Problem 42 - Algebra and Trigonometry

Question

7–52 Find the period and graph the function. $$y=\frac{1}{2} \tan (\pi x-\pi)$$

Step-by-Step Solution

Answer

The period of the function is 1.

Step 1: Identify the Base Function of Tangent

The given function is based on the standard tangent function, which is typically of the form \( y = a \tan(bx + c) \). The standard period of the tangent function \( \tan(x) \) is \( \pi \).

Step 2: Identify the Transformation Parameters

The function \( y = \frac{1}{2} \tan(\pi x - \pi) \) includes a scaling factor \( \frac{1}{2} \) and is altered by \( \pi x - \pi \), suggesting a transformation.

Step 3: Determine the Period of the Function

The tangent function's period is derived from \( \pi = \frac{\pi}{b} \), where \( b \) is the coefficient of \( x \). Here, since \( b = \pi \), the period is \( \pi / \pi = 1 \).

Step 4: Identify Vertical and Horizontal Shifts

The \( -\pi \) in \( \pi x - \pi \) suggests a phase shift, but it simplifies to \( \tan(\pi(x-1)) \), equivalent to a horizontal shift of 1 unit to the right. Since there is no vertical shift in this function, the graph retains its central point along the x-axis.

Step 5: Graph the Function

To graph this function, note that the function \( y = \frac{1}{2} \tan(\pi(x-1)) \) compresses vertically by a factor of \( 0.5 \) from the standard \( \tan \) graph, shifts right by 1 unit, and has a new period of 1. Draw the graph between each asymptote occurring at multiples of 1.

Key Concepts

Tangent Function

The tangent function is one of the basic trigonometric functions and is represented as \( \tan(x) \). It is crucial in understanding triangles and circles, especially in right-angle triangle trigonometry. In its simplest form, the tangent of an angle \( \theta \) is the ratio of the opposite side to the adjacent side in a right-angled triangle.

• Properties: The tangent function is periodic, with vertical asymptotes where the function is undefined since the tangent approaches infinity at odd multiples of \( \frac{\pi}{2} \).
• Range: The tangent function has a range from negative infinity to positive infinity (\(-\infty, \infty\)).

When graphing the tangent function, it's useful to plot several key points and note where these vertical asymptotes lie. In its basic form, the graph rises and falls between these asymptotes.

Function Period

The period of a function is the distance over which a function repeats its values. The tangent function, \( \tan(x) \), unlike sine and cosine, has a standard period of \( \pi \). This means that after a shift of \( \pi \), the graph of the tangent function looks exactly the same as it did at the beginning.
The period of a transformed tangent function \( \tan(bx) \) is determined by the formula \( \frac{\pi}{|b|} \). In our example, \( b = \pi \), so the period is \( \frac{\pi}{\pi} = 1 \). This indicates that every 1 unit along the x-axis, the function's behavior repeats itself, effectively compressing the graph horizontally compared to the standard tangent function.
Understanding the concept of period allows us to predict and graph the behavior of trigonometric functions over different intervals.

Phase Shift

Phase shift indicates how a function is horizontally shifted from its standard position. It is derived from the function form \( \tan(bx + c) \). To calculate phase shift, solve the equation \( bx + c = 0 \), where the function's mathematical operation transforms \( c \) into a visible shift on the x-axis.
For \( y = \frac{1}{2} \tan(\pi x - \pi) \), rewrite it as \( \tan(\pi(x-1)) \), showing a phase shift of 1 unit to the right. This transformation moves each point on the tangent graph rightward by 1 unit, altering its position but not its overall shape or periodic behavior.

• Recognizing phase shifts helps in accurately sketching the function's graph, ensuring all peaks, troughs, and asymptotes are correctly placed.

Graphing Transformations

Graphing transformations refer to the changes to the graph of a function due to alterations in its equation. For the tangent function, transformation affects its amplitude, period, and phase.
The modified function \( y = \frac{1}{2} \tan(\pi x - \pi) \) suggests a few transformations:

• Vertical Scaling: The factor \( \frac{1}{2} \) compresses the graph vertically; this means that the tangent values become half of what they usually are.
• Horizontal Translation: The substitution \( \pi x - \pi \) indicates a 1-unit shift to the right.
• Period Adjustment: With the period reduced to 1, more oscillations occur within a given x-interval compared to the standard tangent graph.

These transformations require careful attention to detail when sketching the function. Identifying and applying each separately will help in drawing the function accurately, reflecting all behavior changes due to these mathematical alterations.