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

To write a cosine function with the given parameters, we need to use the general form of a cosine function:<br /><br />$f(x) = A \cos(B(x - C)) + D$<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: $y = 5$<br />- Amplitude: 4<br />- Period: $\frac{\pi}{2}$<br />- Horizontal shift: $\frac{\pi}{4}$ to the right<br /><br />Step 1: Determine the amplitude.<br />The amplitude is given as 4, so $A = 4$.<br /><br />Step 2: Determine the frequency.<br />The period is given as $\frac{\pi}{2}$, which means the frequency is $B = \frac{2\pi}{\frac{\pi}{2}} = 4$.<br /><br />Step 3: Determine the horizontal shift.<br />The horizontal shift is given as $\frac{\pi}{4}$ to the right, so $C = \frac{\pi}{4}$.<br /><br />Step 4: Determine the midline.<br />The midline is given as $y = 5$, so $D = 5$.<br /><br />Putting it all together, the cosine function is:<br /><br />$f(x) = 4 \cos(4(x - \frac{\pi}{4})) + 5$<br /><br />Simplifying the expression inside the cosine function:<br /><br />$f(x) = 4 \cos(4x - \pi) + 5$<br /><br />Therefore, the cosine function that satisfies the given conditions is:<br /><br />$f(x) = 4 \cos(4x - \pi) + 5$