Graph f(x)=cos(x)-sin(x)

Problem

ƒ(x)=cos(x)−sin(x)

Solution

1. Identify the function as a linear combination of sine and cosine, which can be rewritten as a single shifted cosine or sine wave.

2. Apply the harmonic addition theorem to find the amplitude R and phase shift α using the form R*cos(x+α)

3. Calculate the amplitude R=√(,1+(−1)2)=√(,2)

4. Determine the phase shift by solving cos(α)=1/√(,2) and sin(α)=1/√(,2) which gives α=π/4

5. Rewrite the function as ƒ(x)=√(,2)*cos(x+π/4)

6. Identify key features for graphing: the amplitude is √(,2)≈1.414 the period is 2*π and the graph is shifted left by π/4

7. Find the intercepts by setting ƒ(x)=0 which occurs when cos(x)=sin(x) resulting in x=π/4+n*π

Final Answer

ƒ(x)=√(,2)*cos(x+π/4)

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