Problem 8 - Algebra and Trigonometry

Question

7–52 Find the period and graph the function. $$y=-4 \tan x$$

Step-by-Step Solution

Answer

The period of the function \(y = -4 \tan x\) is \(\pi\).

Step 1: Recall the Basic Formula for Period of Tangent

The period of the tangent function \(y = \tan x\) is \(\pi\). The period of any function of the form \(y = a \tan(bx + c)\) is \(\frac{\pi}{|b|}\). This means the period is affected by the coefficient \(b\) that multiplies the variable \(x\).

Step 2: Identify the Coefficients in the Given Function

In the function \(y = -4 \tan x\), we have \(a = -4\), \(b = 1\), and \(c = 0\). The coefficient \(b\) that affects the period is \(1\).

Step 3: Calculate the Period of the Function

Since \(b = 1\), the period of the function is calculated as \(\frac{\pi}{|1|} = \pi\). Therefore, the period of the function \(y = -4 \tan x\) is \(\pi\).

Step 4: Graph the Function

To graph \(y = -4 \tan x\), note that the basic shape of the tangent graph repeats every \(\pi\) units.1. **Amplitude and Stretch/Shrink:** The coefficient \(-4\) scales the graph vertically, creating a steeper slope compared to \(\tan x\). The negative sign reflects it across the x-axis.2. **Vertical Asymptotes:** Occur at \(x = \frac{\pi}{2} + k\pi\) for integer \(k\).Plot the basic tangent curve while applying this vertical stretch and reflection over the x-axis for periods defined as \({x = -\frac{\pi}{2}, \frac{\pi}{2}, \frac{3\pi}{2}, \ldots}\).

Key Concepts

Tangent Function Graph

When we explore the tangent function graph, we primarily recognize its unique characteristic: a repeating wave-like pattern. This graph originates from the basic function \( y = \tan x \), which generates a series of rising and falling sections within each period before starting anew. The tangent graph, unlike sine or cosine, doesn't oscillate between a maximum and minimum. Instead, it ranges from -∞ to ∞ and passes through the origin.

• The graph has distinct "waves" that appear in cycles every \( \pi \) units.
• Tangent graphs are well-known for their asymmetric "snake-like" shape, which continually increases within each cycle.

For functions such as \( y = -4 \tan x \), the basic tangent graph is affected by both the negative sign and the coefficient \(-4\).

• The negative sign inverts the graph, reflecting it over the x-axis, transforming rises to falls.
• The coefficient \( -4 \) amplifies the slope of the graph, making the ascent and descent sharper than that of the basic \( \tan x \) graph.

Understanding these changes is crucial when graphing the tangent function effectively.

Period Calculation

Calculating the period of a tangent function involves identifying how weight or modifications applied to \( x \) adjust the graph's cyclical behavior.
The standard period for the equation \( y = \tan x \) is \( \pi \). Altering this function to the form \( y = a \tan(bx + c) \) shifts the period to \( \frac{\pi}{|b|} \). This formula comes from the idea that changes to \( x \)'s coefficient compress or stretch the graph horizontally.
Consider the function \( y = -4 \tan x \), where the coefficients are \( a = -4 \), \( b = 1 \), and \( c = 0 \).

• Here, \( b = 1 \), so the formula simplifies to \( \frac{\pi}{1} = \pi \), confirming the period is still \( \pi \).
• Notice the period remains unchanged despite the vertical modification \(-4\); it is only influenced by the variable \( x \)’s coefficient \( b \).

Remember, period calculation helps predict where each cycle of the tangent graph starts and concludes, fundamentally aiding in the overall sketching of the graph.

Vertical Asymptotes of Tan Function

Vertical asymptotes are crucial indicators defining the points where the tangent graph approaches infinity and re-emerges from negative infinity. For the standard tangent function \( y = \tan x \), these appear regularly.

• Asymptotes are positioned at \( x = \frac{\pi}{2} + k \pi \), where \( k \) is any integer, indicating they exist at regular intervals.
• They split the graph into sections that predictably rise from negative infinity, cross zero, and approach positive infinity again.

For the function \( y = -4 \tan x \):

• The vertical asymptotes are still located at these regular intervals, unaffected by the vertical stretching or reflection.
• In practical terms, these are at \( x = -\frac{\pi}{2}, \frac{\pi}{2}, \frac{3\pi}{2}, \ldots \).

These asymptotes guide how and where to draw the tangent graph's characteristic steep inclines and restrict the cycle.Understanding these helps in accurate plotting of any tangent function, no matter the transformations applied.