Question

Graph the function. \(g(x)=\sin 2(x+\pi)\)

Short answer

The function \(g(x)=\sin 2(x+\pi)\) is a sine curve that is shifted to the left by \(\pi\) and had its frequency multiplied by 2.

Step-by-step solution

Identify the horizontal shift

The term inside the brackets with x, \(x+\pi\), represents a shift of the basic sine function. It is a horizontal shift to the left by \(\pi\) units. This happens because \(x+\pi\) is equivalent to \(x-\(-\pi\)\), which represents a shift to the left.

Identify the frequency change

The coefficient of 2 in front of the \(x+\pi\) is responsible for the change in the frequency of the sine function. This makes the function to complete two cycles in the interval \([0, 2\pi]\), unlike the basic sine function which completes one cycle in the same interval.

Sketch the function

First, sketch a basic sine curve on an interval \([0, 2\pi]\). Then, compress it horizontally by a factor of 2 to account for the frequency change. Finally, shift it to the left by \(\pi\) units to account for the horizontal shift. The resulting curve is the graph of the function \(g(x)=\sin 2(x+\pi)\).

Key concepts

Graph Transformations

Graph transformations are techniques to alter the shape and position of a graph without changing its basic form. These transformations can include:

• Vertical shifts, where the graph moves up or down.
• Horizontal shifts, where the graph moves left or right.
• Vertical and horizontal stretches or compressions, where the graph stretches or compresses along the x or y-axis.
• Reflections, where the graph flips over a line.

Understanding these transformations helps in visualizing how the basic graph of a function, like the sine function, can be manipulated to fit different scenarios. With the function given in the exercise, we look specifically at horizontal shifts and frequency changes, both of which are common transformations used in trigonometry to adjust or fit periodic functions in real-world applications.

Frequency Change

Frequency in trigonometric functions refers to the number of times a function repeats within a certain interval. For the sine function, the frequency can be altered by multiplying the angle variable by a constant factor.
In the function, \(g(x) = \sin 2(x + \pi)\), the coefficient 2 changes the frequency. This means the sine wave will complete its cycle twice as fast within the interval \([0, 2\pi]\) compared to the basic sine function. Imagine listening to a song where the melody loop repeats twice as quickly; this is similar to our altered sine wave environment.
To adjust for the frequency change, the graph will appear horizontally compressed. Such transformations are crucial in fields like signal processing and sound engineering, where frequency manipulation is a key component.

Sine Function

The sine function, \(\sin(x)\), is one of the fundamental trigonometric functions. It is periodic with a period of \(2\pi\), meaning it repeats its pattern every \(2\pi\) units.
The sine function's graph is a smooth, wave-like curve that oscillates above and below the x-axis, with peaks at \(\pi/2\) and troughs at \(3\pi/2\), within its standard period.
This wave is fundamental in describing various natural phenomena, including sound and light waves. It serves as a basis for understanding more complex trigonometric transformations.
When we transform the sine function, as seen in the exercise, we alter its graph through various modifications, such as shifts and stretches, which affect its amplitude, period, and phase.

Horizontal Shift

Horizontal shifts in a graph occur when all points on a graph move left or right by the same fixed distance. In the function \(g(x)=\sin 2(x+\pi)\), the term \((x+\pi)\) indicates a horizontal shift to the left by \(\pi\) units. The expression \(x+\pi\) can be rewritten as \(x - (-\pi)\), reinforcing that it's a leftward move.
This horizontal shift means each point on the sine wave begins its pattern \(\pi\) units earlier along the x-axis. Imagine walking along a road where every landmark is shifted back by a certain measure as you move along; this is how horizontal shifts impact function graphs.
Recognizing horizontal shifts is vital for accurately sketching functions and understanding how changes to the function's equation affect its behavior and appearance.