Problem 18 - Precalculus: Functions and Graphs

Question

Find the period and sketch the graph of the equation. Show the asymptotes. $$y=-3 \tan \left(\frac{1}{3} x-\frac{\pi}{3}\right)$$

Step-by-Step Solution

Answer

The period is \(3\pi\) with asymptotes at \(x = (3n+3)\pi/2 + \pi\).

Step 1: Identify the Function Type

The given function is a tangent function, specifically in the form \( y = a \tan(bx - c) \). The tangent function is defined and has undefined values (asymptotes) at certain points.

Step 2: Determine the Period of the Function

The period of the tangent function \( \tan(bx) \) is given by \( \frac{\pi}{|b|} \). For the function \( y = -3 \tan\left(\frac{1}{3}x - \frac{\pi}{3}\right) \), \( b = \frac{1}{3} \). Therefore, the period is \( \frac{\pi}{\frac{1}{3}} = 3\pi \).

Step 3: Identify the Asymptotes

The asymptotes occur at the points where the argument of the tangent is an odd multiple of \( \frac{\pi}{2} \), i.e., \( \frac{1}{3}x - \frac{\pi}{3} = (2n+1)\frac{\pi}{2} \), where \( n \) is an integer. Solve for \( x \):\( \frac{1}{3}x = (2n+1)\frac{\pi}{2} + \frac{\pi}{3} \)\( x = 3\left((2n+1)\frac{\pi}{2} + \frac{\pi}{3}\right) \)Simplifying gives asymptotes at:\( x = (3n+3)\pi/2 + \pi \) for each integer \( n \).

Step 4: Sketch the Graph

1. Plot a period on the graph, using the determined period \( 3\pi \).2. Mark the vertical asymptotes, which repeat every \( 3\pi \).3. The basic tangent shape would run between these asymptotes.4. Factor in that if \( a = -3 \), the graph is vertically stretched along the y-axis by a factor of 3 and reflected over the x-axis.5. Set \( bx-c=0 \) to find the phase shift, resolves to \( x = \pi \).The graph will repeat this pattern for each period centered at \( x = (3k+2)\pi/2\) for an integer \( k \).

Step 5: Analyze Properties and Behavior

The function exhibits vertical asymptotes at calculated positions, the amplitude does not influence tangent (yet affects stretch factor), and the negative sign indicates a vertically flipped graph. Use these properties to ensure the periodic behavior repeats every \( 3\pi \).

Key Concepts

Periodicity

Periodicity in the context of trigonometric functions refers to the repeating nature of these functions over a specific interval. For the tangent function, periodicity is a fundamental feature. The basic form of the tangent function, written as \( \tan(bx) \), typically has a periodic interval of \( \pi \). However, when you introduce a coefficient \( b \) inside the tangent function, the period becomes adjusted by this factor.

• For our specific function \(-3 \tan\left(\frac{1}{3} x - \frac{\pi}{3}\right)\), the coefficient \( b = \frac{1}{3} \).
• The new period can be calculated by the formula \( \frac{\pi}{|b|} \), which results in \( 3\pi \).

Understanding the period helps determine how often the function pattern repeats. This characteristic is crucial when sketching the graph because it informs us about the span over the x-axis within which one complete cycle of the tangent function occurs. Highlighting periodicity, the tangent graph starts a new cycle every \( 3\pi \) units, ensuring that the pattern is uniformly maintained throughout the x-axis.

Asymptotes

Asymptotes are key features of the tangent function where the function becomes undefined. These are vertical lines that the graph approaches but never crosses or touches. Recognizing the points at which these occur is necessary when dealing with tangent functions, as they signal breaks in the graph where the function heads towards infinity.To find the asymptotes for the function \(-3 \tan\left(\frac{1}{3} x - \frac{\pi}{3}\right)\):

• The general form for tangent asymptotes occurs where the argument \( \frac{1}{3}x - \frac{\pi}{3} = (2n+1)\frac{\pi}{2} \), with \( n \) being any integer.
• Solving this for \( x \) gives the asymptotes as \( x = (3n+3)\frac{\pi}{2} + \pi \).

These asymptotes materialize at regular intervals styled by the adjusted period, alternately breaking the graph. While sketching, the asymptotes indicate the vertical lines between which each section of the tangent's cycle sits. As a repeated feature, they help in setting boundaries for parts of the graph, showing where the tangent dips towards positive or negative infinity.

Phase Shift

Phase shift is the horizontal displacement from the normal position where a function starts its cycle. For trigonometric functions like tangent, identifying the phase shift is essential for accurately positioning the graph along the x-axis.To determine the phase shift of the function \(-3 \tan\left(\frac{1}{3} x - \frac{\pi}{3}\right)\):

• The formula \( bx - c = 0 \) aids in finding when the function starts.
• Solving \( \frac{1}{3}x - \frac{\pi}{3} = 0 \) implies \( x = \pi \).

Thus, a phase shift of \( \pi \) means the tangent graph shifts horizontally by \( \pi \) units to the right compared to where it typically starts. In graphing this function, you begin plotting after accounting for this shift, ensuring that the main features of the graph like asymptotes and cycles align properly. This modification is crucial as it directly affects where each cycle of the tangent graph starts on the x-axis, guaranteeing the function's behavior is correctly represented.