Problem 38 - Precalculus: Functions and Graphs

Question

Sketch the graph of the equation. $$y=2 \cos ^{-1} x$$

Step-by-Step Solution

Answer

Sketch the line from ( -1, 2π ) to (1, 0).

Step 1: Understand the domain of the function

The function given is based on the inverse cosine function, denoted as \( \cos^{-1} x \). The inverse cosine function is defined for \( x \) values in the interval \([-1, 1]\). This means that \( x \) must be within this range for the function \( y = 2 \cos^{-1} x \) to be defined.

Step 2: Determine the range of the function

The range of \( \cos^{-1} x \) is \([0, \pi]\). Since the function is \( y = 2 \cos^{-1} x \), you multiply each value of \( \cos^{-1} x \) by 2. Therefore, the range for \( y \) becomes \([0, 2\pi]\). This means \( y \) values will vary from 0 to \( 2\pi \).

Step 3: Plot key points from the function

Select key points within the domain \([-1, 1]\) to evaluate and plot on the graph:- When \( x = -1 \), \( \cos^{-1}(-1) = \pi \), so \( y = 2\pi \).- When \( x = 0 \), \( \cos^{-1}(0) = \frac{\pi}{2} \), so \( y = \pi \).- When \( x = 1 \), \( \cos^{-1}(1) = 0 \), so \( y = 0 \).

Step 4: Connect the points and analyze the behavior

Since the function \( \cos^{-1} x \) is a decreasing function from \( 0 \) to \( \pi \) as \( x \) increases from \(-1\) to \( 1 \), the graph of \( y = 2\cos^{-1} x \) will be a straight line if plotted on a linear scale. It descends from point \(( -1, 2\pi )\) to point \(( 1, 0 )\).

Key Concepts

Domain and Range of Functions

When working with functions, understanding the domain and range is crucial. The domain of a function is the complete set of possible values of the independent variable, usually represented by \( x \). For the inverse cosine function, \( \cos^{-1} x \), the domain is the interval \([-1, 1]\). This is because cosine itself is defined within these bounds, and inverse functions reverse that process. For any input \( x \), the output must be attainable within the real numbers that cosine hits first.The range, on the other hand, is the set of possible output values. For \( \cos^{-1} x \), the range is \([0, \pi]\), as the inverse function can only yield angles within this set. When modifying the function to \( y = 2 \cos^{-1} x \), the range expands to \([0, 2\pi]\). This transformation occurs because each \( y \) value is twice the original \( \cos^{-1} x \) values, reflecting an elongation of the graph vertically.

Transformations of Functions

Transformations are changes made to the basic graph of a function. They affect the position and shape of graphs on a coordinate plane. In the function \( y = 2\cos^{-1} x \), we see a vertical transformation.Vertical transformations occur when a function is multiplied by a constant. In our case, \( 2\cos^{-1} x \) indicates a vertical stretch. This doubles each value of \( y \), expanding the original range of the inverse cosine function from \([0, \pi]\) to \([0, 2\pi]\). Such multipliers impact the rate at which \( y \) values change, altering the function's visual output.Another concept is a shift. While \( y = 2\cos^{-1} x \) doesn't involve any shifts now, knowing that adding or subtracting a value to \( x \) could move a graph along the x-axis is helpful. Similarly, adding or subtracting from the function itself slides the graph up or down in the y-direction. Understanding these transformations helps when graphing not only simple inverse trigonometric functions but all variety of more complex expressions.

Graphing Functions

Graphing a function is essentially drawing its representation in a coordinate system. This visual interpretation provides insights into behavior, patterns, and tendencies of the function.To accurately sketch the graph of \( y = 2 \cos^{-1} x \), you begin by plotting key points. These points are derived through evaluating \( y \) at specific \( x \) values within the domain. From our example:

• When \( x = -1 \), \( y = 2\pi \)
• When \( x = 0 \), \( y = \pi \)
• When \( x = 1 \), \( y = 0 \)

Connecting these dots forms a segment of a line that descends steadily. This linear drop indicates that as \( x \) moves from \(-1\) to \(1\), \( y \) lessens from \( 2\pi \) to \( 0 \). Graphing accurately can capture this smooth descent, reflective of the inverse cosine's properties. By visualizing such functions, students can better discern the nature of trigonometric transformations and their broader implications.