Problem 110 - Algebra and Trigonometry

Question

Graph each pair of functions in the same viewing rectangle. Use your knowledge of the domain and range for the inverse trigonometric function to select an appropriate viewing rectangle. How is the graph of the second equation in cach exercise related to the graph of the first equation? $$ y=\sin ^{-1} x \text { and } y=\sin ^{-1} x+2 $$

Step-by-Step Solution

Answer

The graph of the equation \(y = \sin^{-1}(x) + 2\) is identical to the graph of the equation \(y = \sin^{-1}(x)\), except that it is shifted two units upward due to the addition of the constant 2.

Step 1: Graph the function \(y = \sin^{-1}(x)\)

First graph the function \(y = \sin^{-1}(x)\). This is the inverse sine function, also known as arcsine function. The domain of this function is -1 ≤ x ≤ 1, and its range is \(-\pi/2 ≤ y ≤ \pi/2\). In the graph, this will appear as a curve from \((-1, -\pi/2)\) to \((1, \pi/2)\).

Step 2: Graph the function \(y = \sin^{-1}(x) + 2\)

Next, graph the function \(y = \sin^{-1}(x) + 2\). Adding a constant to a function shifts the graph up or down. In this case, adding 2 shifts the graph 2 units up. Therefore, the graph of this function is the same as the previous graph but shifted upwards by two units.

Step 3: Determine the relationship between the two graphs

The functions \(y = \sin^{-1}(x)\) and \(y = \sin^{-1}(x) + 2\) are identical, except that the latter is shifted up by 2 units. This vertical shift results from the addition of the constant 2 to the \(y = \sin^{-1}(x)\) function.

Key Concepts

Domain and Range of Inverse Trigonometric Functions

Every function has a domain and range, which define the permissible inputs and possible outputs, respectively. When dealing with inverse trigonometric functions, such as the inverse sine function, it is crucial to understand these concepts. For \( y = \sin^{-1}(x) \), also known as the arcsine function, the domain represents all the possible \( x \) values that can be used as input. For arcsine, these values are restricted to \(-1 \leq x \leq 1\). This is because the sine function, which arcsine is derived from, has a range of \(-1\) to \(1\).

The range of \( y = \sin^{-1}(x) \) specifies the potential \( y \) values, which are outputted by the function. This range is limited to \( -\pi/2 \leq y \leq \pi/2 \), reflecting the principal values for the inverse sine function. It helps to always visualize the domain and range on a graph to see where the function can and cannot go. Knowing these limitations ensures that when graphing or analyzing the function, you're working within the correct boundaries.

Graph Transformations

Graph transformations include adjustments like translations, reflections, stretches, and compressions. They help us understand how the graph of a function can change. In the realm of trigonometric functions, these transformations are especially useful for analyzing variations of standard functions, like our arcsine function. When you apply a transformation to a graph, you adjust it according to specific rules.

For example, when considering the graph of \( y = \sin^{-1}(x) \) and \( y = \sin^{-1}(x) + 2 \), we observe a vertical shift. Here, the parent graph, \( y = \sin^{-1}(x) \), remains identical in shape and size but is adjusted vertically. Adding a constant to the function results in a transformation, without altering the function’s underlying structure.

Understanding these transformations allows you to quickly predict how the graph of a function will appear without recalculating every point. This skill is invaluable in both simple exercises and more complex real-world problems where visualizing data changes is crucial.

Vertical Shifts in Graphs

Vertical shifts occur when a constant value is added to or subtracted from a function, resulting in the whole graph moving up or down by that constant amount. This is a straightforward yet powerful transformation that directly affects a graph's position on the plane. For the given function \( y = \sin^{-1}(x) + 2 \), the graph experiences a vertical shift upwards by 2 units relative to \( y = \sin^{-1}(x) \).

Imagine you have the graph of \( y = \sin^{-1}(x) \). Each point on this graph moves vertically by two units, creating a parallel graph that's just 2 units higher. Importantly, this type of shift doesn't affect the domain and range of the function, but it does change the range of the graph output. Instead of \( -\pi/2 \leq y \leq \pi/2 \), the range becomes \(-\pi/2 + 2 \leq y \leq \pi/2 + 2\).

Visualizing vertical shifts helps with understanding transformations and their impact on graphical representations of functions, ensuring a clear comprehension of how equations translate directly to images.