[4(x + 135^\circ)\right] + 2 \)[/tex]
#### a) Amplitude
- The amplitude of the function [tex]\( y = A \cos(Bx + C) + D \)[/tex] is given by [tex]\(|A|\)[/tex].
- For the function [tex]\( y = -\frac{1}{2} \cos \left[4(x + 135^\circ)\right] + 2 \)[/tex], the amplitude [tex]\( A \)[/tex] is [tex]\( \left| -\frac{1}{2} \right| = \frac{1}{2} \)[/tex].
- Amplitude: [tex]\(\frac{1}{2}\)[/tex]
#### b) Period
- The period of a cosine function [tex]\( y = A \cos(Bx + C) + D \)[/tex] is given by [tex]\( \frac{360^\circ}{|B|} \)[/tex] when using degrees.
- For [tex]\( y = -\frac{1}{2} \cos \left[4(x + 135^\circ)\right] + 2 \)[/tex], the coefficient [tex]\( B = 4 \)[/tex].
- The period is [tex]\( \frac{360^\circ}{4} = 90^\circ \)[/tex].
- Period: [tex]\( 90^\circ \)[/tex]
#### c) Phase Shift
- The phase shift of the function [tex]\( y = A \cos(Bx + C) + D \)[/tex] is [tex]\(-\frac{C}{B}\)[/tex].
- In this case, [tex]\( C = 4 \cdot 135^\circ = 540^\circ \)[/tex].
- Hence, phase shift [tex]\( = -\frac{540^\circ}{4} = -135^\circ \)[/tex].
- Phase Shift: [tex]\( -135^\circ \)[/tex] (to the left)
#### d) Vertical Shift
- The vertical shift of the function [tex]\( y = A \cos(Bx + C) + D \)[/tex] is [tex]\( D \)[/tex].
- Here, [tex]\( D = 2 \)[/tex].
- Vertical Shift: 2
### e) Graph and Compare
To compare the given characteristics of each function with their graphs, you can use a graphing tool like Desmos or a graphing calculator.
Here's what you should expect:
- For [tex]\( y = 4 \sin \left[\frac{1}{3}(x + 30^\circ)\right] - 1 \)[/tex]:
- A wavy sine curve oscillating between 3 and -5 (due to amplitude of 4, vertically shifted down by 1).
- The wave completes one full cycle over an interval of [tex]\( 1080^\circ \)[/tex].
- The curve starts shifted 30 degrees to the left.
- For [tex]\( y = -\frac{1}{2} \cos \left[4(x + 135^\circ)\right] + 2 \)[/tex]:
- A wavy cosine curve oscillating between [tex]\( 2.5 \)[/tex] and [tex]\( 1.5 \)[/tex] (due to amplitude of [tex]\( \frac{1}{2} \)[/tex], vertically shifted up by 2).
- The wave completes one full cycle over an interval of [tex]\( 90^\circ \)[/tex].
- The curve starts shifted 135 degrees to the left.
When you graph these, make sure to analyze if the graphs match these properties in terms of amplitude, period, phase shift, and vertical shift.
