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 Attemptiout of 2 f(x)=

To write a sine function with the given parameters, we need to use the general form of a sine function:<br /><br />$f(x) = A \sin(B(x - C)) + D$<br /><br />where:<br />- $A$ is the amplitude<br />- $B$ is the frequency<br />- $C$ is the horizontal shift<br />- $D$ is the midline<br /><br />Given:<br />- Midline: $y = 2$<br />- Amplitude: 5<br />- Period: $\frac{5\pi}{4}$<br />- Horizontal shift: $\frac{5\pi}{8}$ to the left<br /><br />Step 1: Determine the value of $A$.<br />The amplitude is given as 5, so $A = 5$.<br /><br />Step 2: Determine the value of $B$.<br />The period is given as $\frac{5\pi}{4}$. The period of a sine function is given by $\frac{2\pi}{B}$, so we can set up the equation:<br />$\frac{2\pi}{B} = \frac{5\pi}{4}$<br />Solving for $B$, we get:<br />$B = \frac{8}{5}$<br /><br />Step 3: Determine the value of $C$.<br />The horizontal shift is given as $\frac{5\pi}{8}$ to the left. This means $C = \frac{5\pi}{8}$.<br /><br />Step 4: Determine the value of $D$.<br />The midline is given as $y = 2$, so $D = 2$.<br /><br />Putting it all together, the sine function is:<br />$f(x) = 5 \sin\left(\frac{8}{5}\left(x - \frac{5\pi}{8}\right)\right) + 2$<br /><br />Therefore, the sine function with the given parameters is:<br />$f(x) = 5 \sin\left(\frac{8}{5}\left(x - \frac{5\pi}{8}\right)\right) + 2$