sketch a graph of the function $f(x)=cos(x + \frac{pi}{4})$.

Explanation:

Step1: Identify the basic cosine - function properties

The basic cosine function is $y = \cos(x)$ with amplitude $A = 1$, period $T=2\pi$, and phase - shift for $y=\cos(x + c)$ is $c$ units to the left. For $y=\cos(x+\frac{\pi}{4})$, the amplitude $A = 1$, period $T = 2\pi$, and phase - shift is $\frac{\pi}{4}$ units to the left.

Step2: Find key points of the basic cosine function

The key points of $y = \cos(x)$ are: maximum points at $(2k\pi,1)$ for $k\in\mathbb{Z}$, minimum points at $((2k + 1)\pi,-1)$ for $k\in\mathbb{Z}$, and x - intercepts at $(2k\pi+\frac{\pi}{2},0)$ and $(2k\pi+\frac{3\pi}{2},0)$ for $k\in\mathbb{Z}$.

Step3: Adjust key points for the shifted function

For $y=\cos(x+\frac{\pi}{4})$, we substitute $x'=x+\frac{\pi}{4}$.

• Maximum points: $x+\frac{\pi}{4}=2k\pi\Rightarrow x = 2k\pi-\frac{\pi}{4}$, so maximum points are $(2k\pi-\frac{\pi}{4},1)$.
• Minimum points: $x+\frac{\pi}{4}=(2k + 1)\pi\Rightarrow x=(2k + 1)\pi-\frac{\pi}{4}=2k\pi+\frac{3\pi}{4}$, so minimum points are $(2k\pi+\frac{3\pi}{4},-1)$.
• X - intercepts: $x+\frac{\pi}{4}=2k\pi+\frac{\pi}{2}\Rightarrow x = 2k\pi+\frac{\pi}{4}$ and $x+\frac{\pi}{4}=2k\pi+\frac{3\pi}{2}\Rightarrow x=2k\pi+\frac{5\pi}{4}$. So x - intercepts are $(2k\pi+\frac{\pi}{4},0)$ and $(2k\pi+\frac{5\pi}{4},0)$.

Step4: Sketch the graph

Plot the key points for one period (e.g., when $k = 0$: maximum at $(-\frac{\pi}{4},1)$, minimum at $(\frac{3\pi}{4},-1)$, x - intercepts at $(\frac{\pi}{4},0)$ and $(\frac{5\pi}{4},0)$) and then repeat the pattern for other periods. The graph of $y = \cos(x)$ is shifted $\frac{\pi}{4}$ units to the left.

Answer:

Sketch the graph by plotting the key - points (maximums, minimums, and x - intercepts) obtained above and connecting them with a smooth curve, repeating the pattern every $2\pi$ units. The graph of $y=\cos(x)$ is shifted $\frac{\pi}{4}$ units to the left.