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The period of a function tells us how often the function repeats its pattern over the x-axis. For trigonometric functions like sine, cosine, and tangent, understanding the period is crucial for graphing and identifying behavior over time.

The standard period of the tangent function, \( \tan x \), is \( \pi \). This means that it repeats every \( \pi \) units along the x-axis. Unlike sine and cosine, which have periods of \( 2\pi \), tangent's period is shorter because it completes a full cycle of rising to positive infinity and dropping to negative infinity more rapidly.

In our example, the function is \( y = 3 \tan x \). The multiplier 3 affects the vertical stretch but does not change the period. Hence, the period of \( y = 3 \tan x \) remains \( \pi \). Recognizing this invariance is important when planning the graphing strategy for tangent-related exercises.

#### Key Points about Periods of Tangent:

• Tangent's standard period: \( \pi \).
• It completes a cycle more quickly compared to sine and cosine.
• Coefficient before \( \tan x \) (vertical stretch) does not alter the period.

Vertical stretch refers to the factor by which a function's values (heights) are increased or decreased. For the tangent function, a vertical stretch alters its steepness but doesn't affect its general asymptotic behavior.

In the equation \( y = 3 \tan x \), the coefficient 3 is responsible for the vertical stretch. This means every value of \( \tan x \) is multiplied by 3, resulting in the graph being three times taller or steeper than the basic function \( \tan x \). However, the position and spacing of vertical asymptotes and the x-intercepts remain unchanged.

#### Effects of Vertical Stretch on Tangent:

• Increased steepness of the curve.
• Unchanged period (still \( \pi \)).
• Asymptotes and zero crossings remain at their standard positions.

Recognizing changes in vertical stretch aids in accurately sketching trigonometric functions on the coordinate plane, ensuring heights are properly scaled.

Graphing a trigonometric function involves plotting its key characteristics, like the period, vertical stretch, and important points like asymptotes and x-intercepts. For \( y = 3 \tan x \), one needs to capture these elements correctly.

To graph this function:

• First, identify vertical asymptotes, which are at \( x = \frac{-\pi}{2}, \frac{\pi}{2}, \frac{3\pi}{2}, \ldots \). These are important because \( \tan x \) is undefined at these points and the curve approaches positive or negative infinity.
• Next, points where the curve crosses the x-axis are crucial: \( x = -\pi, 0, \pi, 2\pi, \ldots \). At these points, the function returns to the midline (baseline).
• Finally, understand that between each pair of asymptotes, the function creates an 'S'-shaped curve. This curve will be steeper due to the vertical stretch factor of 3.

When sketching, extend the pattern across the graph, ensuring each segment between asymptotes is consistent. Always label the axes, key points, and asymptotes to help visualize the periodic and stretched nature of the function.