Translation of the Graph of f

Oct 13, 2025 by windailys.com 28 views

Introduction

In mathematics, absolute value functions are a fundamental concept in algebra and calculus. These functions are used to model real-world problems, such as distance, speed, and temperature. In this article, we will explore the graphs of two absolute value functions, f(x) = |x| and g(x) = |x + 3| + 3, and discuss how the graph of g is a translation of the graph of f.

The Graph of f(x) = |x|

The graph of f(x) = |x| is a V-shaped graph that opens upwards. It has a minimum point at (0, 0) and is symmetric about the y-axis. The graph consists of two linear segments: one from (-∞, 0) to (0, 0) and another from (0, 0) to (∞, 0). The absolute value function f(x) = |x| can be written as:

f(x) = |x| = x if x ≥ 0 f(x) = |x| = -x if x < 0

The Graph of g(x) = |x + 3| + 3

The graph of g(x) = |x + 3| + 3 is also a V-shaped graph that opens upwards. However, it is shifted 3 units to the left and 3 units upwards compared to the graph of f(x) = |x|. This means that the minimum point of the graph of g(x) = |x + 3| + 3 is at (-3, 3), and it is also symmetric about the y-axis. The graph of g(x) = |x + 3| + 3 can be written as:

g(x) = |x + 3| + 3 = x + 3 if x ≥ -3 g(x) = |x + 3| + 3 = -x - 3 if x < -3

Translation of the Graph of f

The graph of g(x) = |x + 3| + 3 is a translation of the graph of f(x) = |x| 3 units to the right and 3 units upward. This means that if we shift the graph of f(x) = |x| 3 units to the right and 3 units upward, we will obtain the graph of g(x) = |x + 3| + 3.

Proof of Translation

To prove that the graph of g(x) = |x + 3| + 3 is a translation of the graph of f(x) = |x|, we can use the following steps:

Shift the graph of f(x) = |x| 3 units to the right: To do this, we replace x with x - 3 in the equation f(x) = |x|, which gives us f(x - 3) = |x - 3|.
Shift the graph of f(x - 3) = |x - 3| 3 units upward: To do this, we add 3 to the equation f(x - 3) = |x - 3|, which gives us f(x - 3) + 3 = |x - 3| + 3.
Simplify the equation f(x - 3) + 3 = |x - 3| + 3: By simplifying the equation, we get f(x - 3) + 3 = |(x - 3) + 3| + 3 = |x| + 3.

Therefore, we have shown that the graph of g(x) = |x + 3| + 3 is a translation of the graph of f(x) = |x| 3 units to the right and 3 units upward.

Conclusion

In conclusion, the graph of g(x) = |x + 3| + 3 is a translation of the graph of f(x) = |x| 3 units to the right and 3 units upward. This means that if we shift the graph of f(x) = |x| 3 units to the right and 3 units upward, we will obtain the graph of g(x) = |x + 3| + 3. The graph of g(x) = |x + 3| + 3 is also a V-shaped graph that opens upwards and is symmetric about the y-axis.

Real-World Applications

The concept of translations of graphs is used in various real-world applications, such as:

Physics: Translations of graphs are used to model the motion of objects in physics. For example, the graph of the position of an object as a function of time can be translated to represent the motion of the object.
Engineering: Translations of graphs are used in engineering to design and analyze systems. For example, the graph of the stress on a material as a function of strain can be translated to represent the behavior of the material under different loads.
Computer Science: Translations of graphs are used in computer science to represent and analyze data. For example, the graph of the number of users of a website as a function of time can be translated to represent the growth of the website.

Final Thoughts

Introduction

In our previous article, we explored the graphs of two absolute value functions, f(x) = |x| and g(x) = |x + 3| + 3, and discussed how the graph of g is a translation of the graph of f. In this article, we will answer some frequently asked questions about the graphs of absolute value functions.

Q: What is the graph of f(x) = |x|?

A: The graph of f(x) = |x| is a V-shaped graph that opens upwards. It has a minimum point at (0, 0) and is symmetric about the y-axis. The graph consists of two linear segments: one from (-∞, 0) to (0, 0) and another from (0, 0) to (∞, 0).

Q: What is the graph of g(x) = |x + 3| + 3?

A: The graph of g(x) = |x + 3| + 3 is also a V-shaped graph that opens upwards. However, it is shifted 3 units to the left and 3 units upwards compared to the graph of f(x) = |x|. This means that the minimum point of the graph of g(x) = |x + 3| + 3 is at (-3, 3), and it is also symmetric about the y-axis.

Q: How do you prove that the graph of g(x) = |x + 3| + 3 is a translation of the graph of f(x) = |x|?

A: To prove that the graph of g(x) = |x + 3| + 3 is a translation of the graph of f(x) = |x|, we can use the following steps:

Shift the graph of f(x) = |x| 3 units to the right: To do this, we replace x with x - 3 in the equation f(x) = |x|, which gives us f(x - 3) = |x - 3|.
Shift the graph of f(x - 3) = |x - 3| 3 units upward: To do this, we add 3 to the equation f(x - 3) = |x - 3|, which gives us f(x - 3) + 3 = |x - 3| + 3.
Simplify the equation f(x - 3) + 3 = |x - 3| + 3: By simplifying the equation, we get f(x - 3) + 3 = |(x - 3) + 3| + 3 = |x| + 3.

Q: What are some real-world applications of translations of graphs?

A: The concept of translations of graphs is used in various real-world applications, such as:

Physics: Translations of graphs are used to model the motion of objects in physics. For example, the graph of the position of an object as a function of time can be translated to represent the motion of the object.
Engineering: Translations of graphs are used in engineering to design and analyze systems. For example, the graph of the stress on a material as a function of strain can be translated to represent the behavior of the material under different loads.
Computer Science: Translations of graphs are used in computer science to represent and analyze data. For example, the graph of the number of users of a website as a function of time can be translated to represent the growth of the website.

Q: How do you determine the translation of a graph?

A: To determine the translation of a graph, you need to identify the horizontal and vertical shifts of the graph. The horizontal shift is the distance the graph is shifted to the left or right, and the vertical shift is the distance the graph is shifted up or down.

Q: What are some common types of translations of graphs?

A: Some common types of translations of graphs include:

Horizontal shift: A horizontal shift is a translation of the graph to the left or right.
Vertical shift: A vertical shift is a translation of the graph up or down.
Reflection: A reflection is a translation of the graph across a line.
Rotation: A rotation is a translation of the graph around a point.

Conclusion

In conclusion, the graph of g(x) = |x + 3| + 3 is a translation of the graph of f(x) = |x| 3 units to the right and 3 units upward. This concept is used in various real-world applications, such as physics, engineering, and computer science. The understanding of translations of graphs is essential in mathematics and has many practical applications.

Final Thoughts

In this article, we have answered some frequently asked questions about the graphs of absolute value functions. We have discussed the graph of f(x) = |x|, the graph of g(x) = |x + 3| + 3, and how the graph of g is a translation of the graph of f. We have also discussed some real-world applications of translations of graphs and how to determine the translation of a graph.