Please graph the following function: y = tan (2(theta) + (pi)) + 1
To graph the function y = tan(2θ + π) + 1, we need to follow a few steps:
Step 1: Determine the period of the function.
The period of the tangent function is π, which means it repeats itself every π units.
Step 2: Determine the vertical asymptotes.
The vertical asymptotes occur when tan(2θ + π) is undefined, which happens when 2θ + π = (n + 1/2)π, where n is an integer. Solving for θ, we have θ = (n + 1/4)π. These are the vertical asymptotes.
Step 3: Determine the x-intercepts.
To find the x-intercepts, we need to solve the equation tan(2θ + π) + 1 = 0. By subtracting 1 from both sides, we get tan(2θ + π) = -1. The x-intercepts occur when the tangent of the angle is -1, which happens at θ = (n + 1/8)π.
Step 4: Sketch the graph.
Using the information from steps 1-3, we can now sketch the graph of the function y = tan(2θ + π) + 1.
Note: The tan function has a vertical asymptote at θ = (n + 1/4)π, where n is an integer. It also has x-intercepts at θ = (n + 1/8)π, where n is an integer.
Please keep in mind that the graph of the tangent function repeats itself every period π, so when sketching the graph, you only need to show one period.