Problem 42 - Precalculus : Building Concepts and Connections

Question

Sketch the graph of each of the given expressions. $$f(x)=-3 \arcsin x$$

Step-by-Step Solution

Answer

The sketch of the function \(f(x)=-3 \arcsin x\) is obtained by plotting the points (0,0), (-1, 3*\(\pi/2\)), and (1,-3*\(\pi/2\)), then creating a smooth, decreasing curve that connects these points.

Step 1: Define the region

The inverse sine function \(\arcsin x\) is only defined when \(-1 \leq x \leq 1\). Hence, any value of \(x\) outside this range will not be part of the domain of the function.

Step 2: Identify the input-output relation

For any \(x\) in the interval [-1,1], the output of the function \(\arcsin x\) is the value whose sine is \(x\). The range of \(\arcsin x\) is normally \(-\pi/2 \leq y \leq \pi/2\). However, the given function is \(f(x) = -3 \arcsin x\), so this output will be multiplied by -3. Therefore, the range will change as well: \(-3*\pi/2 \leq -3*y \leq 3*\pi/2\). Notice that the direction of the inequality changes due to the multiplication of -3. Hence, the range of \(f(x)\) is \(-3*\pi/2 \leq f(x) \leq 3*\pi/2\).

Step 3: Create the graph

There are a few significant points to note at this stage: \((0,0)\), \((-1, 3*\pi/2)\), and \((1,-3*\pi/2)\). Now create the cartesian plane with x-axis ranging from -1 to 1 and y-axis from -3*\(\pi/2\) to 3*\(\pi/2\). Plot these points and sketch the function. The function decreases as \(x\) increases. We can connect the points with a smooth curve, maintaining the decreasing trend. Due to function properties, the graph will be a reflection of the typical \(\arcsin x\) graph around the x-axis, stretched vertically by a factor of 3.

Key Concepts

Arcsin Function Graph

Understanding the graph of the arcsin function, or inverse sine function, is crucial for visualizing how angles are related to ratios in right triangles. It's the arc whose sine is the number you're given. When graphing the arcsin function, remember that it only takes values from -1 to 1 and returns angles (in radians), which reflect the principal values of the inverse sine.

The key features of the standard arcsin graph include a domain of \[ -1, 1 \], and a range of \[ -\frac{\pi}{2}, \frac{\pi}{2} \]. It's a curve that starts at (-1, -\frac{\pi}{2}), rises to (0, 0), and then ascends to (1, \frac{\pi}{2}). When graphing \(f(x) = -3 \arcsin x\), we implement a vertical stretch by a factor of three and reflect it across the x-axis, leading to a steeper decline from left to right.

To plot \(f(x)\), we identify significant points, such as those where x is -1, 0, or 1, and note that the curve follows a smooth, continuous path. By preserving the general shape of the arcsin curve and adapting it based on the transformation applied (in this case, the stretch and reflection), we can sketch an accurate graph of \(f(x)\).

Domain and Range of Inverse Trigonometric Functions

Inverse trigonometric functions invert the process of their respective trigonometric functions, which means that they output an angle when given a ratio. The most important aspect to remember regarding these functions is their domain (input values) and range (output values).

Let's break down the specifics:

• The domain of the arcsin function is from -1 to 1. This is because the sine of an angle can only be within this interval.
• The range of arcsin is limited to \[ -\frac{\pi}{2}, \frac{\pi}{2} \] because of the definition of the principal value of this inverse function, which only outputs angles in this range to ensure a single value for each input.

Moreover, when transformations like scaling or reflecting are applied to the function, the range and sometimes the domain may shift accordingly. In the specific function of \(f(x) = -3 \arcsin x\), while the domain remains unchanged, the range is affected by the vertical stretch by a factor of 3 and the reflection, resulting in a new range of \[ -3\frac{\pi}{2}, 3\frac{\pi}{2} \].

Transformation of Trigonometric Graphs

Trigonometric graphs can be transformed through various operations, such as translations, reflections, and scaling. These transformations change the appearance of the graph but maintain the original function's overall shape.

Here's how transformations typically work:

• Translations move the graph up, down, left, or right, without altering the graph's shape.
• Reflections flip the graph across the x-axis or y-axis. For instance, multiplying the function by -1 reflects it across the x-axis.
• Scaling involves stretching or compressing the graph. Multiplying the function by a constant greater than one stretches the graph, while a constant between zero and one compresses it.

In the example \(f(x) = -3 \arcsin x\), we notice a reflection over the x-axis accompanied by a vertical stretch by a factor of 3. This means that for every point on the standard arcsin graph, you move it three times further away from the x-axis and flip it across that axis. Recognizing these transformations allows students to predict the general behavior and shape of the graph, even before plotting specific points.