Problem 1 - Algebra and Trigonometry with Analytic Geometry

Question

Find the period and sketch the graph of the equation. Show the asymptotes. $$ y=4 \tan x $$

Step-by-Step Solution

Answer

The period is \(\pi\); asymptotes are at \(x=\frac{\pi}{2} + n\pi\).

Step 1: Understand the Tangent Function

The tangent function, \(\tan x\), has a period of \(\pi\). This means the function repeats itself every \(\pi\) units along the x-axis. For the function \(y = 4 \tan x\), the graph will also have a period of \(\pi\) because the coefficient of \(x\) in \(\tan x\) is 1 (the period is given by \(\frac{\pi}{|b|}\), where \(b\) is the coefficient of \(x\)).

Step 2: Identifying Asymptotes

The function \(\tan x\) has vertical asymptotes where it is undefined, which occurs at \(x = \frac{\pi}{2} + n\pi\), where \(n\) is an integer. Since there is no horizontal scaling or horizontal shift in the function \(y = 4 \tan x\), the asymptotes also occur at the same points: \(x = \frac{\pi}{2} + n\pi\).

Step 3: Adjusting the Amplitude

The coefficient in front of \(\tan x\) is 4, which means the function's value (though not its amplitude, as tangents don't have a standard amplitude due to their behavior at asymptotes) is scaled by 4. This affects the steepness of the graph but not the period or the asymptote locations.

Step 4: Sketch the Graph

On a graph, plot the function \(y = 4 \tan x\). The graph should repeat every \(\pi\) units. Mark the vertical asymptotes at \(x = \frac{\pi}{2} + n\pi\). Between asymptotes, \ \(y = 4 \tan x\) will cross the x-axis at the origin and be symmetric within each period, steeply increasing towards the asymptotes.

Key Concepts

Period of Function

The period of a function is the horizontal length over which a function repeats its pattern. For the tangent function, \(\tan x\), the period is \(\pi\). This means that every \(\pi\) units along the x-axis, the tangent function graph will look identical to the previous segment.
For the function given, \(y = 4 \tan x\), the period remains \(\pi\) because the coefficient of \(x\) remains 1, and thus, \(\frac{\pi}{|b|} = \pi\).

• The "4" multiplier alters the steepness but not the period.
• Periods are essential for predicting the behavior of periodic functions over different intervals.

Understanding periods helps in sketching the graph quickly and accurately, ensuring you can replicate its shape across any length of the x-axis.

Vertical Asymptotes

Vertical asymptotes are lines that the graph of a function approaches but never touches. For the tangent function \(\tan x\), asymptotes occur where the function is undefined. The standard tangent function has asymptotes at \(x = \frac{\pi}{2} + n\pi\), where \(n\) is any integer.
In the case of our function, \(y = 4 \tan x\), there are no shifts or scalings that affect the asymptotes' positions. They will remain at \(x = \frac{\pi}{2} + n\pi\).

• Vertical asymptotes signify where the graph will shoot to positive or negative infinity.
• These are crucial points to identify when sketching or analyzing the behavior of tangent functions.

The presence of the number "4" in front of \(\tan x\) does not affect where these asymptotes lie on the x-axis.

Graphical Representation

Graphical representation of the tangent function involves clearly understanding its periodic nature and asymptotic behavior. To graph \(y = 4 \tan x\), you need to plot a repeating pattern every \(\pi\) units.

• Start by placing vertical asymptotes at \(x = \frac{\pi}{2} + n\pi\).
• The graph will cross the x-axis at multiples of \(\pi\) (e.g., \(x = 0, \pm\pi, \pm 2\pi\)).
• Between each pair of asymptotes, the graph will rise steeply from negative infinity, cross the x-axis, and then head towards positive infinity.

This graphical behavior remains structured and symmetric across each period, creating a recognizable pattern.

Scaling Transformations

Scaling transformations affect the steepness or "stretch" of a function graph vertically or horizontally. In the equation \(y = 4 \tan x\), the multiplier "4" is a vertical scaling factor.

• It scales the output values of the tangent function by 4.
• This makes the graph steeper, meaning it rises or falls faster as it approaches asymptotes.

Vertical scaling does not change the period or location of asymptotes. Horizontal scaling, which is not present here, would affect these aspects by altering the coefficient of \(x\). This understanding of transformations enables more precise control over graph shapes and their representation.