graph the trigonometric function.\n$y = \\frac{1}{2}\\cos(x - \\frac{\\pi}{4})$ \nplot all points…

Explanation:

Step1: Identify Amplitude, Period, and Phase Shift

The function is $y = \frac{1}{2} \cos(x - \frac{\pi}{4})$. The amplitude is $A = |\frac{1}{2}| = \frac{1}{2}$. The period is $T = \frac{2\pi}{|1|} = 2\pi$. The phase shift is $C = \frac{\pi}{4}$ (shift to the right).

Step2: Determine the Interval for One Cycle

One cycle of the cosine function starts at its maximum. For $y = \cos(u)$, the cycle starts at $u=0$ and ends at $u=2\pi$. Here, $u = x - \frac{\pi}{4}$. Start of cycle: $x - \frac{\pi}{4} = 0 \implies x = \frac{\pi}{4}$. End of cycle: $x - \frac{\pi}{4} = 2\pi \implies x = 2\pi + \frac{\pi}{4} = \frac{9\pi}{4}$. The interval for one cycle is $[\frac{\pi}{4}, \frac{9\pi}{4}]$.

Step3: Find the Maximum Points

Maximum value is $A = \frac{1}{2}$. This occurs when $\cos(x - \frac{\pi}{4}) = 1$. $x - \frac{\pi}{4} = 0 \implies x = \frac{\pi}{4}$. Point: $(\frac{\pi}{4}, \frac{1}{2})$. $x - \frac{\pi}{4} = 2\pi \implies x = \frac{9\pi}{4}$. Point: $(\frac{9\pi}{4}, \frac{1}{2})$.

Step4: Find the Minimum Point

Minimum value is $-A = -\frac{1}{2}$. This occurs when $\cos(x - \frac{\pi}{4}) = -1$. $x - \frac{\pi}{4} = \pi \implies x = \pi + \frac{\pi}{4} = \frac{5\pi}{4}$. Point: $(\frac{5\pi}{4}, -\frac{1}{2})$.

Step5: Find the X-intercepts

X-intercepts occur when $y=0$, so $\cos(x - \frac{\pi}{4}) = 0$. $x - \frac{\pi}{4} = \frac{\pi}{2} \implies x = \frac{\pi}{2} + \frac{\pi}{4} = \frac{3\pi}{4}$. Point: $(\frac{3\pi}{4}, 0)$. $x - \frac{\pi}{4} = \frac{3\pi}{2} \implies x = \frac{3\pi}{2} + \frac{\pi}{4} = \frac{7\pi}{4}$. Point: $(\frac{7\pi}{4}, 0)$.

Step6: List Key Points for Graphing

The key points for one cycle starting from $x=\frac{\pi}{4}$ are: Maximum: $(\frac{\pi}{4}, \frac{1}{2})$ X-intercept: $(\frac{3\pi}{4}, 0)$ Minimum: $(\frac{5\pi}{4}, -\frac{1}{2})$ X-intercept: $(\frac{7\pi}{4}, 0)$ Maximum: $(\frac{9\pi}{4}, \frac{1}{2})$

Answer:

To graph the function $y = \frac{1}{2} \cos(x - \frac{\pi}{4})$, plot the following points corresponding to one cycle and connect them with a smooth curve: Maximum: $(\frac{\pi}{4}, \frac{1}{2})$ X-intercept: $(\frac{3\pi}{4}, 0)$ Minimum: $(\frac{5\pi}{4}, -\frac{1}{2})$ X-intercept: $(\frac{7\pi}{4}, 0)$ Maximum: $(\frac{9\pi}{4}, \frac{1}{2})$

The graph is a cosine curve with amplitude $\frac{1}{2}$, period $2\pi$, and phase shift $\frac{\pi}{4}$ to the right. (The final answer is the graph generated by plotting these points using the provided tool).