Problem 15 - Precalculus: Functions and Graphs

Question

Find the period and sketch the graph of the equation. Show the asymptotes. $$y=2 \tan \left(2 x+\frac{\pi}{2}\right)$$

Step-by-Step Solution

Answer

The period is \(\frac{\pi}{2}\); asymptotes at \(x = \frac{k\pi}{2}\). Sketch shows tan waves repeating every \(\frac{\pi}{2}\).

Step 1: Identify the standard form

The equation is given as \(y = 2 \tan \left(2x + \frac{\pi}{2}\right)\). To find the period of the tangent function, let's first identify the transformation from the standard form, which is \(y = a \tan(bx + c)\). In this case, \(a = 2\), \(b = 2\), and \(c = \frac{\pi}{2}\).

Step 2: Calculate the period

The period of a tangent function \(y = a \tan(bx + c)\) is given by \(\frac{\pi}{|b|}\). Therefore, the period of \(y = 2 \tan(2x + \frac{\pi}{2})\) is \(\frac{\pi}{2}\).

Step 3: Find the phase shift

To find the phase shift, divide \(c\) by \(b\), given by \(-\frac{c}{b}\). For our equation, this is \(-\frac{\frac{\pi}{2}}{2} = -\frac{\pi}{4}\).

Step 4: Determine the asymptotes

The vertical asymptotes of a tangent function are at \(bx + c = \frac{\pi}{2} + k\pi\) for \(k \in \mathbb{Z}\). Solving \(2x + \frac{\pi}{2} = \frac{\pi}{2} + k\pi\), we get \(x = \frac{k\pi}{2}\). So asymptotes will be at \(x = \frac{k\pi}{2}\) for \(k\) being any integer.

Step 5: Sketch the graph

1. Draw the x-axis and y-axis. Locate the phase shift at \(-\frac{\pi}{4}\).2. Mark the asymptotes at \(x = \frac{k\pi}{2}\) by drawing dashed vertical lines.3. The period of the function is \(\frac{\pi}{2}\), showing that the function repeats every \(\frac{\pi}{2}\).4. Plot points where the function crosses the x-axis, midway between asymptotes.5. Sketch the tan wave between the asymptotes, repeating every period, scaling the y-values by the factor of 2.

Key Concepts

Period of Functions

The period of a function is important because it tells us how often the function repeats itself. In trigonometric functions, like sine, cosine, and tangent, the period defines the interval after which the graph of the function starts to repeat. For a standard tangent function, the period is \(\pi\)but if there is a transformation, such as a change in the coefficient before the variable, the period will alter accordingly.
To find the new period of a transformed tangent function, such as \(y = 2 \tan(2x + \frac{\pi}{2})\), you use the formula for the period of \(y = a \tan(bx + c)\). This formula is given by \(\frac{\pi}{|b|}\).

• Calculate the period by replacing \(b\) with the coefficient of \(x\), which is 2.
• The period becomes \(\frac{\pi}{2}\).

This means that the function repeats its pattern every \(\frac{\pi}{2}\) units along the x-axis. It's crucial to understand how to calculate the period as it guides how to sketch the function correctly on a graph.

Transformations of Functions

Transformations bring up the versatility of trigonometric functions. These include shifts, reflections, stretches and compressions that change how the function looks on a graph.

• Vertical stretch or compression: This is dictated by the coefficient \(a\). In our function \(y = 2 \tan(2x + \frac{\pi}{2})\), \(a = 2\), which indicates a vertical stretch by a factor of 2. It means the amplitude of the tangent wave is twice as high as usual, but note that tangent doesn't have a true amplitude like sine or cosine.
• Horizontal shift (Phase shift): This is given by \(-\frac{c}{b}\). With \(c = \frac{\pi}{2}\) and \(b = 2\), the shift is \(-\frac{\pi}{4}\). This means the entire graph is shifted left by \(\frac{\pi}{4}\) units.

Knowing these transformations helps identify where the function is located with respect to the usual position, and how the function’s shape and period differ from its parent function. This understanding is vital for accurately plotting the graph.

Trigonometric Asymptotes

Trigonometric asymptotes occur when the function approaches infinity; this commonly appears in functions like tangent, where certain x-values cause the function to be undefined.
For the function \(y = 2 \tan(2x + \frac{\pi}{2})\), the vertical asymptotes can be found by solving the equation:

• The asymptotes occur at points where the angle inside the tangent function equals odd multiples of \(\frac{\pi}{2}\). The general form to find these points is \(bx + c = \frac{\pi}{2} + k\pi,\)
• Plug in \(2x + \frac{\pi}{2} = \frac{\pi}{2} + k\pi\), solve it as \(x = \frac{k\pi}{2}\), with \(k\) being any integer, to get the locations of the asymptotes.

When sketching the graph, these asymptotes are drawn as vertical dotted lines on the graph. They highlight areas where the function shoots upwards and downwards rapidly and are invaluable for accurately portraying the behavior of a tangent function on a graph. Understanding the placement and nature of asymptotes ensures that the graph reflects the true behavior of the function.