Writing the Equation of the Original Function

Oct 13, 2025 by windailys.com 96 views

The Equation y = 2x - 1: Understanding Translation and Graphing

In mathematics, functions are essential tools for modeling real-world phenomena. A function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range). The equation y = 2x - 1 is a linear function that has been translated from its original form. In this article, we will explore the concept of translation in functions, write the equation of the original function, determine the number of units and direction of translation, and sketch the graphs of the two functions.

Translation is a fundamental concept in mathematics that involves shifting a function's graph horizontally or vertically. When a function is translated, its graph moves to a new position without changing its shape or size. There are two types of translations: horizontal and vertical.

Horizontal Translation: A horizontal translation occurs when the graph of a function is shifted to the left or right. This type of translation is represented by the equation y = f(x - h), where h is the number of units the graph is shifted.
Vertical Translation: A vertical translation occurs when the graph of a function is shifted up or down. This type of translation is represented by the equation y = f(x) + k, where k is the number of units the graph is shifted.

Writing the Equation of the Original Function

To write the equation of the original function, we need to reverse the translation. Since the given equation is y = 2x - 1, which is a vertical translation of the original function, we can write the equation of the original function as y = 2x + 1.

Determining the Number of Units and Direction of Translation

To determine the number of units and direction of translation, we need to compare the given equation y = 2x - 1 with the equation of the original function y = 2x + 1. By comparing the two equations, we can see that the given equation has been translated 1 unit down.

Sketching the Graphs of the Two Functions

To sketch the graphs of the two functions, we need to plot the points on a coordinate plane. The original function y = 2x + 1 is a linear function with a slope of 2 and a y-intercept of 1. The given function y = 2x - 1 is also a linear function with a slope of 2, but it has been translated 1 unit down.

Graph of the Original Function

The graph of the original function y = 2x + 1 is a straight line with a positive slope. The y-intercept is 1, which means that the graph passes through the point (0, 1). The graph has a slope of 2, which means that for every 1 unit increase in x, the graph increases by 2 units in y.

Graph of the Given Function

The graph of the given function y = 2x - 1 is also a straight line with a positive slope. However, it has been translated 1 unit down, which means that the graph passes through the point (0, -1). The graph still has a slope of 2, but it is shifted down by 1 unit.

In conclusion, the equation y = 2x - 1 describes a function that has been translated. By writing the equation of the original function and determining the number of units and direction of translation, we can understand the concept of translation in functions. Sketching the graphs of the two functions helps us visualize the translation and its effect on the graph.

Key Takeaways

Translation is a fundamental concept in mathematics that involves shifting a function's graph horizontally or vertically.
The equation y = 2x - 1 is a linear function that has been translated 1 unit down from its original form.
To write the equation of the original function, we need to reverse the translation.
Sketching the graphs of the two functions helps us visualize the translation and its effect on the graph.

Q: What is translation in functions?

A: Translation is a fundamental concept in mathematics that involves shifting a function's graph horizontally or vertically. When a function is translated, its graph moves to a new position without changing its shape or size.

Q: What are the two types of translations?

A: There are two types of translations: horizontal and vertical.

Horizontal Translation: A horizontal translation occurs when the graph of a function is shifted to the left or right. This type of translation is represented by the equation y = f(x - h), where h is the number of units the graph is shifted.
Vertical Translation: A vertical translation occurs when the graph of a function is shifted up or down. This type of translation is represented by the equation y = f(x) + k, where k is the number of units the graph is shifted.

Q: How do I determine the number of units and direction of translation?

A: To determine the number of units and direction of translation, you need to compare the given equation with the equation of the original function. By comparing the two equations, you can see the number of units and direction of translation.

Q: What is the equation of the original function?

A: To write the equation of the original function, you need to reverse the translation. Since the given equation is y = 2x - 1, which is a vertical translation of the original function, the equation of the original function is y = 2x + 1.

Q: How do I sketch the graphs of the two functions?

A: To sketch the graphs of the two functions, you need to plot the points on a coordinate plane. The original function y = 2x + 1 is a linear function with a slope of 2 and a y-intercept of 1. The given function y = 2x - 1 is also a linear function with a slope of 2, but it has been translated 1 unit down.

Q: What are some real-world applications of translation in functions?

A: Translation in functions has many real-world applications, such as:

Physics: Translation is used to describe the motion of objects in physics.
Engineering: Translation is used to design and optimize systems in engineering.
Computer Science: Translation is used in computer graphics and game development.

In conclusion, translation in functions is a fundamental concept in mathematics that involves shifting a function's graph horizontally or vertically. By understanding the concept of translation, we can better analyze and solve problems involving functions and their graphs. We hope this Q&A article has helped you understand the concept of translation in functions.

Key Takeaways

Translation is a fundamental concept in mathematics that involves shifting a function's graph horizontally or vertically.
There are two types of translations: horizontal and vertical.
To determine the number of units and direction of translation, you need to compare the given equation with the equation of the original function.
Translation in functions has many real-world applications.