Problem 44 - Algebra and Trigonometry Real Mathematics, Real People

Question

Compare the graph of the function with the graph of \(f(x)=\arctan x\) \(g(x)=\arctan (-x)+4\)

Step-by-Step Solution

Answer

The graph of \( g(x)=\arctan(-x)+4 \) is identical to the graph of \( f(x)=\arctan x \), except it is reflected over the y-axis and shifted upwards by 4 units. The function \( g(x) \) decreases instead of increasing and the y-values tend to \( 4 -\frac{\pi}{2} \) and \( 4 + \frac{\pi}{2} \) as \( x \) approaches \( \infty \) and \( -\infty \) respectively.

Step 1: Sketch the Graph of the Parent Function

Start by sketching the graph of the parent function \(f(x) = \arctan x\). The function \( \arctan x \) is an increasing function that tends to \( \frac{\pi}{2} \) as \( x \rightarrow \infty \) and to \( -\frac{\pi}{2} \) as \( x \rightarrow -\infty \).

Step 2: Sketch the Graph of the Transformed Function

The next step is to sketch the transformed function \( g(x) = \arctan (-x)+4 \). Realize that \( -x \) will reflect the graph of \( \arctan x \) over the y-axis because each x-coordinate value is negated. Furthermore, adding 4 will shift the graph up by 4 units. Therefore, the shape of the graph will be the same as \( \arctan x \) but reflected over the y-axis and shifted upwards by 4 units.

Step 3: Analyze the Comparison

Analyzing the comparison, the reflection over the y-axis of the graph of \( \arctan x \) results in an overall inversion of the graph about the y-axis. The graph of \( \arctan(-x) \) is a decreasing function that tends to \( -\frac{\pi}{2} \) as \( x \rightarrow \infty \) and to \( \frac{\pi}{2} \) as \( x \rightarrow -\infty \). Adding the 4 shifts all these y-values up by 4, so the graph of \( g(x) \) tends to \( 4 -\frac{\pi}{2} \) as \( x \rightarrow \infty \) and to \( 4 + \frac{\pi}{2} \) as \( x \rightarrow -\infty \).

Key Concepts

Graph Transformations

Graph transformations involve changing the appearance of a graph without altering its fundamental shape. They are a powerful tool to study function behavior and manipulate graphs to meet specific criteria. There are several types of graph transformations, including:

• Translations: Shifting the graph either horizontally or vertically.
• Reflections: Flipping the graph over a specified axis.
• Stretching and Shrinking: Altering the dimensions of the graph.

When working with trigonometric functions like the arctangent function, transformations can provide a deeper insight into their properties and behavior. In the case of the function given, transformations make it easier to understand related behavior, such as shifts and reflections, which are necessary to grasp the full range of the modified function \(g(x) = \arctan(-x) + 4\). This highlights how the function's general shape stays consistent, even when it's modified through reflections and vertical shifts.

Reflection of Functions

Reflection in functions is a kind of transformation where the graph is flipped over an axis. For the function \( \arctan(-x) \), reflection occurs over the y-axis.

• Reflection over the y-axis: Each x-value of the original function is negated, yielding \( -x \). This changes the direction in which the graph moves. If a function is increasing, as \( \arctan x \) is, its reflection \( \arctan(-x) \) becomes a decreasing function.
• For the arctangent function, reflection turns it from tending towards \( \frac{\pi}{2} \) as \( x \rightarrow \infty \) to \( -\frac{\pi}{2} \) in the same condition once reflected.

Reflections are fundamentally about direction changes. They make an increasing behavior decrease and vice versa. This explains why \( \arctan(-x) \) looks flipped compared to \( \arctan x \). It offers a reversed view on approaching the same asymptotes from opposite directions.

Vertical Shifts

Vertical shifts adjust the graph of a function by moving it up or down without changing its orientation or shape.

• In the case of \( g(x) = \arctan(-x)+4 \), the graph of \( \arctan(-x) \) is raised vertically by 4 units. This means every point on the graph is shifted upward by 4 units along the y-axis.
• Effect on asymptotes: The shift affects asymptotic behavior as it moves the asymptotes up by four units — instead of \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\), they become \( 4 - \frac{\pi}{2} \) and \( 4 + \frac{\pi}{2} \) respectively.

Vertical shifts do not invert or stretch the graph; they only alter its vertical position. This is particularly useful when working with the arctangent function, as simple shifts can have significant implications on the graphical representation of the function in its altered form.

Arctangent Function

The arctangent function, represented as \( \arctan x \), is one of the inverse trigonometric functions. It is the inverse of the tangent function, meaning it provides the angle whose tangent is a given number.

• Graphical characteristics: The graph of \( \arctan x \) is a smooth curve that increases from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\).
• Asymptotic behavior: As \( x \rightarrow \infty \), the function approaches \( \frac{\pi}{2} \); as \( x \rightarrow -\infty \), it approaches \(-\frac{\pi}{2} \).

Understanding the arctangent function is key when performing transformations, as its behavior strongly influences the behavior of related functions derived from it, such as \( g(x) = \arctan(-x) + 4 \). Each transformation (reflection, shift) builds on the basic increase and asymptotic nature of \( \arctan x \), making it vital to understand its inclination and general path across the Cartesian plane.