Graph f(x)=tan(x)-3

Problem

ƒ(x)=tan(x)−3

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 transformation applied to the parent function. The expression tan(x)−3 indicates a vertical shift downward by 3 units.

3. Locate the key points for one period. In the parent function y=tan(x) the center point is (0,0) After the shift, the new center point is (0,−3)

4. Identify the asymptotes, which remain unchanged by a vertical shift. The vertical asymptotes for this function are located at x=−π/2 and x=π/2

5. Plot additional points to define the shape. For x=π/4 tan(π/4)−3=1−3=−2 For x=−π/4 tan(−π/4)−3=−1−3=−4

6. Sketch the curve by drawing the characteristic tangent shape through the points (−π/4,−4) (0,−3) and (π/4,−2) approaching the vertical asymptotes.

Final Answer

ƒ(x)=tan(x)−3

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