Question

A sine function has the following key features:

Answer 1

Answer:

Hence the function is given by:

$y=4\sin (\dfrac{\pi x}{6})+1$

Step-by-step explanation:

Let the general function be given by:

$y=a\sin (bx)+c$

As we know that period of $\sin x$ is $2\pi$.

Hence for period to be 12 we have to consider the function as:

$\sin (\dfrac{\pi x}{6})$

Hence $b=\dfrac{\pi}{6}$

Also the midline is y=1. this means that the minimum and maximum value are at the same distance from y=1.

also the amplitude of the function is 4.

i.e. the maximum value of the function is 4+1=5.

and the minimum value is -4+1=-3

Also y-intercept is (0,1) i.e. when x=0 y takes the value as 1.

i.e. c=1.

Hence the function is given by:

$y=4\sin (\dfrac{\pi x}{6})+1$

Answer 2

Answer: 4 sin[ (π/6)x ] + 1

Compared with the parent function y= sin(x), the graph of 4 sin[ (π/6)x ] + 1 will be strecthed vertically by a scale factor of 4, translated 1 unit up, and with a shorter distance between the peaks.

Soluton:

1) The parent function is sin(x)

2) sin(x) has:

Middle line: y = 0

Amplitude: 1 because the function goes from 1 unit up to 1 unit down the midddle line.

Period: 2π because sine function repeats every 2π units.

y-intercept: (0,0) because sin(0) = 0.

Now look how these changes in the function reflect on the parameters:

A sin (ωx + B) + C:

That function will have:

 amplitude A, becasue the amplitude is scaled by that factor

Period: 2π / ω, because the function is compressed horizontally by that factor.

It will be translated B units to the left

It will be translated C units up.

And you need

Period = 12 => 2π / ω = 12 => ω = π/6

 A = 4

Translate the midline from y = 0 to y = 1 => shift the function 1 unit up => C = 1.

Translate the y-intercept from y = 0 to y = 1, which is already accomplished when you translate the function 1 unit up.

So, the is the function searched>

y = A sin (ωx + B) + C = 4 sin[ (π/6)x ] + 1

Now you can check the amplitude, the period, the middle line and the y-intercept of that y = 4 sin[ (π/6)x ] + 1.

I strongly suggest that you graph it with a graphing program or calculator.