Problem 19 - 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 $$f(t)=\sin t ; \quad g(t)=3 \sin t+2$$

Step-by-Step Solution

Answer

Answer: The transformations are a vertical stretch by a factor of 3 and a vertical shift upwards by 2 units.

Step 1: Identify the given functions

We are given the following functions: $$f(t) = \sin t$$ $$g(t) = 3\sin t + 2$$

Step 2: Analyze the differences between the two functions

To determine the transformations, we need to compare \(f(t)\) and \(g(t)\). Let's analyze each term in \(g(t)\) to understand its effect on the graph of \(f(t)\): 1. In \(g(t)\), the \(\sin t\) term is multiplied by a constant, \(3\). This has the effect of stretching the graph of \(f(t)\) vertically by a factor of \(3\). 2. In \(g(t)\), a constant, \(2\), is added to the transformed term, \(3\sin t\). This has the effect of shifting the graph of \(f(t)\) vertically upwards by \(2\) units.

Step 3: List the transformations

Based on our analysis, we can list the following transformations to be applied to the graph of \(f(t)\) in order to obtain the graph of \(g(t)\): 1. Vertical stretch by a factor of \(3\). 2. Vertical shift upwards by \(2\) units.

Key Concepts

Trigonometric Functions

Trigonometric functions, like the sine function in our exercise, are fundamental elements of mathematics that describe relationships between the sides and angles of triangles. They also model periodic phenomena in many areas such as physics, engineering, and even music. The function \(f(t) = \text{sin } t\) represents the sine wave, which has a standard period of \(2\pi\) radians, with its values oscillating between -1 and +1.

When working with trigonometric functions, it's important to understand how various transformations can alter their graphs. These transformations can stretch, compress, shift, or reflect the graph without changing its fundamental shape or periodic nature. Comprehending these effects is key to mastering the analysis and application of trigonometric functions in various fields.

Vertical Stretch

A vertical stretch is one of the transformations that can be applied to the graph of a trigonometric function. It alters the amplitude, which is the height of the peaks and depths of the troughs, from the baseline value.

In our exercise, the presence of the coefficient \(3\) in the function \(g(t) = 3\sin t\) indicates a vertical stretch by a factor of three. This means that each point on the graph of \(f(t) = \text{sin } t\) will be pulled away from or pushed towards the \(t\)-axis by three times the original value of the sine function. The result is a taller graph, with peaks and troughs thrice as far from the centerline as before, but the period remains unchanged.

Vertical Shift

In addition to stretching or compressing graphs, trigonometric functions can also be shifted vertically. A vertical shift moves the entire graph up or down along the vertical axis without affecting its shape or periodicity.

The term \(+2\) in the function \(g(t) = 3\sin t + 2\) of our exercise demonstrates a vertical shift. The number \(2\) added to the trigonometric function indicates that every point of the original sinusoidal graph is to be moved up by two units. This shift doesn't change where the graph starts its cycle (the phase) or how frequently the cycles occur (the frequency), but it does translate the midpoint up by two units, relocating the central axis of the sine wave.