Problem 45 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

Graph the function. \(g(x)=-\sin (x-\pi)+4\)

Step-by-Step Solution

Answer

The graph of \(g(x)=-\sin(x-\pi)+4\) is a sine wave, oscillating from 3 to 5, with a horizontal shift of \(\pi\) units to the right and an inversion due to the negative factor.

Step 1: Identify the Basic Function

The basic function here is the sine function, \(\sin x\), which is a periodic function oscillating between -1 and 1. Its graph is a wave that crosses the origin (0,0) and repeats itself every \(2\pi\) units.

Step 2: Identify the Transformations

The given function \(g(x)=-\sin (x - \pi) + 4\) contains three transformations: a reflection about the x-axis, a horizontal shift to the right by \(\pi\) units, and a vertical shift upward by 4 units. Reflection about x-axis changes the sine wave from a hump to a valley or vice versa. An upward shift moves the wave upward in such a way it oscillates around a line other than the x-axis.

Step 3: Apply the Transformations and Graph the Function

Start by reflecting the basic sine function about the x-axis. Then shift all points \(\pi\) units to the right, representing the phase shift, then shift the graph upwards by 4 units. Maintain the periodicity of \(\sin x\) which is \(2\pi\).

Key Concepts

Graph Transformations

Graph transformations are a way to modify the appearance of a graph without altering its fundamental nature. These transformations involve shifting, stretching, compressing, or reflecting a graph. For the function, \(g(x)=-\sin(x-\pi)+4\), three transformations are involved:

• Reflection about the x-axis: This takes the sine wave and flips it upside down, turning peaks into troughs and vice versa. Mathematically, multiplying the function by -1 accomplishes this.
• Horizontal Shift: A horizontal shift involves moving the entire graph left or right along the x-axis. For our function, the sine wave shifts to the right by \(\pi\) units due to the \(x-\pi\) term.
• Vertical Shift: This moves the graph up or down along the y-axis. In this function, the wave is moved upward by 4 units as indicated by the \(+4\).

Each transformation distinctly alters the graph's position but preserves its general shape as a sine wave. Practicing these transformations helps in mastering the creation of more complex graphs.

Periodic Functions

Periodic functions repeat their values at regular intervals, known as the period. The sine function is a prominent example, characterized by its regular, repeating pattern.
In the case of \(g(x)=-\sin(x-\pi)+4\), the periodic nature remains unchanged even after transformations. The sine function has a period of \(2\pi\), meaning the wave repeats every \(2\pi\) units. When transformations occur:

• Reflection and Shifts: These changes affect the position and orientation on a graph but do not alter the fundamental period.
• Amplitude and Midline: The vertical shift to 4 does not impact the periodicity, but it changes the average value around which the function oscillates.

Understanding periodic functions and their transformations helps in analyzing real-world phenomena like sound waves, tides, and seasonal temperatures, all of which exhibit a periodic behavior.

Sine Waves

Sine waves are smooth, periodic oscillations that occur widely in nature and technology. The basic function \(\sin x\) produces a wave-like graph, essential in the study of trigonometric functions. Sine waves have distinct characteristics:

• Amplitude: The height from the centerline to the peak of the wave. Transformations like reflection and shifting do not alter amplitude but change other properties.
• Frequency: Related to the period, it indicates how often the wave repeats in a given interval. For \(\sin(x)\), the frequency is determined by dividing \(2\pi\) by the period.

When transforming \(\sin x\) to \(g(x)=-\sin(x-\pi)+4\), the wave's appearance shifts, but it remains a sine wave. Its peaks, valleys, and repeated nature make it reliable for understanding phenomena like sound and light waves. Grasping sine waves equips you with knowledge applicable to diverse scientific disciplines.