Problem 24 - Calculus with Concepts in Calculus

Question

Sketch the graph of the function. Indicate any intercepts and symmetry, and determine whether the function is even, odd, or neither. $$ \cos \frac{x-\pi}{2} $$

Step-by-Step Solution

Answer

The function \( f(x) = \cos \frac{x-\pi}{2} \) is even, with x-intercepts at \( x = \pi + 2k\pi, \, k \in \mathbb{Z} \). No y-intercept.

Step 1: Understand the function

The given function is \( f(x) = \cos \frac{x-\pi}{2} \). It is a transformation of the cosine function.

Step 2: Identify the transformations

The expression \( \frac{x-\pi}{2} \) represents a horizontal shift and scaling of the cosine function. Specifically, it implies a shift of \( \pi \) units to the right and a horizontal stretching by a factor of 2.

Step 3: Determine intercepts

To find the intercepts, set \( x \) such that \( f(x) = 0 \). For cosine, \( \cos \theta = 0 \) when \( \theta = \frac{\pi}{2} + k\pi \), where \( k \) is an integer. Solving \( \frac{x-\pi}{2} = \frac{\pi}{2} + k\pi \) gives the x-intercepts. The function does not cross the y-axis because as \( x = 0 \), \( f(x) = \cos(-\frac{\pi}{2}) = 0 \).

Step 4: Check symmetry

Substitute \(-x\) into \( f(x) \): \( f(-x) = \cos \left( \frac{-x-\pi}{2} \right) \). Notice that \( \cos(\theta) \) is an even function, hence \( f(-x) = f(x) \). This implies the function is even.

Step 5: Sketch the graph

Using the information from the previous steps, sketch the graph. The graph is a horizontally stretched cosine wave shifted \( \pi \) units to the right. It retains the cosine wave pattern but has a period doubled due to the factor of 2 transformation.

Key Concepts

Cosine Function

The cosine function is one of the fundamental trigonometric functions often written as \( \cos \theta \). It represents the x-coordinate of a point on the unit circle corresponding to an angle \( \theta \).
The range of the cosine function is between \(-1\) and \(1\), and it is periodic, repeating its values every \(2\pi\) radians. The basic cosine function, \(y = \cos x\), starts at \(y = 1\) when \(x = 0\), reaches \(y = 0\) at \(x = \frac{\pi}{2}\), goes to \(y = -1\) at \(x = \pi\), and returns to \(y = 1\) at \(x = 2\pi\).
Cosine is also symmetric with respect to the y-axis, which is a key characteristic of even functions. Understanding these properties helps us predict and sketch transformations of the cosine graph, such as a shift or stretch.

Graph Transformations

Graph transformations change the look of a graph by altering its shape, position, or size. They include shifts, stretches, compressions, and reflections.
For example, consider the function \( f(x) = \cos \frac{x-\pi}{2} \). This involves two types of transformations on the basic cosine graph:

• **Horizontal shift:** Subtracting \(\pi\) inside the cosine function shifts the graph \(\pi\) units to the right.
• **Horizontal stretching:** Dividing the input by \(2\) stretches the function horizontally, doubling its period to \(4\pi\).

Together, these transformations result in a graph that is stretched and shifted. This is important in trigonometry as it allows us to model real-world phenomena such as sound waves and light waves, which may not always follow a simple cosine pattern.

Symmetry in Functions

Symmetry in functions describes how a graph can be mirrored or rotated in a way that it remains unchanged. For trigonometric functions like cosine, symmetry helps determine if a function is even or odd, pivotal for simplifying complex calculations.
An **even function** is one that satisfies \( f(-x) = f(x) \) for all \( x \). The cosine function is a classic example of an even function, exhibiting symmetry about the y-axis.
This means that if you flip the graph over the y-axis, it would look the same. In the function \( f(x) = \cos \frac{x-\pi}{2} \), substituting \(-x\) results in the same function, confirming its even nature.Recognizing symmetry can aid in graphing, solving equations, and integrating functions more easily. Understanding symmetry helps to grasp the intrinsic patterns of these functions, paving the way for deeper insights into mathematical relationships.