Problem 67 - Precalculus: Functions and Graphs

Question

Graph the function \(f\) in the viewing rectangle \([-2 \pi, 2 \pi, \pi / 2]\) by \([-4,4] .\) Use the graph of \(f\) to predict the graph of \(g .\) Verify your prediction by graphing \(g\) in the same vlewing rectangle. $$f(x)=\tan 0.5 x ; \quad g(x)=\tan \left[0.5\left(x+\frac{\pi}{2}\right)\right]$$

Step-by-Step Solution

Answer

Graph of \( g(x) \) is a left shift of \( f(x) \) by \( \frac{\pi}{2} \).

Step 1: Understand the Function

First, we examine the function \( f(x) = \tan(0.5x) \). The \( \tan(x) \) function typically has a period of \( \pi \), but for \( \tan(0.5x) \), the period is stretched to \( 2\pi \). So the period of \( f(x) \) becomes \( 4\pi \). This means we will expect to see one full cycle within the range \([-2\pi, 2\pi]\). Assess the behavior: it has vertical asymptotes at every \( x = (4n+1)\pi \) where \( n \) is an integer.

Step 2: Sketch the Graph of f(x)

Plot \( f(x) = \tan(0.5x) \) within the viewing rectangle \([-2\pi, 2\pi]\) by \([-4, 4]\). Identify key points like where \( f(x) = 0 \) at \( x = -4\pi, -2\pi, 0, 2\pi, 4\pi \). Draw the curve between these points, and note the vertical asymptotes within the viewing rectangle at \( x = -3\pi, -\pi, \pi, 3\pi \).

Step 3: Predict the Graph of g(x)

Assess the transformation for \( g(x) = \tan \left[0.5 \left(x + \frac{\pi}{2}\right) \right] \). Compare this to \( f(x) \): \( g(x) \) is a horizontal translation of \( f(x) \) by \( -\frac{\pi}{2} \) units. Therefore, the graph of \( g(x) \) will look like the graph of \( f(x) \) shifted to the left by \( \frac{\pi}{2} \) units.

Step 4: Verify by Graphing g(x)

Graph \( g(x) = \tan \left[0.5 \left(x + \frac{\pi}{2}\right) \right] \) within the same viewing rectangle. The graph should be the graph of \( f(x) = \tan(0.5x) \) shifted left by \( \frac{\pi}{2} \). Check by plotting the key points and asymptotes positions shifted accordingly.

Key Concepts

Tangent Function

The tangent function, denoted as \(\tan(x)\), is a fundamental trigonometric function with several unique properties. It represents the ratio of the sine and cosine functions as \(\tan(x) = \frac{\sin(x)}{\cos(x)}\). Unlike the sine and cosine functions, which oscillate between -1 and 1, the tangent function can take on any real number value.
Since the cosine function is in the denominator, \(\tan(x)\) has vertical asymptotes where \(\cos(x) = 0\). These occur at odd multiples of \(\pi/2\), resulting in the tangent function having no maximum or minimum points, only vertical asymptotes in its range. This gives the tangent graph a series of repeating vertical gaps.

• The basic period of \(\tan(x)\) is \(\pi\), meaning it repeats every \(\pi\) units.
• At the beginning of each period, \(\tan(x)\) crosses through the origin \((0,0)\), where it returns to zero every \(\pi\) units.

Understanding these characteristics is crucial when graphing any tangent function, especially when transformations are applied.

Period of Trigonometric Functions

The concept of the period in trigonometric functions refers to the interval over which the function completes one full cycle. For the tangent function, the standard period is \(\pi\), meaning every \(\pi\) units along the x-axis, the graph repeats its pattern.
When the function is modified, such as \(\tan(0.5x)\) from our example, the period changes accordingly. With this particular function, we see the period stretching. This happens because of the coefficient 0.5 multiplied by \(x\), affecting the frequency of oscillations.
The modified period \(a\) can be calculated using the formula:
\[\text{New period} = \frac{\text{Original period}}{|\text{Multiplier of }x|}\]

• For \(\tan(0.5x)\): \(\frac{\pi}{0.5} = 2\pi\), doubling the standard period.

Hence, this stretched period results in fewer cycles visible within a given interval, such as from \([-2\pi, 2\pi]\). This manipulation is an essential tool to predict a graph's behavior over a specific domain.

Horizontal Translation of Graphs

A horizontal translation in graphs means shifting the graph left or right along the x-axis. This form of transformation alters the graph's position without changing its shape. Mathematically, this can be expressed through functions like \(g(x) = f(x + c)\), where \(c\) is a constant.
In the context of the exercise with \(f(x) = \tan(0.5x)\) and \(g(x) = \tan\left[0.5(x+\frac{\pi}{2})\right]\), the latter is a horizontally translated version of the former. The value \(c = \frac{\pi}{2}\) indicates a leftward shift.

• This means the graph of \(g(x)\) takes each point of \(f(x)\) and moves it \(\frac{\pi}{2}\) units to the left.
• All asymptotes and intercepts of \(f(x)\) also shift accordingly.

Horizontal translations do not affect the period; they only reposition the graph. Recognizing these shifts allows us to accurately predict and graph transformed functions by merely applying this displacement to the original function.