Amplitude: No amplitude Period: π Phase Shift: h = π/4 Graph:
Practice makes perfect
We want to state the amplitude, period, and phase shift for the given function. Then, we will use this information to draw the graph of the function. Let's do it!
Amplitude and Period
Let's start by recalling the general form of a tangent function.
y= a tan b(θ - h)+ k
In this form, the period is π| b|. Since the graph of a tangent function does not have maximum or minimum points, it has no amplitude. Now, let's consider the given function.
y=tan (θ - π/4)
⇕
y= 1tan 1(θ - π/4)+ 0
We can see that b= 1. With this information, we will find the period of the function.
Period: π/| b| ⇒ π/| 1| = π
Therefore, the period is π.
Phase Shift
Once again, let's compare the general form of a tangent function with the given function.
General Form
y= a tan b(θ - h)+ k [0.8em]
Given Function
y= 1tan 1(θ - π/4)+ 0
The phase shift is shown by the value of h. Since our function has h= π4, we know that the phase shift is π4.
Graph
We will start by drawing the graph of the parent function y=tan θ. The first asymptote for positive values of θ is located at θ= π2. Since the period of this function is π, to find other asymptotes, we add and subtract multiples of π.
Finally, we will shift the graph π4 units to the right. This will result in the graph of our function, y = tan (θ - π4).
To clearly see the desired graph, let's remove the graph of y=tan θ.
