Problem 87 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the function. \(f(x)=\arcsin (x-2)\)

Step-by-Step Solution

Answer

The graph of the function \(f(x) = \arcsin(x-2)\) is a shift of the graph of \(f(x) = \arcsin(x)\) 2 units to the right. It is defined for x between 2 and 3.

Step 1: Understand the arcsine function

The arcsine of x, denoted as \(\arcsin(x)\) or \(\sin^{-1}(x)\), is the inverse function of the sine function. It returns the angle whose sine is \(x\). The domain is the set of all real numbers between -1 and 1, and the output is all real numbers between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\). It has its vertical asymptotes at \(x = -1\) and \(x = 1\).

Step 2: Understand the shift of the function

The graph of \(f(x) = \arcsin(x)\) is shifted to the right by 2 units to get the graph of \(f(x) = \arcsin(x-2)\). This is because the graph of \(f(x) = \arcsin(x-h)\) is a horizontal shift of the graph \(f(x) = \arcsin(x)\) h units to the right.

Step 3: Graph the function

Use your graphing utility to graph the function \(f(x) = \arcsin(x-2)\). Make sure to set the window to show the entire graph. The x-values should include -3 to 3, and y-values should include \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\).

Key Concepts

Inverse Trigonometric Functions

The arcsine function is part of a group known as inverse trigonometric functions. These include \(\arcsin(x)\), \(\arccos(x)\), and \(\arctan(x)\), among others. The purpose of these functions is to determine the angle that produces a given trigonometric ratio. For example, \(\arcsin(x)\) finds the angle whose sine is \(x\). The domain of \(\arcsin(x)\) is limited to values between -1 and 1 due to the nature of the sine function, which also has this range. Its range is similarly restricted to values between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \) radians, which corresponds to angles in quadrants I and IV. This limitation ensures that \(\arcsin(x)\) is a proper function with single outputs for each input.

Horizontal Shifts

Horizontal shifts change the position of the graph along the x-axis without affecting its shape. When a function like \(f(x) = \arcsin(x-h)\) is given, the parameter \(h\) dictates the shift. If \(h\) is positive, the graph shifts to the right; if negative, to the left. For example, \(f(x) = \arcsin(x-2)\) signifies a right shift by 2 units from \(f(x) = \arcsin(x)\). This means that if the original graph started at x = -1, the shifted graph commences at x = 1. Despite this shift, the peak and trough of the graph remain between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\), as they are determined by the input to \(\arcsin\), not simply \(x\).

Graphing Utilities

Graphing utilities are tools that assist us in visually interpreting functions, providing a clearer understanding than what mathematical expressions alone can offer. These tools range from calculators to sophisticated software. When graphing a function like \(f(x) = \arcsin(x-2)\), the steps involved include:

• Setting an appropriate window: For this function, set x-values to span from -3 to 3 to capture the horizontal shift.
• Adjusting y-values to range from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) ensures the full vertical stretch is visualized.

These setups help in portraying the complete behavior of the function, highlighting shifts and even asymptotes. Engaging with graphing utilities can greatly boost your understanding of mathematical concepts by seeing them in action on a graph.