Problem 43 - Algebra and Trigonometry Real Mathematics, Real People

Question

Compare the graph of the function with the graph of \(f(x)=\arctan x\) \(g(x)=-\arctan x-3\)

Step-by-Step Solution

Answer

The transformation of the function \(f(x) = \arctan x\) to \(g(x) = -\arctan x - 3\) involves reflecting over the x-axis and shifting downwards by 3 units. These transformations make the curve of \(g(x)\) fall as x goes from \(-\infty\) to \(+\infty\) with asymptotes at \(y = -3\).

Step 1: Identify the transformations

First, recognize that the function \(g(x)\) is obtained from the parent function \(f(x) = \arctan x\) by reflecting it over the x-axis (due to the negative sign) and shifting it downwards by 3 (due to the '-3'). These are the two transformations applied to \(f(x)\) to get \(g(x)\).

Step 2: Reflected graph

Reflecting the graph of \(f(x) = \arctan x\) over the x-axis results in a curve that falls from \(+\infty\) to 0 as \(x\) goes from \(-\infty\) to \(+\infty\). This is due to the negative sign on \(\arctan x\).

Step 3: Shifted graph

After reflection, shift the graph downwards by 3 units to account for the '-3'. Now, the curve still falls as \(x\) goes from \(-\infty\) to \(+\infty\), but the asymptotes occur at \(y = 0 - 3 = -3\) instead of \(y = 0\).

Key Concepts

Function Transformations

For instance, the exercise provided presents the transformation of the function \(f(x)=\arctan x\) into \(g(x)=-\arctan x-3\). In the steps solution, two main transformations were detected: reflecting the graph across the x-axis and shifting it vertically. Understanding these operations and their effects can significantly assist students in visualizing and plotting the behavior of the transformed function on a coordinate plane.

Reflecting Graphs

Reflecting graphs is a type of transformation that involves flipping a graph over a specific line, such as the x-axis or y-axis. Reflection over the x-axis changes the sign of all the y-coordinates of points on the parent graph, which essentially flips the graph upside down, while reflection over the y-axis changes the sign of all the x-coordinates, mirroring the graph.

Let's take the function \(g(x)=-\arctan x\) as an example from the exercise. The negative sign in front of \(\arctan x\) indicates a reflection over the x-axis. Imagining the original graph of \(f(x)=\arctan x\) flipping vertically gives us the reflected graph's shape. This reflection does not change the x-intercepts or horizontal asymptotes, but it does invert the rise and fall of the curve, which is crucial for accurately sketching the graph.

Vertical Shifts

Vertical shifts are straightforward yet powerful transformations. They occur when we add or subtract a constant value to the function's output. This causes the entire graph to move up or down on the coordinate plane. In the context of our exercise, the \(\arctan x\) function is being shifted down by 3 units due to the '-3' at the end of the function \(g(x)=-\arctan x-3\).

It's essential to realize that vertical shifts do not affect the shape of the graph; they only alter its position. The asymptotes, axis intercepts, and general direction of the curve all move in the same manner. Knowing how vertical shifts work can be helpful when drawing the new function's graph relative to the original parent function.

Inverse Trigonometric Functions

Inverse trigonometric functions serve as the counterpart to the familiar trigonometric functions such as sine, cosine, and tangent. They are used to find the angles when the ratios of the sides in a right triangle are known. The \(\arctan x\) function, for instance, gives the angle whose tangent is \(x\).

These functions come with a specific set of properties such as domain and range restrictions, which result in their unique graphs. The function \(f(x)=\arctan x\) gradually increases and has horizontal asymptotes, indicative of the fact that tangent has an infinite number of periods but arctangent is restricted to provide only one value for each x. When transforming the graph of \(\arctan x\), such as by reflecting and shifting as seen in the exercise, it's critical to apply these modifications while keeping in mind the nature of the original graph to maintain its integrity.