91Ó°ÊÓ

The period of a trigonometric function indicates how long it takes for the function to complete one full cycle. For the basic secant function, \( \sec(x) \), the period is \( 2\pi \), meaning that it repeats its values every \( 2\pi \) units along the x-axis.

To determine the period of a transformed secant function, such as \( r(x) = -\sec\left(\frac{2\pi}{3}x\right)+2 \), we examine the coefficient \( b \) in the argument of the secant function. The period \( P \) of the transformed function is given by \( P = \frac{2\pi}{|b|} \). Hence, with \( b = \frac{2\pi}{3} \), the period is \( P = 3 \).

This interval \( P \) is the length on the x-axis over which the secant function graph completes its peaks and troughs once before starting the same pattern again. Recognizing the period of the function allows us to predict its behavior and also determines the intervals we use when graphing the function.

A vertical shift in trigonometric graphs describes the movement of the graph up or down along the y-axis. This transformation doesn't affect the shape of the graph but changes its position.

In the function \( r(x) = -\sec\left(\frac{2\pi}{3}x\right)+2 \), the '+2' at the end signifies a vertical shift. Here, the entire graph of the secant function is raised 2 units above its original position. Graphically, every point \( (x, y) \) on the unshifted graph is moved to \( (x, y+2) \) on the shifted graph.

Identifying Vertical Shift

• Locating the vertical shift is straightforward. It's given by the constant \( d \) added or subtracted at the end of the trigonometric function.
• In \( r(x) = -\sec\left(\frac{2\pi}{3}x\right)+2 \), the vertical shift is '+2', which means every point on the graph moves two units upwards.
• If \( d \) were negative, the graph would shift downwards by that amount.

By considering the vertical shift, we can accurately place the graph on the coordinate system, ensuring it mirrors the function's properties correctly.

When we talk about transformations of trigonometric graphs, we're referring to changes in the graph's position, shape, or orientation due to modifications in the function's equation. This includes reflections, stretches or compressions, vertical and horizontal shifts, and period changes.

In our example, \( r(x) = -\sec\left(\frac{2\pi}{3}x\right)+2 \), several transformations are at work:

• Reflection: The negative sign in front of the secant indicates a reflection across the x-axis. This means that peaks become troughs and vice versa.
• Vertical Shift: The '+2' at the end of the function indicates a shift upward by 2 units.
• Period Change: The coefficient \( \frac{2\pi}{3} \) in the function modifies the period from its standard \( 2\pi \) to 3 units.

These transformations collectively alter the original \( \sec(x) \) graph into its new form. To successfully sketch the graph of \( r(x) \) by hand, we plot the secant function without transformations, and step by step, add each transformation. This systematic approach ensures a clear understanding of how the final graph is developed from the parent function and helps students visualize and grasp the effects of each transformation.