Question

Graph the function. \(g(x)=-\sin \frac{1}{2} x-2\)

Short answer

The graph of the function \(g(x)=-\sin \frac{1}{2} x-2\) is a sine wave reflected over the x-axis, stretched horizontally by a factor of 2, and shifted down by 2 units. It oscillates between -3 and -1 with a period from 0 to \(4\pi\).

Step-by-step solution

Understand the Basic Graph of Sine Function

The graph of the base sine function, \(y = sin(x)\), is a wave that oscillates between -1 and 1, crossing the x-axis at \(x = 0\), \( x = \pi\), \(x = 2\pi\), and so on.

Apply the Reflection

Because of the negative sign in front of the sine function, the graph will be reflected over the x-axis. This means that peaks become valleys and valleys become peaks.

Apply The Horizontal Stretch

Since there is a coefficient of \(\frac{1}{2}\) on the x term inside the function, it causes the graph to be 'stretched' horizontally by a factor of 2. This will double the period of the function, showing one complete wave cycle from \(x = 0\) to \(x = 4\pi\). So instead of crossing the x-axis at \(x = 0\), \( x = \pi\), \(x = 2\pi\), it now will cross at \(x = 0\), \( x = 2\pi\), \(x = 4\pi\), and so on.

Apply the Vertical Shift

The -2 at the end of the function represents a vertical shift. The whole graph lowers by 2 units. This means the peaks and valleys of the wave will now oscillate between -3 and -1, instead of -1 and 1.

Graph the Function

After applying all the transformations, draw the function's graph, making sure to accurately depict the reflection, the horizontal stretch, and the vertical shift.

Key concepts

Understanding the Sine Function

The sine function, represented as \(y = \sin(x)\), is one of the fundamental functions in trigonometry. This function produces a smooth, wavelike pattern known as a sinusoidal wave. Here are some key characteristics of the sine function:

• Amplitude: The amplitude of the sine function is 1, meaning it oscillates between -1 and 1.
• Period: The sine wave completes a full cycle every \(2\pi\) radians.
• Intersections: It crosses the x-axis at multiples of \(\pi\), such as \(0, \pi, 2\pi\), and so on.
• Extrema: The function reaches its maximum at \(x = \frac{\pi}{2} + 2k\pi\) and minimum at \(x = \frac{3\pi}{2} + 2k\pi\), where \(k\) is an integer.

Understanding these basic features helps in predicting the behavior of the sine wave and in recognizing its transformations.

Function Transformation Essentials

Function transformation refers to changes made to the graph of a function. They include various modifications such as shifting, stretching, and reflecting the graph:

• Reflection: If a function is negatively multiplied, such as \(y = -\sin(x)\), it reflects over the x-axis, flipping the graph upside down.
• Horizontal Stretch: A coefficient inside the function's argument, like \(\sin\left(\frac{1}{2}x\right)\), horizontally stretches the graph. It adjusts the period of the function. A factor of \(\frac{1}{2}\) means the graph will stretch, doubling the period.
• Vertical Shift: Adding or subtracting a constant, as in \(y = \sin(x) - 2\), shifts the function up or down vertically, moving the entire graph up or down along the y-axis by the constant's amount.

These transformations change the graph's appearance but the fundamental properties of the sine wave, like smooth oscillations, remain.

Grasping Periodic Functions

Periodic functions are functions that repeat their values at regular intervals. The sine function is a classic example of a periodic function with several notable traits:

• Repeating Pattern: Sine functions repeat every \(2\pi\). This is known as the function’s period.
• Predictability: Due to their repeating nature, knowing the behavior of one cycle can help predict any value in subsequent cycles.
• Utility in Real Life: Sinusoidal patterns describe many real-world phenomena, from sound waves to tides, thanks to this inherent periodicity.

When understanding periodic functions like the sine, it’s vital to grasp how modifications such as a horizontal stretch or shift alter the period while maintaining the periodic nature.