Problem 19 - Calculus for Biology and Medicine

Question

sketch the graph of each function. Do not use a graphing calculator. (Assume the largest possible domain.) $$ y=2 \sin (x+\pi / 4) $$

Step-by-Step Solution

Answer

The graph is a sine wave with amplitude 2, a left phase shift of \(\pi/4\), and a period of \(2\pi\).

Step 1: Identify the Function Type

The function given is \( y = 2 \sin(x + \pi/4) \), which is a sine function. This means the graph will exhibit the characteristic wave-like pattern of a sine function.

Step 2: Determine the Amplitude

The coefficient in front of the sine function is 2, which means the amplitude of the graph is 2. This results in the sine wave oscillating between -2 and 2 on the y-axis.

Step 3: Find the Phase Shift

The expression \( x + \pi/4 \) indicates a phase shift. The \( +\pi/4 \) inside the function results in a phase shift to the left by \( \pi/4 \) units. This shifts the entire graph \( \pi/4 \) units to the left.

Step 4: Identify the Period

The standard period of the sine function is \( 2\pi \). Because there is no coefficient affecting \( x \) (other than 1), the period remains the same at \( 2\pi \).

Step 5: Plot Key Points

Using the properties found, plot key points of the sine wave. Start from the left-shifted origin at \( x = -\pi/4 \). The peaks occur at \( y = 2 \), and the troughs occur at \( y = -2 \), repeating every \( 2\pi \). Initial key points will be \( (-\pi/4, 0) \), \( (\pi/4, 2) \), \( (3\pi/4, 0) \), \( (5\pi/4, -2) \), and continue similarly.

Step 6: Sketch the Graph

Connect the plotted key points in a smooth, continuous wave pattern. Ensure that each complete cycle from one peak to the next covers a distance of \( 2\pi \), and that the graph oscillates between -2 and 2. Repeat the pattern across the x-axis to indicate the graph continues indefinitely in both directions.

Key Concepts

Graphing Sine Functions

When tackling a sine function like \( y = 2 \sin(x + \pi/4) \), it's essential to understand its shape and structure. Sine functions depict wave-like patterns starting from the origin. These waves typically oscillate about the x-axis. The function you're working with, however, involves a transformation. The transformation creates a shift in how the wave appears or how high or low it oscillates.

To sketch this sine function by hand:

• Identify key characteristics like amplitude, period, and phase shift.
• Plot these details accurately on a set of axes.

The resulting graph will show a smooth, continuous wave that cycles at regular intervals, represented in this case by a cycle of complete waves repeating over intervals of \( 2\pi \). Each wave crosses the x-axis, hits a peak, then descends back, forming troughs before rising again.

Amplitude and Period

Amplitude and period are critical components of trigonometric functions, particularly when graphing. For the given sine function, \( y = 2 \sin(x + \pi/4) \), the amplitude and period define the vertical and horizontal extent of the wave, respectively.

The amplitude is determined by the coefficient in front of the sine, in this case, 2. This means our sine wave will reach a maximum height of 2 and a minimum height of -2, relative to the central axis (the x-axis).

Next, let's tackle the period. It's essential to know how long it takes the function to complete one full cycle. In simple sine functions such as this one, absent other modifying coefficients, the period is naturally \( 2\pi \). This means after \( 2\pi \) units along the x-axis, the wave will repeat its familiar pattern.

In summary, understanding amplitude and period helps us anticipate the repetitive motion and vertical sway of the sine wave, allowing for an accurate graph sketch.

Phase Shift in Trigonometric Functions

Phase shift is the horizontal translation of a trigonometric function on the graph. For our function \( y = 2 \sin(x + \pi/4) \), understanding phase shift allows us to grasp how the wave is moved along the x-axis.

Unlike vertical shifts which are easier to visualize, phase shifts can be a bit more intricate. Here, the expression \( x + \pi/4 \) suggests a leftward shift by \( \pi/4 \) units. This shift moves the starting point of the sine wave from what would normally be its origin at \( x = 0 \) to \( x = -\pi/4 \).

This positional change affects all key points in the wave, displacing the entire set of oscillations leftwards. For graphing:

• Recognize this shift as it dictates where your initial plotting points lie.
• Start with the new origin, ensuring consistency in the wave's appearance.

Mastering phase shifts is vital, as it determines how shifts in formula parameters affect the actual position of the sine wave on your graph.