Write a sine function that has a midline of y=2 an amplitude of 5 , a period of (5pi )/(4) , and a horizontal shift of (5pi )/(8) to the left. Answer Attempt 2 out of 2 f(x)=

To write a sine function with the given parameters, we need to consider the following information:<br /><br />1. Midline: The midline is the horizontal line that divides the graph of the function into two equal parts. In this case, the midline is $y=2$, which means that the function will have a vertical shift of 2 units upwards.<br /><br />2. Amplitude: The amplitude is the distance between the midline and the maximum or minimum value of the function. In this case, the amplitude is 5, which means that the function will have a vertical stretch of 5 units.<br /><br />3. Period: The period is the distance between two consecutive points where the function repeats itself. In this case, the period is $\frac{5\pi}{4}$, which means that the function will repeat itself every $\frac{5\pi}{4}$ units.<br /><br />4. Horizontal shift: A horizontal shift moves the graph of the function left or right. In this case, the function is shifted $\frac{5\pi}{8}$ units to the left.<br /><br />Using this information, we can write the sine function as:<br /><br />$f(x) = 5\sin\left(2\pi\left(x+\frac{5\pi}{8}\right)-2\pi\right)+2$<br /><br />Simplifying the equation, we get:<br /><br />$f(x) = 5\sin\left(2\pi x+\frac{5\pi}{4}-2\pi\right)+2$<br /><br />$f(x) = 5\sin\left(2\pi x-\frac{7\pi}{4}\right)+2$<br /><br />Therefore, the sine function with the given parameters is:<br /><br />$f(x) = 5\sin\left(2\pi x-\frac{7\pi}{4}\right)+2$