Problem 33 - Algebra 2

Question

Write an equation for each translation. \(y=\sin x, 3\) units up

Step-by-Step Solution

Answer

The equation for the sine function translated 3 units up is \(y=\sin x + 3\)

Step 1: Identify original function

The original function is given by \(y=\sin x\)

Step 2: Apply translation

A vertical translation is an addition or subtracting of a constant to the original function. Here the function is translated 3 units up, meaning we add 3 to the original function.

Step 3: Write the translated function

The new function will be written as \(y = \sin x + 3\), which is the original function translated 3 units up

Key Concepts

Sine Function

The sine function is a fundamental concept in trigonometry. It is typically introduced as a function of an angle, producing values that oscillate between -1 and 1. This oscillation makes it ideal for modeling periodic phenomena such as waves. The sine function is defined for any real number, and it's periodic with a period of

• Amplitude: It denotes the height of the wave from its central axis, usually 1 in its simple form ( \(y = \sin x\)).
• Period: The length to complete one full cycle of the wave, which is \(2\pi\) for the sine function.
• Basic Form: The simplest form of the sine function is \(y = \sin x\), where \(x\) is the angle in radians.

Understanding the sine function's properties is crucial for analyzing transformations and their effects on the graph.

Function Translation

Function translation involves shifting the entire graph of a function in a specific direction. This can happen either horizontally or vertically. Horizontal translation affects the \(x\)-coordinates, while vertical translation affects the \(y\)-coordinates of the points on a graph. Here's how it works:

• Horizontal Translation: Adjusts the graph left or right by modifying the \(x\) values. For example, \( y = \sin(x - c) \) translates the sine function \(c\) units to the right.
• Vertical Translation: Shifts the graph up or down by modifying the \(y\) values. For instance, \(y = \sin x + c\) translates the graph \(c\) units upward.

Translations are straightforward yet powerful transformations, allowing flexibility in modeling how functions move along the coordinate plane.

Vertical Translation

Vertical translation modifies the vertical position of a graph. In the case of the sine function, a vertical translation shifts the curve up or down without altering its shape.To vertically translate a function:

• Add a positive constant to move the graph up. For example, \(y = \sin x + 3\) raises the sine wave by 3 units.
• Subtract a constant to move the graph down. For example, \(y = \sin x - 3\) lowers it by 3 units.

Vertical translation is useful for aligning functions to meet specific criteria, such as matching a graph to a particular set of data points.

Graph Transformations

Graph transformations include several techniques applied to modify a graph's size, shape, or position on the coordinate plane. These transformations can be categorized mainly into translations, stretches, compressions, and reflections. For sine functions:

• Translation involves shifting the graph up, down, left, or right.
• Stretching/Compressing affects the amplitude, making the wave taller or shorter. This is handled by multiplying the sine function by a constant.
• Reflection flips the graph across an axis, changing the direction of the wave.

Each of these transformations can be combined to create a variety of different-looking graphs from the basic sine wave. Such techniques are not only crucial in mathematics but also widely applicable in fields like physics and engineering for modeling real-world behavior.