Graph each function over a two-period interval.
y = tan(2x - π)

Verified step by step guidance

1. Identify the basic form of the tangent function: $y=\tan (kx+c)$. Here, $k=2$ and $c=-\pi$.
2. Determine the period of the function. The period of $\tan (kx)$ is $\frac{\pi }{k}$. For $y=\tan (2x)$, the period is $\frac{\pi }{2}$.
3. Calculate the phase shift using $c$. The phase shift is given by $-\frac{c}{k}$. For $y=\tan (2x-\pi )$, the phase shift is $\frac{\pi }{2}$.
4. Determine the vertical asymptotes. For $y=\tan (2x-\pi )$, the vertical asymptotes occur at $2x-\pi =\frac{\pi }{2}+n\pi$, where $n$ is an integer.
5. Graph the function over a two-period interval, taking into account the period, phase shift, and vertical asymptotes. The interval will be from $x=\frac{\pi }{4}$ to $x=\frac{5\pi }{4}$.

Key Concepts

Periodicity of Trigonometric Functions

Trigonometric functions, such as tangent, are periodic, meaning they repeat their values in regular intervals. The period of the tangent function is typically π. However, when the function is transformed, such as by multiplying the variable by a coefficient, the period changes. For the function y = tan(2x - π), the period is π/2, which is derived from dividing the standard period by the coefficient of x.

Phase Shift

Phase shift refers to the horizontal shift of a trigonometric function along the x-axis. In the function y = tan(2x - π), the term -π indicates a phase shift. To find the phase shift, we set the inside of the function equal to zero, which gives us the shift value. This shift affects where the function starts on the graph, moving it to the right by π/2.

Graphing Tangent Functions

Graphing tangent functions involves understanding their asymptotes and behavior. The tangent function has vertical asymptotes where the function is undefined, occurring at odd multiples of π/2. For y = tan(2x - π), the asymptotes will be located at x = π/4 + n(π/2), where n is an integer. This knowledge is crucial for accurately sketching the graph over the specified interval.