Graph y=tan(x-pi/4)

Problem

y=tan(x−π/4)

Solution

1. Identify the parent function and its properties. The parent function is y=tan(x) which has a period of π and vertical asymptotes at x=π/2+n*π for any integer n

2. Determine the phase shift by looking at the argument of the tangent function. The expression x−π/4 indicates a horizontal shift to the right by π/4 units.

3. Calculate the new vertical asymptotes by setting the argument equal to the original asymptote locations.

(x−π/4)=−π/2⇒x=−π/4

(x−π/4)=π/2⇒x=(3*π)/4

4. Find the x-intercept by setting the argument equal to zero.

(x−π/4)=0⇒x=π/4

5. Plot key points within one period. Since the tangent function has a coefficient of 1 the points halfway between the intercept and the asymptotes will be at y=−1 and y=1

x=0⇒y=tan(−π/4)=−1

x=π/2⇒y=tan(π/4)=1

6. Sketch the curve passing through (0,−1) (π/4,0) and (π/2,1) approaching the vertical asymptotes at x=−π/4 and x=(3*π)/4

Final Answer

y=tan(x−π/4)

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