Problem 12 - 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 $$f(t)=\cos t ; \quad g(t)=\cos t-2$$

Step-by-Step Solution

Answer

Answer: The transformation needed is a vertical shift downwards by 2 units.

Step 1: Identify the main transformation from \(f(t)\) to \(g(t)\).

Comparing the two functions, we can see that \(g(t) = f(t) - 2\). This indicates that there is a vertical shift downwards by 2 units involved in the transformation.

Step 2: List the transformations

Considering the transformation found in Step 1, we can list the transformations needed to change the graph of \(f(t)\) into the graph of \(g(t)\): 1. Vertical shift downwards by 2 units. There are no other transformations applied to the function \(f(t) = \cos t\) in order to obtain \(g(t) = \cos t - 2\).

Key Concepts

Vertical Shift

Understanding the concept of a vertical shift is crucial when studying the behavior of functions, especially in the realm of trigonometry. A vertical shift refers to moving a function's graph up or down along the y-axis, without altering its shape. This transformation is expressed as adding or subtracting a constant value to the function.
For example, if we have the cosine function, denoted by \( f(t) = \text{cos}(t) \), a vertical shift can be applied by subtracting 2, resulting in a new function \( g(t) = f(t) - 2 \). This transformation will move the entire graph of the cosine function down by two units on the y-axis. No matter the value of \( t \), every corresponding y-value of \( f(t) \) will be decreased by 2, creating the graph of \( g(t) \).

Trigonometric Functions

Trigonometric functions are a foundational concept in mathematics that describe the relationships between the angles and sides of triangles as well as representing periodic phenomena. The primary trigonometric functions include sine, cosine, and tangent, each with unique graphs showcasing their periodic nature.
Trigonometric functions are particularly important when analyzing waves, oscillations, and circular motion in various fields such as physics, engineering, and even economics. These functions are defined on the unit circle, where the angle \( t \) corresponds to the length of the arc on the circle. Understanding how to transform these functions, including through vertical shifts, allows us to model and predict a wide range of real-world behaviors.

Cosine Function

The cosine function, one of the most significant trigonometric functions, is traditionally written as \( \text{cos}(t) \). It represents the x-coordinate of a point on the unit circle as it corresponds to an angle \( t \), measured in radians. The cosine function is even, periodic, and oscillates between 1 and -1.

Properties of the Cosine Function

• Period: \( 2\text{π} \) (or 360 degrees)
• Amplitude: 1
• Range: [ -1, 1 ]
• Principal period: \( t \text{ from } 0 \text{ to } 2\text{π} \)

Modifying the cosine function graphically can be done in several ways including horizontal shifts, reflections, stretching, compressing, and the vertical shift. These transformations morph the function to fit specific requirements, like in the transformation from \( f(t) \text{ to } g(t) \) in our exercise.

Function Transformation

Function transformation encompasses the various ways a function's graph can be altered to produce a new graph. These transformations include shifting, stretching, compressing, and reflecting the graph both horizontally and vertically.

Common Transformations Include:

• Vertical Shifts: Moving the graph up or down.
• Horizontal Shifts: Moving the graph left or right.
• Vertical Stretching: Stretching the graph away from the x-axis.
• Vertical Compression: Squeezing the graph toward the x-axis.
• Reflections: Flipping the graph over an axis or line.

In the context of our exercise, we observe a vertical shift in the transformation from \( f(t) = \text{cos}(t) \) to \( g(t) = \text{cos}(t) - 2 \). This vertical displacement by 2 units downwards demonstrates a simple yet profound instance of function transformation.