Math / Trigonometry

QuestionSketch a graph of the function $f(x)=\cos \left(x+\frac{5 \pi}{4}\right)$ .
Clear All Draw: $\sqrt[M]{ } \sqrt{ }$

Studdy Solution

STEP 1

1. The function $f(x) = \cos \left(x + \frac{5\pi}{4}\right)$ is a cosine function with a horizontal phase shift.
2. The graph of $\cos(x)$ is known and can be used as a reference for transformations.

STEP 2

1. Identify the basic properties of the cosine function.
2. Determine the phase shift of the function.
3. Sketch the graph of the transformed function.

STEP 3

The basic cosine function $\cos(x)$ has a period of $2\pi$ , an amplitude of 1, and oscillates between -1 and 1.

STEP 4

The function $f(x) = \cos \left(x + \frac{5\pi}{4}\right)$ involves a horizontal phase shift.
The phase shift is determined by the expression inside the cosine function: $x + \frac{5\pi}{4}$ .
The phase shift is $-\frac{5\pi}{4}$ , which means the graph of $\cos(x)$ is shifted to the left by $\frac{5\pi}{4}$ .

STEP 5

To sketch the graph of $f(x) = \cos \left(x + \frac{5\pi}{4}\right)$ :
1. Start with the basic cosine graph $\cos(x)$ .
2. Shift the entire graph to the left by $\frac{5\pi}{4}$ .
3. The new graph will still have the same amplitude and period as the original cosine graph.
The graph of the function $f(x) = \cos \left(x + \frac{5\pi}{4}\right)$ is a cosine wave shifted to the left by $\frac{5\pi}{4}$ .

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