Problem 86 - Precalculus

Question

For the following exercises, find and graph two periods of the periodic function with the given strething factor, \(|A|\) period, and phase shift. A tangent curve, \(A=-2,\) period of \(\frac{\pi}{4},\) and phase shift \((h, k)=\left(-\frac{\pi}{4},-2\right)\)

Step-by-Step Solution

Answer

Graph two periods of \(y = -2 \cdot \tan(4(x + \frac{\pi}{4})) - 2\).

Step 1: Understand the Tangent Function

The standard form of a tangent function is given by \(y = A \cdot \tan(B(x - h)) + k\), where \(A\) is the amplitude, \(B\) relates to the period, \(h\) is the horizontal shift (phase shift), and \(k\) is the vertical shift. For a tangent function \(A\) represents vertical stretching or compression but is not technically an amplitude because it doesn't affect the range of the tangent function.

Step 2: Calculate the Value of B using the Period

The period of the tangent function is given by \( \frac{\pi}{B} \). Given that the period is \(\frac{\pi}{4}\), we can solve for \(B\): \[\frac{\pi}{B} = \frac{\pi}{4} \B = 4.\]

Step 3: Create the Equation of the Tangent Function

Using the values given: \(A = -2\), \(B = 4\), \(h = -\frac{\pi}{4}\), and \(k = -2\), substitute them into the tangent function equation: \[y = -2 \cdot \tan(4(x + \frac{\pi}{4})) - 2.\]

Step 4: Determine Key Points and Plot Two Periods

The period of the tangent function is \(\frac{\pi}{4}\), so one period ranges from \(-\frac{\pi}{4} + \frac{\pi}{16}\) to \(\frac{\pi}{16}\) (since the period is centered around the phase shift \(h = -\frac{\pi}{4}\)). Plot key points:- At \(x = -\frac{\pi}{4}\), \(y = -2\).- At \(x = -\frac{\pi}{8}\), the function is undefined because \(\tan\) reaches its vertical asymptote.- At \(x = 0\), \(y = -6\).- The next period, start plotting similarly from \(\frac{\pi}{16}\).Remember to translate these by the vertical shift \(k = -2\). The critical values and asymptotes move left by \(\frac{\pi}{4}\) and down by 2.

Step 5: Plot the Graph and Confirm Satisfying Conditions

Graph the points found in step 4 and ensure there are repeating patterns every \(\frac{\pi}{4}\). Ensure the curve fits between vertical asymptotes which occur every period \(\frac{\pi}{8}\). The tangent curve commonly goes through its midline at intervals of \(\frac{\pi}{8}\) and has vertical asymptotes at \(\pm \frac{\pi}{8} + k\cdot\frac{\pi}{4}\).

Key Concepts

Periodic Function

The tangent function is a classic example of a periodic function. A periodic function is one that repeats its values in regular intervals or periods. For the tangent function, this repetition occurs every specific interval known as its period. In simpler terms, this means that if you move to the right or left on the graph by one complete period, the shape of the graph repeats itself.

For the tangent function, in general, the period is determined using the formula \( \frac{\pi}{B} \). This is due to the fact that the standard period of the basic tangent function is \( \pi \). Here, \( B \) is a factor that compresses or stretches the graph horizontally.

• If \( B \) is greater than 1, the graph compresses and the period is shorter than \( \pi \).
• If \( B \) is less than 1, the graph stretches and the period is longer than \( \pi \).

In our exercise, the calculated period is \( \frac{\pi}{4} \), which means the function reoccurs every \( \frac{\pi}{4} \) units on the x-axis, making it repeat more frequently than the standard tangent function with a period \( \pi \).

Phase Shift

A phase shift in a trigonometric function is essentially a horizontal shift along the x-axis. For the tangent function, the standard form is expressed as \( y = A \cdot \tan(B(x - h)) + k \), where \( h \) represents the phase shift.

A negative value for \( h \) shifts the entire graph to the left, whereas a positive \( h \) moves it to the right. In our example, we have a phase shift of \( h = -\frac{\pi}{4} \), which translates the graph to the left by \( \frac{\pi}{4} \) units.

• Horizontal shifts can significantly influence the starting point of graph cycles.
• This feature is crucial for determining where the first vertical asymptote will occur.

In conjunction with the period, understanding the phase shift helps in accurately plotting repeated cycles of the tangent function, ensuring that the pattern aligns as it should horizontally.

Vertical Asymptotes

When dealing with trigonometric functions like the tangent function, understanding vertical asymptotes is vital. Vertical asymptotes are lines where the function approaches but never actually reaches; they are like invisible barriers.

For the tangent function, these asymptotes occur at regular intervals and are located where the function is undefined since the tangent function is undefined at odd multiples of \( \frac{\pi}{2} \). However, with transformations, these intervals change.

• In the modified tangent function, vertical asymptotes arise due to the compressed or stretched period with the factor \( B \).
• For our problem, after considering the transformations, the new asymptotes occur at intervals of \( \frac{\pi}{8} \) along the x-axis.

Understanding these asymptotes ensures that while graphing, these undefined points and intervals where the function heads toward infinity are clearly marked, dictating where the graph "shoots off" from the coordinate plane.

Trigonometric Graphing

Graphing trigonometric functions like the tangent function requires attention to a set pattern and specific markers such as periods, phase shifts, and vertical asymptotes. This graphing process helps visualize the repetitive nature and shape of the function.