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

Explanation:

Step1: Determine the amplitude

The amplitude $A$ of a cosine - function $y = A\cos(B(x - C))+D$ is half the difference between the maximum and minimum values of the function. The maximum value is $y_{max}=4.2$ and the minimum value is $y_{min}=0.4$. So, $A=\frac{y_{max}-y_{min}}{2}=\frac{4.2 - 0.4}{2}=\frac{3.8}{2}=1.9$.

Step2: Determine the phase - shift

The general form of a cosine function is $y = A\cos(B(x - C))+D$. For a cosine function $y = \cos x$, the maximum occurs at $x = 0$. In the given function, the maximum occurs at $x = 0.3$. So the phase - shift $C = 0.3$.

Step3: Determine the vertical shift

The vertical shift $D$ is the average of the maximum and minimum values of the function. So, $D=\frac{y_{max}+y_{min}}{2}=\frac{4.2 + 0.4}{2}=\frac{4.6}{2}=2.3$.

Step4: Determine the period and $B$

The period of the cosine function is not given explicitly, but since the standard form of the cosine function is $y = A\cos(B(x - C))+D$ and the coefficient of $(x - C)$ inside the cosine is $B$, and we assume the period is not changed from the standard period of $2\pi$ for $\cos x$, so $B = 1$. The function is $y=1.9\cos((x - 0.3))+2.3$.

Answer:

$y = 1.9\cos((x - 0.3))+2.3$