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

Problem

ƒ(x)=sin(x)+cos(x)

Solution

1. Rewrite the function using a trigonometric identity to combine the terms into a single sine function.

2. Multiply and divide the expression by √(,1+1)=√(,2) to prepare for the angle sum identity.

ƒ(x)=√(,2)*(1/√(,2)*sin(x)+1/√(,2)*cos(x))

3. Substitute cos(π/4) for 1/√(,2) and sin(π/4) for 1/√(,2)

ƒ(x)=√(,2)*(sin(x)*cos(π/4)+cos(x)*sin(π/4))

4. Apply the sine addition formula sin(A+B)=sin(A)*cos(B)+cos(A)*sin(B)

ƒ(x)=√(,2)*sin(x+π/4)

5. Identify the key features of the graph: the amplitude is √(,2)≈1.414 the period is 2*π and there is a horizontal phase shift of π/4 to the left.

6. Determine the critical points: the maximum occurs at x=π/4 the minimum occurs at x=(5*π)/4 and the x-intercepts occur at x=(3*π)/4 and x=(7*π)/4

Final Answer

ƒ(x)=√(,2)*sin(x+π/4)

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