Question

Need help on these !! 13 and 14! Please!!

Answer 1

Question # 13

Answer:

The required equation for the given function is y = 4sin(x/2+2π/3) -2 , as shown attached graph diagram.

Step-by-step explanation:

As the general sine function is given by

$y=asin(bx+c)+d$.......[A]

• amplitude = a
• period = 2π ÷ b
• Phase shift = -c ÷ b
• Vertical shift = d

As in the question,  

• amplitude = a = 4
• period = 4π
• phase shift = -4π/3
• Vertical shift = d = -2

As  

period = 2π ÷ b  

b = 2π/period

b = 2π/4π ∵ period = 4π

b = 1/2  

Also

Phase shift = -c/b

-4π/3 = -c/b ∵ phase shift = -4π/3

4π/3 = c/b  

c = b × 4π/3  

c = 1/2 × 4π/3  

c = 4π/6  

c = 2π/3

So, putting Amplitude ⇒ a = 4, Vertical shift ⇒ d = -2, b = 1/2 ,  

and c = 2π/3 in Equation [A] would bring us the required equation for the given function.

$y=asin(bx+c)+d$

y = 4sin(x/2+2π/3)+(-2)

y = 4sin(x/2+2π/3) -2            

Note: The graph is also shown in attached diagram.

                                             Question # 14

Answer:

The required equation for the given function is y = cot(x+π/3)+2, as shown in attached graph diagram.

Step-by-step explanation:

As the general cotangent function is given by

$y=acot(bx+c)+d$.......[A]

• amplitude = a
• period = π ÷ b
• Phase shift = -c ÷ b
• Vertical shift = d

As in the question,  

• period = π
• phase shift = -π/3
• Vertical shift = d = 2

As  

period = π ÷ b  

b = π/period

b = π/π ∵ period = 4π

b = 1  

Also

Phase shift = -c/b

-π/3 = -c/b ∵ phase shift = -π/3

π/3 = c/b  

c = b × π/3  

c = 1 × π/3  

c = π/3

So, putting vertical shift ⇒ d = 2, b = 1 and   c = π/3 in Equation [A] would bring us the required equation for the given function.

$y=acot(bx+c)+d$

y = cot(x+π/3)+2

Note: The graph is also shown in attached diagram.

Keywords: amplitude, period , phase shift , vertical shift

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