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

Answer

Explanation:

Step1: Find the amplitude (A)

The amplitude is half the distance between the maximum and minimum values. The maximum value is 2.1 and the minimum is 0.3. So, ( A=\frac{2.1 - 0.3}{2}=\frac{1.8}{2} = 0.9 ).

Step2: Find the period (T) and then the coefficient for the denominator

The period is the distance between two consecutive maximum points. From ( x = 0.6 ) to ( x = 2.2 ), the period ( T=2.2 - 0.6 = 1.6 ). The formula for the period of a cosine function is ( T=\frac{2\pi}{B} ), so ( B=\frac{2\pi}{T}=\frac{2\pi}{1.6}=\frac{2\pi}{\frac{8}{5}}=\frac{5\pi}{4} )? Wait, no, wait. Wait, the general form is ( y = A\cos\left(\frac{2\pi}{T}(x - C)\right)+D ). So ( \frac{2\pi}{T} ), so ( T = 1.6 ), so the denominator in the fraction is ( T = 1.6=\frac{8}{5} )? Wait, 2.2 - 0.6 = 1.6, which is ( \frac{8}{5} )? Wait, 0.6 to 2.2 is 1.6, which is ( \frac{8}{5} )? Wait, 1.6*5=8, so ( T = \frac{8}{5} ), so ( \frac{2\pi}{T}=\frac{2\pi}{\frac{8}{5}}=\frac{5\pi}{4} )? No, wait, no. Wait, let's recalculate the period. Wait, 2.2 - 0.6 is 1.6, which is 8/5? Wait, 1.6 = 8/5? 8 divided by 5 is 1.6, yes. So ( T=\frac{8}{5} ), so the coefficient for the denominator (the value in the square under the 2π) is ( T=\frac{8}{5}=1.6 ). Wait, but let's check the phase shift and vertical shift.

Step3: Find the vertical shift (D)

The vertical shift is the midline, which is the average of the maximum and minimum values. ( D=\frac{2.1 + 0.3}{2}=\frac{2.4}{2}=1.2 ).

Step4: Find the phase shift (C)

The cosine function ( y = \cos(x) ) has a maximum at ( x = 0 ). Here, the maximum is at ( x = 0.6 ), so the phase shift ( C = 0.6 ) (since the maximum is at ( x = 0.6 ), so the function is shifted right by 0.6 units).

So putting it all together:

Amplitude ( A = 0.9 )
Period ( T = 1.6=\frac{8}{5} ), so ( \frac{2\pi}{T}=\frac{2\pi}{\frac{8}{5}}=\frac{5\pi}{4} )? Wait, no, wait. Wait, the general form is ( y = A\cos\left(\frac{2\pi}{T}(x - C)\right)+D ). So ( \frac{2\pi}{T} ), so ( T = 1.6 ), so ( \frac{2\pi}{1.6}=\frac{2\pi}{\frac{8}{5}}=\frac{5\pi}{4} )? Wait, no, 2π divided by 1.6. Let's compute 1.6 as 8/5, so 2π divided by (8/5) is 2π*(5/8)=5π/4? Wait, but let's check the graph. Wait, the first maximum is at x=0.6, so the phase shift C is 0.6. The vertical shift D is 1.2, amplitude A is 0.9. The period T is 1.6, so the denominator in the fraction (the value under 2π) is T=1.6=8/5.

So the formula is ( y = 0.9\cos\left(\frac{2\pi}{1.6}(x - 0.6)\right)+1.2 ). Let's verify:

Amplitude 0.9, vertical shift 1.2, so maximum is 0.9 + 1.2 = 2.1, minimum is -0.9 + 1.2 = 0.3, which matches. The period is 1.6, so from x=0.6, adding 1.6 gives 2.2, which is the next maximum, correct. The phase shift is 0.6, so the maximum is at x=0.6, which matches the graph.

So:

The first box (amplitude) is 0.9.
The denominator under 2π is the period, which is 1.6 (or 8/5, but 1.6 is 8/5? Wait, 2.2 - 0.6 = 1.6, so T=1.6, so the denominator is 1.6.
The phase shift (the value in the (x - [ ]) part) is 0.6.
The vertical shift (the last box) is 1.2.

Let's check:

( y = 0.9\cos\left(\frac{2\pi}{1.6}(x - 0.6)\right)+1.2 )

Let's compute ( \frac{2\pi}{1.6}=\frac{2\pi}{\frac{8}{5}}=\frac{5\pi}{4} )? Wait, no, 2π divided by 1.6: 1.6 is 8/5, so 2π divided by (8/5) is (2π)(5/8)=5π/4? Wait, but 5π/4 is 3.927, but 2π/1.6 is approximately 3.927, which is 5π/4. Wait, but let's check with x=0.6: ( \cos\left(\frac{2\pi}{1.6}(0.6 - 0.6)\right)=\cos(0)=1 ), so y=0.91 + 1.2=2.1, correct. At the minimum, the cosine is -1, so y= -0.9 + 1.2=0.3, correct. The period: the distance between x=0.6 and x=2.2 is 1.6, which is the period, so the function repeats every 1.6 units, correct.

So the values are:

Amplitude (A) = 0.9

Denominator in the fraction (T) = 1.6 (or 8/5, but 1.6 is 8/5? Wait, 1.6 is 8/5? 8 divided by 5 is 1.6, yes. So ( \frac{2\pi}{1.6}=\frac{2\pi}{\frac{8}{5}}=\frac{5\pi}{4} ), but in the formula, it's ( \frac{2\pi}{T} ), so T is 1.6, so the denominator is 1.6.

Phase shift (C) = 0.6

Vertical shift (D) = 1.2

Answer:

The formula is ( y = 0.9\cos\left(\frac{2\pi}{1.6}(x - 0.6)\right)+1.2 ), so the values are:

The first box (amplitude): ( 0.9 )
The denominator under ( 2\pi ): ( 1.6 ) (or ( \frac{8}{5} ))
The phase shift (inside (x - [ ])): ( 0.6 )
The vertical shift (last box): ( 1.2 )

So filling in the boxes:

( y = \boldsymbol{0.9} \cos\left(\frac{2\pi}{\boldsymbol{1.6}}(x - \boldsymbol{0.6})\right) + \boldsymbol{1.2} )