Problem 62 - Precalculus

Question

In Exercises \(61-66,\) sketch the graph of the function over the indicated interval. $$y=\frac{1}{3}+\frac{2}{3} \sin (2 x-\pi),\left[-\frac{3 \pi}{2}, \frac{3 \pi}{2}\right]$$

Step-by-Step Solution

Answer

The graph is a sinusoidal wave with points at specific intervals reflecting its transformations, repeated over the interval \([-\frac{3\pi}{2}, \frac{3\pi}{2}]\).

Step 1: Understand the Function

We have the function \( y = \frac{1}{3} + \frac{2}{3} \sin (2x - \pi) \) to graph. This is a sinusoidal function (a transformation of a basic sine function), which involves changes in amplitude, phase shift, and vertical shift.

Step 2: Identify the Basic Properties

The amplitude is given by \( \frac{2}{3} \). The period is calculated using \( \frac{2\pi}{B} \), where \( B = 2 \), so the period is \( \pi \). The phase shift is \( \frac{\pi}{2} \) to the right, since we solve \( 2x - \pi = 0 \). The vertical shift is \( \frac{1}{3} \) upwards.

Step 3: Determine Key Points

For a sine wave, key points occur at increments of \( \frac{\text{Period}}{4} \). Here, the key points occur at \( x = \left\{ -\frac{3\pi}{2}, -\pi, -\frac{\pi}{2}, 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \right\} \). Evaluate \( y \) at these points using the given function.

Step 4: Calculate Key Values

Calculate \( y \) for each key point: - At \( x = -\frac{3\pi}{2} \), \( y = \frac{1}{3} + \frac{2}{3} \times 0 = \frac{1}{3} \).- At \( x = -\pi \), \( y = \frac{1}{3} - \frac{2}{3} = -\frac{1}{3} \).- At \( x = -\frac{\pi}{2} \), \( y = \frac{1}{3} + \frac{2}{3} \times 0 = \frac{1}{3} \).- At \( x = 0 \), \( y = \frac{5}{3} \).- At \( x = \frac{\pi}{2} \), \( y = \frac{1}{3} \).- At \( x = \pi \), \( y = -\frac{1}{3} \).- At \( x = \frac{3\pi}{2} \), \( y = \frac{1}{3} \).

Step 5: Plot the Graph

Plot the ordered pairs \((x, y)\) from calculated key values: \((-\frac{3\pi}{2}, \frac{1}{3}), (-\pi, -\frac{1}{3}), (-\frac{\pi}{2}, \frac{1}{3}), (0, \frac{5}{3}), (\frac{\pi}{2}, \frac{1}{3}), (\pi, -\frac{1}{3}), (\frac{3\pi}{2}, \frac{1}{3})\).Connect these points smoothly by following the sinusoidal shape of the function over the given interval.

Key Concepts

Amplitude

The amplitude of a trigonometric function describes the height of the wave from its central axis. For the function \( y = \frac{1}{3} + \frac{2}{3} \sin(2x - \pi) \), the amplitude is \( \frac{2}{3} \). This tells us that the peaks and troughs of the sine wave differ by \( \frac{2}{3} \) from the midline of the function.
The amplitude doesn't affect the horizontal position but dictates the vertical stretch of the sine wave. It is always positive and calculated as the coefficient of the sine function. For example, if the coefficient of \( \sin(x) \) changes, the wave stretches or shrinks accordingly. Amplitude is key as it determines the maximum and minimum values the wave reaches possible.

• A larger amplitude results in taller waves.
• A smaller amplitude results in shorter waves.

Understanding this helps quickly visualize how wave properties change with different amplitudes.

Phase Shift

The phase shift of a sinusoidal function tells us how much the function shifts horizontally along the x-axis. In the function \( y = \frac{1}{3} + \frac{2}{3} \sin(2x - \pi) \), the phase shift is found by rewriting the argument of the sine function.
To find the phase shift, we set \( 2x - \pi = 0 \) and solve for \( x \) to get \( x = \frac{\pi}{2} \). This signifies a shift of \( \frac{\pi}{2} \) units to the right.

• A positive phase shift (to the right) moves the wave in the positive x direction.
• A negative phase shift (to the left) moves it toward the negative x direction.

Phase shifts are crucial for pinpointing where the cycle of a wave begins, allowing accurate plotting during graphing.

Vertical Shift

A vertical shift moves a function up or down on the graph, influencing where the wave oscillates around its midline. For \( y = \frac{1}{3} + \frac{2}{3} \sin(2x - \pi) \), the vertical shift is \( \frac{1}{3} \), moving the wave upward by \( \frac{1}{3} \).
This adjustment dictates where the central axis of the wave is located compared to the origin.

• A positive shift moves the wave upwards.
• A negative shift moves it downwards.

This movement does not affect the wave's period or amplitude, but it is vital for graphing purposes to ensure the wave centers around the correct line on the graph.

Sinusoidal Function

A sinusoidal function represents waves typically characterized by their smooth, wave-like curves. The general form for a sinusoidal function is \( y = A \sin(Bx - C) + D \). In our specific function \( y = \frac{1}{3} + \frac{2}{3} \sin(2x - \pi) \), each component has a role:

• \( A = \frac{2}{3} \) is the amplitude.
• \( B = 2 \) affects the period of the wave, calculated as \( \frac{2\pi}{B} \).
• \( C = \pi \) is part of the phase shift, determining its horizontal translation.
• \( D = \frac{1}{3} \) is the vertical shift of the wave.

A sinusoidal function reflects repetitive phenomena such as sound waves, tides, and harmonic motion. Understanding the key features of a sinusoidal function helps in sketching the graph, predicting behavior, and solving problems related to waves and oscillations.