Question

Graph the function. \(g(x)=-\sin x-5\)

Short answer

The graph of the function \(g(x) = -\sin x -5\) starts at (0,-5), falls to (π/2,-6), rises to (π,-5), goes up to (3π/2,-4), and descends back to (2π, -5). The pattern then repeats every \(2π\) interval because of the periodic nature of the sine function.

Step-by-step solution

Understand the Base Function

The sine function, sin(x), is a periodic function that oscillates between -1 and 1 on the y-axis. It has a period of \(2\pi\), meaning it repeats its values every \(2\pi\) interval. The graph starts at the point (0,0), goes to (π/2,1), comes back to (π,0), goes into the third quadrant at (3π/2, -1), and comes back to the x-axis at (2π,0). This pattern is repeated for each successive \(2\pi\) interval.

Apply the Reflection

In given function \(g(x)\) the negative sign in front of the sin function means that the function is reflected about the x-axis. That is, the standard sin(x) function is effectively flipped vertically. This means when sin(x) is at its maximum of +1, -sin(x) will be at its minimum of -1, and vice versa.

Apply the Vertical Shift

-5 shifts the whole function down by 5 units along the y-axis. Thus, the values of -sin(x), which ranged from -1 to 1, will now range from -6 to -4.

Draw the Graph

Start at the origin (0,0) but this time go to (π/2, -6), return to (π, -5), proceed to the third quadrant to (3π/2, -4), and back to the x-axis at (2π, -5). This pattern then repeats all along the x-axis because of the periodic nature of the sine function. You will have the graph of \(g(x) = -\sin x - 5\).

Key concepts

Periodic Functions

A periodic function is one that repeats its values at regular intervals, known as periods. Think of it like the pendulum of a clock swinging back and forth; no matter how long you watch it, the motion is consistent and unchanging. In mathematics, the sine function, represented by sin(x), is a classic example of a periodic function. It repeats itself every 2π radians. This repetition is what allows us, for example, to predict the tides or model sound waves over time. Understanding the periodic nature of functions like the sine wave is crucial for grasping how these functions behave over long stretches of the x-axis.

Amplitude and Period

The amplitude of a sine function refers to the height of its peak or the depth of its trough from the center line of its graph. Essentially, it's a measure of how far the graph stretches above or below the x-axis. For the basic sin(x) function, the amplitude is 1. On the other hand, the period of a sine function is the distance it takes for the function to complete one full cycle. For sin(x), this is always equal to 2π radians. With amplitude and period, we can predict and describe the size and timing of the waves produced by functions, which is an integral part of many scientific and engineering tasks.

Vertical Shifts

A vertical shift moves the graph of a function up or down on the coordinate plane. For the sine function, adding or subtracting a number to the entire function (like -5 in the problem g(x) = -sin x - 5) shifts the graph vertically. Here, the -5 translates to shifting the sine wave five units downward. This doesn't change the period or amplitude; it simply moves the whole wave up or down. Vertical shifts are useful for adjusting the baseline of measurement when comparing functions or adjusting data to a new starting point.

Reflection over the X-axis

A reflection over the x-axis in trigonometric functions like the sine function means flipping the graph upside down. It's as though the wave has been mirrored across the x-axis. In mathematical terms, this is achieved by placing a negative sign in front of the function (for example, g(x) = -sin x). The peaks become troughs, and the troughs become peaks; however, the period of the function remains the same. Reflecting functions is important in scenarios where a 'negative' version of a waveform is required, such as inverting sound waves or electrical signals.