Question

Write the equation of the trigonometricgraph. Try fractional values or I for thebox next to x.

Answer 1

We have the graph of a sine function, and we want to find the following parameters

`y=A\sin(\omega x)+c`

We can easily find c, just do x = 0 and look at the graph

`\begin{gathered} y=A\sin(\omega x)+c\text{ \lparen x = 0\rparen} \\ \\ y=c \end{gathered}`

Therefore when x = 0 the value of y is the value of c, looking at the graph we can see that it's -1, therefore

`y=A\sin(\omega x)-1`

For A, we can say that A is the sum between the max and min, divided by 2, we must take the modulus here for negative values, therefore

`\begin{gathered} A=(|\max|+|\min|)/(2) \\ \\ A=(1+3)/(2) \\ \\ A=2 \end{gathered}`

The amplitude here is 2, but we have a detail! see that the function is flipped over the x-axis, therefore we must do the definition:

`f(x)=2\sin(\omega x)-1`

Flip it

`f(x)\rightarrow-f(x)\Rightarrow-2\sin(\omega x)-1`

Therefore we have

`y=-2\sin(\omega x)-1`

Now the last one, we must find ω, we can find it using the definition of ω

`ω=(2\pi)/(T)`

Where T is the period of the function. Looking at the graph we can see that T = 4π/3, then

`\begin{gathered} \begin{equation*} ω=(2\pi)/(T) \end{equation*} \\ \\ ω=(2\pi)/(1)\cdot(3)/(4\pi)=(3)/(2) \\ \\ ω=(3)/(2) \end{gathered}`

Now we have all the parameters

`y = -2\sin\left((3)/(2)x\right) - 1`

Or writing in decimal

`y=-2\sin\left(1.5x\right)-1`

The final answer is

`\begin{gathered} y = -2\sin\left((3)/(2)x\right) - 1 \\ \end{gathered}`