Math  /  Trigonometry

QuestionFind a function of the form y=Asin(kx)+Cy=A \sin (k x)+C or y=Acos(kx)+Cy=A \cos (k x)+C whose graph matches the ff shown below:
Leave your answer in exact form; if necessary, type pi for π\pi. y=y= \square Question Help: Video

Studdy Solution

STEP 1

1. The function is either of the form y=Asin(kx)+C y = A \sin(kx) + C or y=Acos(kx)+C y = A \cos(kx) + C .
2. The graph is a sinusoidal function with specific characteristics: period, amplitude, and midline.
3. The graph starts at its maximum point, indicating a cosine function.
4. The period of the function is 8 units.
5. The amplitude of the function is 8.
6. The midline of the function is at y=0 y = 0 , indicating no vertical shift.

STEP 2

1. Determine the form of the function (sine or cosine).
2. Calculate the amplitude A A .
3. Determine the period and calculate k k .
4. Identify the vertical shift C C .
5. Write the equation of the function.

STEP 3

Since the graph starts at its maximum point, the function is of the form y=Acos(kx)+C y = A \cos(kx) + C .

STEP 4

The amplitude A A is the distance from the midline to a peak or trough. Given that the peaks are at y=8 y = 8 and the troughs at y=8 y = -8 , the amplitude is:
A=8 A = 8

STEP 5

The period T T of the function is given as 8 units. The formula for the period of a cosine function is T=2πk T = \frac{2\pi}{k} . Solving for k k :
8=2πk 8 = \frac{2\pi}{k}
k=2π8=π4 k = \frac{2\pi}{8} = \frac{\pi}{4}

STEP 6

The midline of the function is at y=0 y = 0 , so there is no vertical shift. Therefore, C=0 C = 0 .

STEP 7

Now that we have determined A=8 A = 8 , k=π4 k = \frac{\pi}{4} , and C=0 C = 0 , we can write the equation of the function:
y=8cos(π4x) y = 8 \cos\left(\frac{\pi}{4}x\right)
The function is:
y=8cos(π4x) y = 8 \cos\left(\frac{\pi}{4}x\right)

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