Problem 23 - Algebra 2

Question

Graph each function in the interval from 0 to 2\(\pi\) $$ y=\cos (x-\pi)-3 $$

Step-by-Step Solution

Answer

The graph of the function \(y=\cos(x-\pi)-3\) in the interval [0, \(2\pi\)] begins at -2 followed by a decrease to -4 at \(x=\pi\) and again increases to -2 at \(x=2\pi\).

Step 1: Understand the base function

Start with the base function, which is \(y=\cos(x)\). The cosine function oscillates between -1 and 1 with a period of \(2\pi\). For the interval \(0 \leq x \leq 2\pi\), it starts from 1 at \(x=0\), reaches -1 at \(x=\pi\), and returns to 1 at \(x=2\pi\) completing one full cycle.

Step 2: Apply horizontal shift

The term inside the cosine function, \(x-\pi\), implies a horizontal shift. The graph is shifted \(\pi\) units to the right as we replace \(x\) with \(x-\pi\). So now, the lowest point (-1) will be at \(x=\pi\) and the function would complete a full cycle at \(x=3\pi\). However, only the part till \(x=2\pi\) is to be considered here.

Step 3: Apply vertical shift

The term -3 outside the cosine function implies a vertical shift. The entire graph is moved 3 units down. Consequently, the function oscillates between -4 and -2.

Step 4: Draw the graph

After considering the transformations, the graph starts from -2 at \(x=0\), reaches -4 at \(x=\pi\), increases to -2 at \(x=2\pi\).

Key Concepts

Cosine Function

The cosine function is a fundamental concept in trigonometry. It is often denoted as \( y = \cos(x) \).
The graph of the basic cosine function shows a wave-like pattern that repeats every \( 2\pi \) radians. This repeating pattern is called periodicity.
The points on this graph oscillate between -1 and 1, starting at 1 when \( x = 0 \). This is known as the amplitude of the function.

• Starts at its maximum point (1) at \( x=0 \).
• Dips to its minimum point (-1) at \( x=\pi \).
• Returns to its maximum point (1) by \( x=2\pi \).

The basic cosine function is symmetric around the y-axis. This symmetry helps in understanding and predicting the behavior of more complex trigonometric functions.

Function Transformations

A function transformation involves modifying a basic function to change its shape, position, or orientation.
For trigonometric functions like cosine, transformations can include translations (shifts), stretching, compressing, and reflections.
In drafting a transformed graph, it's crucial to follow these specific operations step by step:

• Translation: Shifts the graph horizontally and/or vertically.
• Stretching: Changes the amplitude, either expanding or compressing the graph vertically.
• Reflection: Flips the graph across the x-axis or y-axis.

The original exercise primarily focuses on translations. We handle horizontal shifts by adjusting the input variable \( x \), and vertical shifts by directly adding or subtracting from the output value (the function result).
Keep this process in mind when tweaking the base function to get the correct graph.

Vertical and Horizontal Shifts

Vertical and horizontal shifts change the position of a graph in the coordinate plane.
In the equation \( y=\cos(x-\pi)-3 \), we observe both types of shifts:

• Horizontal Shift: The term \( x-\pi \) indicates a rightward shift by \( \pi \) units. This means every x-value on the cosine graph increases by \( \pi \) units, altering the graph's position along the x-axis.


• Vertical Shift: The term \(-3\) outside the cosine function shows a downward shift by 3 units, affecting the entire graph's position along the y-axis. Now, the cosine graph oscillates between -4 and -2.

By understanding these shifts, we can predict how the graph will look without necessarily plotting every point. With horizontal and vertical transformations, it's key to adjust the graph according to both the new starting point and the oscillation boundaries.