Problem 20 - Algebra and Trigonometry

Question

Find the period and graph the function. $$ y=\tan \left(x-\frac{\pi}{4}\right) $$

Step-by-Step Solution

Answer

The period of the function is \( \pi \), shifted \( \frac{\pi}{4} \) to the right.

Step 1: Determine the Period of the Tangent Function

The period of the basic tangent function, \( y = \tan(x) \), is \( \pi \). For a tangent function of the form \( y = \tan(bx - c) \), the period is calculated as \( \frac{\pi}{|b|} \). In the function \( y = \tan(x - \frac{\pi}{4}) \), \( b = 1 \), so the period remains \( \pi \).

Step 2: Identify the Phase Shift

For the function \( y = \tan(x - \frac{\pi}{4}) \), the phase shift is controlled by the expression \( bx - c \). To find the phase shift, solve the equation \( bx - c = 0 \). Here, \( x - \frac{\pi}{4} = 0 \) which gives \( x = \frac{\pi}{4} \). Thus, the graph is shifted \( \frac{\pi}{4} \) units to the right.

Step 3: Graph the Function

Start by marking the phase shift by shifting each point on the graph of \( y = \tan(x) \) by \( \frac{\pi}{4} \) units to the right. Note that the asymptotes also move from \( x = -\frac{\pi}{2} \) and \( x = \frac{\pi}{2} \) to \( x = -\frac{\pi}{4} \) and \( x = \frac{3\pi}{4} \). Then plot key points, such as \( x = \frac{\pi}{4} \) where \( y = 0 \), and draw the characteristic S-shape of the tangent curve within one period, \( x = -\frac{\pi}{4} \) to \( x = \frac{3\pi}{4} \).

Key Concepts

Tangent Function

The tangent function, represented by \( y = \tan(x) \), is one of the fundamental trigonometric functions. It's unique among trig functions due to its distinct properties. The tangent function is defined as the ratio of the sine function to the cosine function, \[y = \frac{\sin(x)}{\cos(x)}\]

• This definition implies that the tangent is undefined wherever the cosine function is zero, leading to discontinuities or breaks in the graph.
• As a periodic function, \( y = \tan(x) \) repeats its values over specific intervals, influencing its visual pattern on the graph.

Initially, students often associate tangent graphs with a continuous wave-like form but realize it has vertical asymptotes due to points where \( \cos(x) = 0 \). Understanding these asymptotes is crucial for grasping the graph's behavior.

Period of Tangent Function

The period of a trigonometric function describes how often the function repeats its values. For the standard tangent function \( y = \tan(x) \), this interval is shorter compared to sine and cosine functions.

• The basic tangent function has a period of \( \pi \), meaning it repeats every \( \pi \) units along the x-axis.
• When transformed into the function form \( y = \tan(bx - c) \), the period is calculated using the formula \( \frac{\pi}{|b|} \).

For \( y = \tan(x - \frac{\pi}{4}) \), the value of \( b \) is 1, so the period remains unchanged at \( \pi \). This understanding helps in accurately plotting the repeated sections and boundaries of the tangent function on a graph.

Phase Shift in Trigonometry

Phase shift refers to the horizontal movement of a trigonometric graph along the x-axis. In the equation \( y = \tan(bx - c) \), the phase shift is captured by changes in the expression \( bx - c \).

• Solving \( bx - c = 0 \) determines how much the graph shifts.
• In this case, solving \( x - \frac{\pi}{4} = 0 \) yields \( x = \frac{\pi}{4} \), indicating a shift of \( \frac{\pi}{4} \) units to the right.

Understanding phase shifts is vital because it affects the starting position of the tangent function's repeating pattern, altering where key features like peaks and zero crossings appear.

Asymptotes in Trigonometric Graphs

Asymptotes are crucial elements of tangent graphs, indicating where the function is undefined and approaching infinity. In \( y = \tan(x) \), these vertical lines arise wherever the cosine of the angle is zero, typically at odd multiples of \( \pi/2 \).

• Originally, for \( y = \tan(x) \), asymptotes occur at \( x = -\pi/2, \pi/2, 3\pi/2 \), and so on.
• In a transformed function like \( y = \tan(x - \frac{\pi}{4}) \), asymptotes also shift according to the phase shift.

For this problem, asymptotes move to \( x = -\frac{\pi}{4} \) and \( x = \frac{3\pi}{4} \). Recognizing asymptotes is essential for sketching the graph since they define boundaries where the function appears to "shoot" towards positive or negative infinity.