Problem 2 - Algebra 2

Question

State the amplitude, period, and phase shift for each function. Then graph the function. $$ y=\tan \left(\theta+60^{\circ}\right) $$

Step-by-Step Solution

Answer

Amplitude is not defined, period is \(180^\circ\), phase shift is \(-60^\circ\).

Step 1: Identify Components of Tangent Function

The general form of a tangent function is given by \( y = a \tan(bx + c) + d \). Here, the given function is \( y = \tan(\theta + 60^\circ) \). Identifying the components, we see: \( a = 1 \), \( b = 1 \), \( c = 60^\circ \), and \( d = 0 \).

Step 2: Determine Phase Shift

The phase shift of the tangent function can be calculated using the formula \( \text{Phase Shift} = -\frac{c}{b} \). Substituting the values we have \( \text{Phase Shift} = -\frac{60^\circ}{1} = -60^\circ \).

Step 3: Find the Period of Tangent Function

For tangent functions, the period is determined by \( \text{Period} = \frac{\pi}{b} \). Since \( b = 1 \), the period is \( \frac{\pi}{1} = \pi \). This means the tangent function repeats every \( 180^\circ \).

Step 4: Graphing the Tangent Function

The graph of \( y = \tan(\theta + 60^\circ) \) involves shifting the standard tangent graph horizontally by \(-60^\circ\). Start by sketching the basic tangent function between its asymptotes at \(-90^\circ\) and \( 90^\circ\) and then shift each point to the left by \(60^\circ\). This function still has vertical asymptotes every \(180^\circ\) due to its period.

Key Concepts

Tangent Function

The tangent function is one of the fundamental trigonometric functions. It is often written as \( an(x) \). Unlike the sine and cosine functions, the tangent function is defined as the ratio of the sine to the cosine function:

• \( an(x) = \frac{ ext{sin}(x)}{ ext{cos}(x)} \)

The tangent function is unique for a few reasons:

• Its range is all real numbers, unlike sine and cosine which range between -1 and 1.
• It features vertical asymptotes where the cosine function is zero, typically occurring at odd multiples of \(90^{\circ}\) (or \(\frac{\pi}{2}\)) for the basic tangent function.

The graph of the tangent function displays a repeating pattern known as a period, where the function repeats its values. These characteristics make the tangent function an important tool in both mathematics and science.

Amplitude

The concept of amplitude is commonly used when discussing the sine and cosine functions. It indicates the maximum extent of a vibration or oscillation, measured from the position of equilibrium. However, unlike sine and cosine, the tangent function does not have an amplitude. This is because its range extends to infinity. The tangent function is unbounded, meaning it can take on any real number as a value, so describing it in terms of amplitude isn't applicable.

Period and Phase Shift

The period of a function refers to the length required for the function to complete one full cycle. For the tangent function, the standard period is \( \pi \) radians or \( 180^{\circ} \). This period is influenced by the coefficient \( b \) in the general form \( y = a \tan(bx + c) + d \) of the tangent function. The formula to find the period is:

• \( ext{Period} = \frac{\pi}{b} \)

In the given function \( y = \tan(\theta + 60^{\circ}) \), since \( b = 1 \), the period remains \( \pi \).

Phase shift refers to the horizontal translation of the function along the x-axis, calculated as:

• \( ext{Phase Shift} = -\frac{c}{b} \)

For our function, \( c = 60^{\circ} \), leading to a phase shift of \(-60^{\circ}\). This means the entire graph of \( \tan(\theta) \) is shifted leftward by \(60^{\circ}\), affecting where each cycle of the function begins and ends.

Graphing Trigonometric Functions

Graphing functions like \( y = \tan(\theta + 60^{\circ}) \) involves a few steps. First, identify the graph's basic pattern. The basic tangent function features curves that approach vertical asymptotes at odd multiples of \( \frac{\pi}{2} \) and repeat every \( \pi \) radians.

For the function given, with a phase shift of \(-60^{\circ}\):

• Start by locating the new position of one cycle's midpoint, located at \(-60^{\circ}\).
• Draw a vertical line for the asymptote at \(-90^{\circ} + (-60^{\circ})\).
• Continue to identify and include additional vertical asymptotes at every \(180^{\circ}\) interval.

Shifting also means the "rise" and "fall" points of the function shift by \(60^{\circ}\), providing guidance on how to draw the graph.

• Ensure that all features of the basic graph, such as the repeating pattern and halfway points, reflect the phase shift.

This approach makes it easier to accurately graph any transformed trigonometric function.