Question

fill in the blanks to correctly complete the sentence below. the graph of y = 6 + 3 cos(x + \frac{2\pi}{3}) is obtained by shifting the graph of y = cos x unit(s) horizontally to the stretching it vertically by a factor of , and then shifting it unit(s) vertically

Explanation:

Step1: Analyze horizontal shift

For the function $y = A\cos(B(x - C))+D$, the horizontal shift is given by $C$. In $y = 3\cos(x+\frac{2\pi}{3})+6$, we have $C=-\frac{2\pi}{3}$. A negative $C$ value means a shift to the left. So the graph of $y = \cos x$ is shifted $\frac{2\pi}{3}$ units horizontally to the left.

Step2: Analyze vertical stretch

The coefficient of the cosine function determines the vertical stretch. For $y = 3\cos(x+\frac{2\pi}{3})+6$, the coefficient of $\cos$ is $3$. So it is stretched vertically by a factor of $3$.

Step3: Analyze vertical shift

The constant added outside the cosine function gives the vertical shift. For $y = 3\cos(x+\frac{2\pi}{3})+6$, the constant is $6$. So it is shifted $6$ units vertically up.

Answer:

$\frac{2\pi}{3}$, left, $3$, $6$ up