Sketch the graphs of the following on [-π, 2 π]. (a) y=sin2 x (b) y=2 sint (c) y=cos(x-(π)/(4))

step 1 The first figure is a graph of y equals sine 2x. Notice that the amplitude is 1, and the period

Sketch the graphs of the following on [-π, 2 π]. (a) y=sin2...

Key Concepts

Reciprocal Functions and Asymptotes

The secant function is the reciprocal of the cosine function. This means its graph has vertical asymptotes at the points where the cosine function is zero, as division by zero is undefined. Recognizing and correctly sketching these asymptotes along with the characteristic curve of the secant function is fundamental in graphing reciprocal trigonometric functions.

Period and Frequency Modifications

The period of a trigonometric function is the length of one complete cycle of the wave. When the input of the sine or cosine function is multiplied by a constant (e.g., sin(2x)), the period changes inversely with that constant. This highlights the relationship between frequency and period, where an increase in frequency results in a decrease in the period of the graph.

Phase Shift or Horizontal Translation

Phase shift describes the horizontal movement of a trigonometric graph. A function such as cos(x - ?/4) is shifted to the right by ?/4 units. Understanding phase shift is essential for accurately sketching the translated graph on the given interval, as it adjusts the position of the key landmarks of the wave.

Trigonometric Functions and Their Graphs

This concept involves understanding the basic shapes and characteristics of sine, cosine, and secant functions. It includes knowing key points such as maximums, minimums, zeros, and the periodic nature of these functions. Drawing these graphs on a given interval requires recognizing their repeating patterns and symmetry properties.

Amplitude and Vertical Scaling

Amplitude refers to the height of the oscillations in a trigonometric function. Vertical scaling, such as multiplying a sine or cosine function by a constant, stretches or compresses the graph vertically. For instance, in the function y = 2 sin(t), the amplitude is 2, meaning the peaks and troughs of the sine wave reach 2 and -2 respectively.

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