Question

I need help 1. Is this graph sine or cosine 2. What is the amplitude of the graph 3. What is the equation of the midline 4. What is the period of the function 5. Write the equation of the function

Answer 1

2.-

The amplitude of a sinusoid can be found as half the difference between its maximum and minimum values. Since the maximum value of the sinusoid in the graph is 2 and the minimum value is -6, then its amplitude is:

`A=(2-(-6))/(2)=(2+6)/(2)=(8)/(2)=4`

Therefore:

`A=4`

3.-

The value of the midline can be found as the average value of the maximum and the minimum of the sinusoid:

`b=(2+(-6))/(2)=(2-6)/(2)=(-4)/(2)=-2`

Then, the equation of the midline is:

`-2`

1.-

The graph of a sine takes the same value as the midline when x=0, while the graph of a cosine takes the maximum value at x=0. We can see that when x=0, the graph has a value of -2, which is the same value as the midline.

Therefore, the graph corresponds to a sine.

4.-

The period of the function is equal to the length over the X-axis required for the function to complete one cycle. The function will repeat the same values for each interval with the same size as one period. In this case, we can see that the function returns to the same value of -2 preserving the same shape after x=0 when it reaches x=2π/3. Then, the period of the function is 2π/3:

`T=(2\pi)/(3)`

5.- The equation of a sinusoid with amplitude A, period T, and midline b is:

`y=A\sin ((2\pi)/(T)x)+b`

Substitute A=4, T=2π/3 and b=-2 to find the equation of the function:

`\begin{gathered} y=4\cdot\sin ((2\pi)/(2\pi/3)x)+(-2) \\ =4\cdot\sin (3x)-2 \end{gathered}`

Therefore, the equation of the function is:

`y=4\sin (3x)-2`