Question

27 - 28. describe the transformation of each function.

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

Explanation:

Step1: Analyze the general form of the cosine function

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

Step2: Identify the values of \(A\), \(B\), \(C\), and \(D\)

For the function \(y=-2\cos(3x+\frac{2\pi}{5}) + 5\), we have \(A=-2\), \(B = 3\), \(C=-\frac{2\pi}{5}\), and \(D = 5\).

Step3: Determine the vertical stretch

The value of \(|A|\) gives the vertical stretch. Here, \(|A|=| - 2|=2\), so there is a vertical stretch by a factor of \(2\).

Step4: Determine the reflection

Since \(A=-2<0\), the graph is reflected about the \(x\) - axis.

Step5: Determine the horizontal compression

The formula for the period of a cosine function is \(T=\frac{2\pi}{|B|}\). With \(B = 3\), the period is \(T=\frac{2\pi}{3}\). Compared to the parent function \(y=\cos(x)\) (with period \(2\pi\)), there is a horizontal compression by a factor of \(\frac{1}{3}\).

Step6: Determine the phase shift

The phase shift is given by \(\frac{C}{B}\). Here, \(\frac{C}{B}=\frac{-\frac{2\pi}{5}}{3}=-\frac{2\pi}{15}\), so the phase shift is \(\frac{2\pi}{15}\) units to the left.

Step7: Determine the vertical shift

The value of \(D\) gives the vertical shift. Since \(D = 5\), there is a vertical shift of \(5\) units up.

Answer:

The function \(y=-2\cos(3x+\frac{2\pi}{5})+5\) is a transformation of the parent function \(y = \cos(x)\) with a vertical stretch by a factor of \(2\), a reflection about the \(x\) - axis, a horizontal compression by a factor of \(\frac{1}{3}\), a phase shift of \(\frac{2\pi}{15}\) units to the left, and a vertical shift of \(5\) units up.