Question

write a cosine function that has a midline of y = 5, an amplitude of 3, a period of 1, and a horizontal shift of 1/4 to the right.

Explanation:

Step1: Recall the general form of cosine function

The general form of a cosine - function is $y = A\cos(B(x - C))+D$.

Step2: Determine the value of $D$ (mid - line)

The mid - line of the cosine function is given by $y = D$. Since the mid - line is $y = 5$, then $D = 5$.

Step3: Determine the value of $A$ (amplitude)

The amplitude of the cosine function is $|A|$. Given that the amplitude is 3, then $A=\pm3$. Let's take $A = 3$ (we could also take $A=-3$).

Step4: Determine the value of $B$ (period)

The period of the cosine function $y = A\cos(B(x - C))+D$ is $T=\frac{2\pi}{|B|}$. Given that $T = 1$, then $1=\frac{2\pi}{|B|}$, so $|B| = 2\pi$. Let's take $B = 2\pi$.

Step5: Determine the value of $C$ (horizontal shift)

The horizontal shift is given by $C$. Given a horizontal shift of $\frac{1}{4}$ to the right, then $C=\frac{1}{4}$.

Answer:

$y = 3\cos(2\pi(x-\frac{1}{4})) + 5$