Problem 34 - Precalculus: Functions and Graphs

Question

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

Step-by-Step Solution

Answer

The graph of \( y = \frac{1}{2} \sin^{-1} x \) is an arc passing through the points \((-1, -\frac{\pi}{4}), (0, 0), (1, \frac{\pi}{4})\) within \( x \in [-1, 1] \).

Step 1: Understand the Inverse Sine Function

The inverse sine function, denoted as \( \sin^{-1}(x) \) or \( \arcsin(x) \), is the function that 'undoes' sine. It gives the angle \( y \) whose sine is \( x \), for \( -1 \leq x \leq 1 \) and \( -\frac{\pi}{2} \leq y \leq \frac{\pi}{2} \). This forms the basis of our given equation.

Step 2: Modify Range through Scalar Multiplication

The given function is \( y = \frac{1}{2} \sin^{-1} x \). This means each value of \( \sin^{-1}(x) \) is multiplied by \( \frac{1}{2} \). Consequently, the range of \( y \) is now \( y \in \left(-\frac{\pi}{4}, \frac{\pi}{4}\right) \).

Step 3: Identify Key Points

It's helpful to determine a few key points to sketch the curve. At \( x = 0 \), \( y = \frac{1}{2} \times \sin^{-1}(0) = 0 \). At \( x = 1 \), \( y = \frac{1}{2} \times \frac{\pi}{2} = \frac{\pi}{4} \). Finally, at \( x = -1 \), \( y = \frac{1}{2} \times \left(-\frac{\pi}{2}\right) = -\frac{\pi}{4} \).

Step 4: Sketch the Graph

Plot the identified points and the range of \( x \, (-1 \leq x \leq 1) \) on a coordinate system. Draw a smooth curve passing through these points, reflecting that the function is continuous from one value to another within its domain. The curve should be increasing because the inverse sine function is increasing.

Key Concepts

Graphing Functions

Graphing a function involves plotting points on a coordinate grid and drawing a curve through these points that represents the equation. For the function \(y = \frac{1}{2} \sin^{-1} x\), understanding the inverse sine function is crucial. This inverse function accepts values from -1 to 1, producing output values or angles between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\).

• Start by determining key points that will guide your graph plotting.
• For \(x = 0\), calculate \(y = 0\), which gives you the origin point.
• For \(x = 1\), calculate \(y = \frac{\pi}{4}\).
• For \(x = -1\), calculate \(y = -\frac{\pi}{4}\).

Using these points:

• Plot them on a coordinate plane.
• Connect these points with a smooth curve.
• Recognize that this function graph is a transformation of the basic inverse sine graph.

Graphing functions helps visualize how inputs relate to outputs. It gives a clear picture of how the function behaves with changes in \(x\).

Scalar Multiplication

Scalar multiplication in the context of functions often refers to multiplying the output of the function by a scalar (constant number). In \(y = \frac{1}{2} \sin^{-1} x\), each value of \(\sin^{-1} x\) is multiplied by \(\frac{1}{2}\).

• This changes the amplitude of the inverse sine function.
• It narrows the range from \(-\frac{\pi}{2}, \frac{\pi}{2}\) to \(-\frac{\pi}{4}, \frac{\pi}{4}\).

Why multiply by a scalar?

• It stretches or compresses the function graph vertically.
• Helps in modeling real-world scenarios where proportionate relationships exist.

When you do scalar multiplication, visualize the graph shrinking or expanding in the vertical direction depending on whether the scalar is less than or greater than one.

Function Range

The range of a function is the set of all possible output values. For the function \(y = \frac{1}{2} \sin^{-1} x\), the basic inverse sine function \(\sin^{-1} x\) originally has a range of \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\).

• Due to the scalar multiplication by \(\frac{1}{2}\), the overall range adapts to \(-\frac{\pi}{4}, \frac{\pi}{4}\).
• Understanding range adjustments helps in mapping a function's output correctly.

The concept of range is critical when dealing with inverse trigonometric functions because these functions have specific beginning and ending values that need close attention.

• Knowing the range ensures that the function behaves as expected within its limits.
• It is essential when determining key points for graph plotting, ensuring you stay within the acceptable output values.

Precalculus

Precalculus serves as the bridge between algebra and calculus. It equips students with essential skills and concepts such as functions, graphs, and equation analysis before tackling calculus topics. In precalculus, students often encounter trigonometric functions and their transformations.

• Inverse trigonometric functions like \(\sin^{-1} x\) are foundational topics.
• Understanding transformations like scalar multiplication is part of precalculus training.
• This helps in analyzing how these change the function's appearance and behavior.

Precalculus builds a toolbox of skills including:

• Graphing a variety of functions, essential for visualizing mathematical equations.
• Understanding and applying mathematical transformations.
• Preparing students for the more complex ideas and calculations encountered in calculus.

Overall, mastering these concepts in precalculus, including inverse trigonometric functions, sets a strong foundation for future success in calculus and beyond.