Problem 72 - A Graphical Approach to Precalculus with Limits

Question

Graph each function over a two-period interval. $$y=3+\frac{1}{2} \tan x$$

Step-by-Step Solution

Answer

Graph \( y = 3 + \frac{1}{2} \tan x \) over \([0, 2\pi]\), with asymptotes at \( x = \frac{\pi}{2}, \frac{3\pi}{2} \), vertically compressed and shifted up by 3.

Step 1 - Understand the Parent Function

The function given is a transformation of the basic tangent function, which is \( y = \tan x \). The period of the tangent function is \( \pi \) radians. This means that the function repeats its pattern every \( \pi \) radians.

Step 2 - Identify the Modifications

The function \( y = 3 + \frac{1}{2} \tan x \) has two modifications to the basic tangent function. The term \( \frac{1}{2} \tan x \) indicates a vertical compression by a factor of \( \frac{1}{2} \), whereas the \( +3 \) indicates a vertical shift upwards by 3 units.

Step 3 - Determine the Two-Period Interval

Since the period of \( \tan x \) is \( \pi \), a two-period interval for this function would be \([0, 2\pi]\). This is because \( 2 \times \pi = 2\pi \).

Step 4 - Plot the Asymptotes and Key Points

In the interval \([0, 2\pi]\), the tangent function has vertical asymptotes at multiples of \( \pi/2 \), specifically at \( x = \frac{\pi}{2}, \frac{3\pi}{2} \). The points where the function intersects the line \( y=3 \) can be found using the zeros of the tangent function which are at \( x = 0, \pi \).

Step 5 - Sketch the Graph

Starting from the origin, sketch the typical shape of the tangent graph within \( [0, \pi] \) and then extend by repeating the behavior in \( [\pi, 2\pi] \). Apply the vertical transformation by compressing vertically by \( \frac{1}{2} \), and then shift the entire graph up by 3 units.

Key Concepts

Tangent Function

The tangent function, denoted as \( y = \tan x \), is one of the fundamental trigonometric functions. It is defined as the ratio of the sine and cosine functions, \( \tan x = \frac{\sin x}{\cos x} \). This means that \( \tan x \) is undefined wherever \( \cos x = 0 \), resulting in vertical asymptotes. These occur at odd multiples of \( \frac{\pi}{2} \).

• The tangent function is periodic, with a period of \( \pi \) radians. This means the function's pattern repeats every \( \pi \) radians.
• Unlike sine and cosine, which oscillate between -1 and 1, the tangent function has no maximum or minimum values. Instead, it increases and decreases without bound within each period.

The graph of \( y = \tan x \) resembles a series of curves that alternate between sloping upwards and downwards within each interval defined by its vertical asymptotes.

Graphing Transformations

Graphing transformations are changes applied to a graph's appearance using various operations like stretching, compressing, and shifting. For \( y = \frac{1}{2} \tan x \), we are dealing with a vertical compression.

• Vertical Compression — occurs when you multiply the tangent function by a coefficient less than 1, like \( \frac{1}{2} \) here. This reduces the steepness of the graph but retains the same basic shape and direction.
• Vertical stretching, in contrast, involves multiplication by a factor greater than 1, which makes the graph steeper.

When you apply a vertical compression, the slope of each repeated tangent curve becomes less extreme. Despite these transformations, the position of asymptotes and zeros remain unchanged, as they depend on the tangent function's original period and locations.

Vertical Shifts

Vertical shifts are another type of graph transformation that involves moving the graph up or down along the y-axis. This transformation does not affect the x-values of the function, such as the period or the locations of asymptotes and zeros. In our example, we have the function \( y = 3 + \frac{1}{2} \tan x \).

• Here, the graph is shifted vertically upward by 3 units due to the addition of the constant term \( +3 \).
• This adjustment affects the entire function equally, moving all points, including any intercepts, higher by 3 units.

Vertical shifts are a straightforward way to modify the position of a graph without altering its shape. It means that the central line or axis of the tangent function, usually at \( y = 0 \), is now positioned at \( y = 3 \). This shift aligns all transformations with the new baseline, ensuring the graph reflects all imposed changes correctly throughout the specified interval.