Problem 16 - Contemporary Precalculus

Question

List the transformations needed to change the graph of \(f(t)\) into the graph of \(g(t) .\) ISee Section 3.4 .1 $$f(t)=\tan t ; \quad g(t)=-\tan t$$

Step-by-Step Solution

Answer

Question: List the transformations needed to change the graph of \(f(t) = \tan t\) into the graph of \(g(t) = -\tan t\). Answer: Reflect the graph of \(f(t) = \tan t\) about the x-axis.

Step 1: Identify the transformation type

Compare the two functions to determine the transformation type. In this case, we can see that \(g(t) = -\tan t\) has a negative sign compared to \(f(t) = \tan t\). This indicates that we need to consider a reflection transformation.

Step 2: Determine the direction of reflection

Since the negative sign is applied to the entire function, we can determine that the reflection will be performed about the x-axis. We can write this transformation as: $$g(t) = -1 \cdot f(t)$$

Step 3: List the transformations

Now that we have identified the transformation type and direction, we can list the transformations needed to change the graph of \(f(t)\) into the graph of \(g(t)\) as follows: 1. Reflect the graph of \(f(t) = \tan t\) about the x-axis to obtain the graph of \(g(t) = -\tan t\).

Key Concepts

Reflection Transformation

A reflection transformation is like flipping a mirror image of a graph. This concept is used when we want to change the position of a graph while maintaining its shape and size. Imagine placing a mirror along a line on the graph. When you could see a new, flipped version of the original graph in this mirror, you are seeing a reflection. The transformation requires identifying the line over which to reflect. Typically, this line could be the x-axis, y-axis, or any other specific line. Reflections change the direction of the graph. If the graph was moving upwards, after a reflection over the x-axis, it will now move downwards. Similarly, reflecting over the y-axis will switch the left and right sides. Reflection transformations are straightforward once you know the line of reflection. Simply negate the coordinates perpendicular to the reflecting line to save time, and you'll have your reflected graph.

Trigonometric Functions

Trigonometric functions are fundamental in describing wave-like oscillations and periodic behavior in mathematics and the natural world. The main trigonometric functions include sine, cosine, and tangent, which are defined based on the ratios of the sides of a right triangle relative to its angles.The tangent function, which we have in our example, oscillates between positive and negative values. It is periodic, repeating its values in regular intervals of \(\pi \). When graphing the tangent function, we see a series of vertical asymptotes where the function is undefined, and these separate the different 'waves' of the tangent graph. Understanding the behavior of trigonometric functions is crucial not only in mathematics but also in engineering, physics, and many fields where cyclic and oscillatory patterns occur.

X-Axis Reflection

The x-axis reflection is a specific type of reflection transformation that involves flipping the graph over the horizontal line known as the x-axis.Here's how it works:

• Take any function, for example, \(f(t)\).
• When reflecting over the x-axis, multiply the entire function by \(-1\) to get the new function \(g(t) = -f(t)\).

This process changes the sign of all the y-values of the graph, effectively flipping it upside down.For instance, the graph of \(\tan t\) becomes \(-\tan t\). Peaks become troughs and vice versa, while the location of the characteristic asymptotes remains the same.Understanding x-axis reflections helps in visualizing how functions behave when inverted around a horizontal line, especially in trigonometric graphs where symmetry plays a large role.