Question

Write a rule for \(g\) that represents the indicated transformations of the graph of \(f\).. f(x)=3 \sin x \text {; translation } 2 \text { units up and } \pi \text { units right }

Short answer

The transformed function that represents the given transformations of the graph of the original function is \(g(x) = 3\sin(x - \pi) + 2\)

Step-by-step solution

Identify the given function

The problem identifies the function to work with as \(f(x)=3\sin x\), an amplified sine function. It's important to note that this function oscillates between -3 and 3 rather than the standard -1 to 1.

Understand the desired transformation

It is desired to transform the function \(f(x)\) by shifting it 2 units upwards and \(\pi\) units to the right. To shift the graph 2 units up, add 2 to the original function. To shift the function \(\pi\) units to the right, subtract \(\pi\) from the original function's argument.

Write the transformed function

Now that there is an understanding of the transformations, we apply those to the given function. Replacing \(x\) by \(x - \pi\) in the argument of the sine function accounts for the shift to the right, and adding 2 to the entire function results in the final transformed function being \(g(x) = 3\sin(x - \pi) + 2\)

Key concepts

Sine Function

The sine function, denoted as \( \sin(x) \), is a fundamental trigonometric function that originates from the study of right-angled triangles and describes a repeating wave-like pattern. This function is periodic, with a period of \( 2\pi \) radians, which means the sine function repeats its values every \( 2\pi \) radians as you move along the x-axis. The standard sine wave oscillates between -1 and 1. In mathematics or physics, the sine function is often used to model periodic phenomena such as sound waves, light waves, or seasonal changes.

When the sine function is multiplied by a coefficient, as in \( f(x)=3\sin x \), the amplitude—or the height of the wave's peak and depth of its trough—is affected. In this example, the amplitude is increased to 3, making the function oscillate between -3 and 3. This alteration plays a critical role in how the function behaves and appears when graphed.

Horizontal Shift

A horizontal shift in mathematics refers to the movement of a graph along the x-axis. To achieve a horizontal shift, you would subtract or add a constant to the input variable, \( x \), of the function. If you subtract a positive value, like \( \pi \) in the case of \( f(x) = 3\sin(x - \pi) \), the graph shifts to the right. Conversely, adding a value would shift the graph to the left.

A key note to remember is that horizontal shifts do not change the shape of the graph; they only move it from side to side. This type of transformation can be critical when aligning graphs to model or fit certain data points in fields such as engineering or data analysis.

Vertical Shift

In contrast to a horizontal shift, a vertical shift moves a graph up or down along the y-axis. This is achieved by adding or subtracting a constant to the entire function. For instance, adding 2 to the sine function in \( g(x) = 3\sin(x - \pi) + 2 \) results in each point on the graph being moved up two units.

Vertical shifts can be seen as changing the baseline from which the function oscillates. They are especially useful when you want to adjust the starting point or the mean position of a wave-like function without affecting its periodic nature.

Function Transformation

Function transformation involves altering a function's graph in various ways, such as shifting, stretching, compressing, or reflecting it. These changes can be understood as modifications to the function's formula which result in a new graph. To transform a function, mathematics offers a set of rules or operations that you can apply to the original function's equation.

In the provided exercise, the sine function went through a horizontal shift to the right and a vertical shift upward. Both these transformations contribute to the new function \( g(x) = 3\sin(x - \pi) + 2 \). Understanding these fundamental transformations is crucial for students tackling more complex mathematical problems, as these are the building blocks for manipulating and interpreting functions in various fields of study and real-world applications.