find a formula for this function.\ny = ? cos(\\frac{2\\pi}{\\quad} (x - \\quad)) + \\quad

Answer

Explanation:

Step1: Find the amplitude (A)

The amplitude is half the distance between the maximum and minimum values. The maximum value is 7.2 and the minimum is 2.4. So, ( A=\frac{7.2 - 2.4}{2}=\frac{4.8}{2} = 2.4)? Wait, no, wait. Wait, the general form of a cosine function is ( y = A\cos\left(\frac{2\pi}{T}(x - h)\right)+k), where ( A) is amplitude, ( T) is period, ( h) is phase shift, and ( k) is vertical shift.

Wait, vertical shift ( k) is the midline, which is the average of max and min. So ( k=\frac{7.2 + 2.4}{2}=\frac{9.6}{2}=4.8). Then amplitude ( A) is max - midline = ( 7.2 - 4.8 = 2.4). Wait, no, wait: amplitude is the distance from midline to max (or min). So ( A = 7.2 - 4.8 = 2.4)? Wait, no, 7.2 - 4.8 is 2.4? Wait 7.2 - 4.8 is 2.4? Wait 4.8 + 2.4 is 7.2, 4.8 - 2.4 is 2.4. So amplitude ( A = 2.4)? Wait, but let's check the period. The period ( T) is the distance between two consecutive maxima. The first maximum is at ( x = 2.5), the next at ( x = 7.5), so ( T=7.5 - 2.5 = 5). So the coefficient of ( x) inside the cosine is ( \frac{2\pi}{T}=\frac{2\pi}{5}).

Phase shift ( h): the cosine function ( \cos(x)) has a maximum at ( x = 0). Here, the maximum is at ( x = 2.5), so the phase shift ( h = 2.5) (since we have ( \cos\left(\frac{2\pi}{5}(x - h)\right)), so when ( x = h), the argument is 0, and cos(0)=1, which is the maximum).

Vertical shift ( k) is the midline, which is ( \frac{7.2 + 2.4}{2}=4.8).

So putting it all together: ( y = 2.4\cos\left(\frac{2\pi}{5}(x - 2.5)\right)+4.8)

Wait, let's verify:

Amplitude: 2.4 (distance from midline 4.8 to max 7.2: 7.2 - 4.8 = 2.4; to min 2.4: 4.8 - 2.4 = 2.4)
Period: 5 (distance between 2.5 and 7.5 is 5, so ( \frac{2\pi}{5}) gives period 5)
Phase shift: 2.5 (max at x=2.5, so when x=2.5, the argument is 0, cos(0)=1, so 2.4*1 + 4.8 = 7.2, correct)
Vertical shift: 4.8 (midline, average of 7.2 and 2.4)

So the formula is ( y = 2.4\cos\left(\frac{2\pi}{5}(x - 2.5)\right)+4.8)

Step1: Determine Amplitude (A)

Amplitude is half the difference between max and min.
( A=\frac{7.2 - 2.4}{2}=\frac{4.8}{2}=2.4)

Step2: Determine Period (T) and ( \frac{2\pi}{T})

Period is distance between two maxima: ( T = 7.5 - 2.5 = 5).
Thus, ( \frac{2\pi}{T}=\frac{2\pi}{5}).

Step3: Determine Phase Shift (h)

The maximum occurs at ( x = 2.5). For ( \cos\left(\frac{2\pi}{5}(x - h)\right)), max at ( x = h), so ( h = 2.5).

Step4: Determine Vertical Shift (k)

Vertical shift is midline (average of max and min):
( k=\frac{7.2 + 2.4}{2}=\frac{9.6}{2}=4.8).

Answer:

( y = \boldsymbol{2.4} \cos\left( \frac{2\pi}{\boldsymbol{5}} (x - \boldsymbol{2.5}) \right) + \boldsymbol{4.8} )