Problem 21 - 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 $$f(t)=\sin t ; \quad g(t)=\sin (t-2)$$

Step-by-Step Solution

Answer

Answer: A horizontal shift of 2 units to the right is needed.

Step 1: Identify the base function of both graphs

Both graphs have the sine function as their base function, \(f(t) = \sin{t}\) and \(g(t) = \sin{(t-2)}\).

Step 2: Identify any horizontal or vertical shifts

Since the sine function in \(g(t)\) has \((t-2)\) inside the parentheses, there is a horizontal shift of 2 units to the right. There are no vertical shifts.

Step 3: Identify any horizontal or vertical scaling

There are no coefficients in front of the \(t\) inside the sine function and no coefficients in front of the sine function itself in both \(f(t)\) and \(g(t)\). Therefore, there are no horizontal or vertical scaling.

Step 4: Identify any reflections

Both functions have a positive sine function. Thus, there are no reflections in the graph.

Step 5: List all transformations

To transform the graph of \(f(t)\) into the graph of \(g(t)\), the following transformation is needed: 1. Horizontal shift of 2 units to the right.

Key Concepts

Sine Function Graph

The sine function, represented as \( y = \sin(x) \), is one of the fundamental waveforms that appears frequently not only in mathematics but also in the sciences, particularly in fields like physics and engineering. The graph of the sine function is characterized by its smooth, periodic waves that repeat every \(2\pi\) radians. Each cycle consists of a crest at \(y = 1\), a trough at \(y = -1\), and intersects the horizontal axis at \(y = 0\).

This wavelike pattern is an example of simple harmonic motion and is used to model many types of periodic phenomena, such as sound waves and the motion of a pendulum. In our scenario, the base function \(f(t) = \sin(t)\) represents a pure sine wave with a period of \(2\pi\) and an amplitude of 1, which means it oscillates between \(1\) and \( -1\) without any scaling or stretching.

Horizontal Shift

A horizontal shift in the context of function graphs is a type of transformation that moves the graph to the left or to the right along the horizontal axis without changing its shape or orientation. When dealing with the sine function, a horizontal shift changes the phase of the wave, meaning it affects where the cycle of the wave starts.

To understand this, consider a function \( g(t) = \sin(t - c) \), where \(c\) is a constant. If \( c > 0 \), the graph of the function will shift \( c \) units to the right. Conversely, if \( c < 0 \), the graph will shift \( |c| \) units to the left. In the case of our exercise, the function \( g(t) = \sin(t-2) \) exhibits a horizontal shift of 2 units to the right, meaning the wave starts its cycle later compared to the original \( f(t) \) function.

Function Transformation

Function transformation encompasses various operations that alter the graph of a function, including shifting, scaling, and reflecting. These modifications enable the creation of a multitude of graphs from a single base function. When a function undergoes a transformation, its general shape is preserved; what changes are its position, size, and orientation.

In the context of precalculus, understanding function transformation is crucial as it allows for the visualization of complex equations and the prediction of their behavior. For a sine function, transformations can adjust the period (how often the wave repeats per unit along the x-axis), amplitude (the height of the waves), phase (where the cycle starts), and vertical position (where the center of the wave sits along the y-axis). However, in the solution provided, our function \(g(t) \) only exhibits a horizontal shift, leaving the amplitude, period, and vertical position unchanged from the base function \(f(t)\).