Write a cosine function that has a midline of y=3 an amplitude of 5, a period of (3)/(4) and a horizontal shift of (3)/(16) to the right. Answer Attempt 2 out of f(x)=

To write a cosine function with the given parameters, we need to consider the following:<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 $y = 3$, so the function will have a vertical shift of 3 units.<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, so 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 $\frac{3}{4}$, so the function will have a horizontal compression of $\frac{4}{3}$.<br /><br />4. Horizontal shift: The horizontal shift moves the graph of the function left or right. In this case, the horizontal shift is $\frac{3}{16}$ to the right, so the function will have a horizontal shift of $\frac{3}{16}$.<br /><br />Using these parameters, we can write the cosine function as:<br /><br />$f(x) = 5\cos\left(2\pi\left(\frac{x - \frac{3}{16}}{\frac{3}{4}}\right)\right) + 3$<br /><br />Simplifying the expression inside the cosine function, we get:<br /><br />$f(x) = 5\cos\left(2\pi\left(\frac{4x - 3}{3}\right)\right) + 3$<br /><br />Therefore, the cosine function with the given parameters is:<br /><br />$f(x) = 5\cos\left(\frac{8\pi x - 6\pi}{3}\right) + 3$