Question

Graph the function. \(g(x)=-5 \sin \left(x-\frac{\pi}{2}\right)+3\)

Short answer

The graph of the function \(g(x)=-5 \sin \left(x-\frac{\pi}{2}\right)+3\) has an amplitude of 5, a regular period of \(2\pi\), shifts \(\frac{\pi}{2}\) unit to the right, and 3 units up.

Step-by-step solution

Identify the amplitude

The amplitude is the absolute value of A. In this equation, A is -5, thus, the amplitude is 5.

Identify the period

In the equation, B equals to 1. To find the period of the function, divide \(2\pi\) by absolute value of B. So, the period is \(2\pi\).

Identify the phase shift

The phase shift is C. This value represents a horizontal shift of the graph. In this equation, C is \(\frac{\pi}{2}\) which means there is a shift of \(\frac{\pi}{2}\) units to the right.

Identify the vertical shift

The vertical shift is D which represents a shift of the graph up or down. In this equation, D is 3, so the graph is shifted up by 3 units.

Graph the function

Based on all the identified parameters, the graph has an amplitude of 5, a regular period of \(2\pi\), shifts \(\frac{\pi}{2}\) units rightward (phase shift), and 3 units upward (vertical shift).

Key concepts

Amplitude of a Sine Function

Understanding the amplitude of a sine function is crucial when graphing it. The amplitude refers to the maximum distance a point on the graph is above or below the midline, which is the axis that the sine wave oscillates around. In simpler terms, it's the 'height' of the wave from its resting position.

For the function given, the equation is of the form \(y = A\sin(B(x - C)) + D\) where the coefficient \(A\) affects the amplitude. Specifically, the amplitude is the absolute value of \(A\), which means it is always a positive number. For our example, \(A = -5\), thus the amplitude is \(5\). It's important to note that whether \(A\) is positive or negative, it only flips the graph around the midline but doesn't change the amplitude.

Period of a Sine Function

The period of a sine function determines how frequently the sine wave repeats along the x-axis. It is the distance required for the function to go through one complete cycle before it starts repeating itself.

Mathematically, the period \(T\) is found using the formula \(T = \frac{2\pi}{|B|}\), where \(B\) is the coefficient of \(x\) inside the sine function. A larger value of \(B\) suggests a quicker cycle, resulting in a shorter period, while a smaller \(B\) indicates a longer and stretched wave. In the function \(g(x) = -5\sin(x - \frac{\pi}{2}) + 3\), \(B = 1\), meaning the period of this sine function is \(2\pi\), representing one complete cycle of the sine wave.

Phase Shift

A phase shift in trigonometric functions such as sine or cosine represents a horizontal displacement of the graph. It adjusts where the wave cycle starts on the x-axis. The formula for determining a phase shift is derived from the \(C\) value in the sine function's equation, \(y = A\sin(B(x - C)) + D\).

To identify the phase shift, look at the value of \(C\). A positive \(C\) will shift the graph to the right, while a negative \(C\) moves it to the left. For the example function \(g(x)\), the phase shift is \(C = \frac{\pi}{2}\), resulting in the graph of the sine function being shifted to the right by \(\frac{\pi}{2}\) units.

Vertical Shift

Vertical shift moves the sine function up or down on a graph, changing the midline around which the wave oscillates. The value \(D\) in the general form \(y = A\sin(B(x - C)) + D\) tells us how much the function is shifted vertically.

Specifically, a positive \(D\) will lift the graph upwards, while a negative \(D\) will push it downwards. For \(g(x) = -5\sin(x - \frac{\pi}{2}) + 3\), the value of \(D\) is \(3\), indicating that the entire graph is shifted up by 3 units. Therefore, this moves the midline of the sine wave above the x-axis by 3 units.