Given:
- The amplitude of the Sine Function:
[tex]A=10[/tex]
- The midline:
[tex]y=4[/tex]
- And the period:
[tex]Period=2[/tex]
- You know that the function does not have a Phase shift.
• You need to remember that, by definition, the General Equation for a Sine Function has this form:
[tex]y=Asin\mleft(B\mleft(x+C\mright)\mright)+D[/tex]
Where "A" is the amplitude, "C" is the phase shift, "D" is the vertical shift and this is the period:
[tex]Period=\frac{2\pi}{B}[/tex]
Since the midline is given by the vertical shift, you can identify that, in this case:
[tex]D=4[/tex]
And, knowing the period, you can set up that:
[tex]2=\frac{2\pi}{B}[/tex]
Solving for "B", you get:
[tex]\begin{gathered} 2B=2\pi \\ \\ B=\frac{2\pi}{2} \\ \\ B=\pi \end{gathered}[/tex]
• It is important to remember the following Transformation Rule for Functions:
When:
[tex]-f(x)[/tex]
The function is reflected over the x-axis.
Therefore, knowing all the data, you can set up this equation:
[tex]f(x)=-10\sin (\pi x)+4[/tex]
Hence, the answer is: First option.
