Properties of the New Function

Oct 13, 2025 by windailys.com 106 views

Introduction

In mathematics, transformations of graphs are essential concepts that help us understand how functions behave under various operations. When we apply transformations to a graph, we can create new functions that have different properties and characteristics. In this article, we will explore the transformation of the graph of a function $f(x) = \sqrt{x} + 7$ to create a new function $g(x)$ .

The Original Function

The original function is given by $f(x) = \sqrt{x} + 7$ . This function represents a square root function shifted 7 units upwards. The graph of this function is a curve that opens upwards, with the vertex at the point (0, 7).

The Transformation

The graph of $g(x)$ is a translation 4 units down of the graph of $f(x)$ . This means that we need to shift the graph of $f(x)$ 4 units downwards to obtain the graph of $g(x)$ . To do this, we need to subtract 4 from the function $f(x)$ .

The New Function

Let's denote the new function as $g(x)$ . Since the graph of $g(x)$ is a translation 4 units down of the graph of $f(x)$ , we can write the function $g(x)$ as:

g(x) = f(x) - 4

Substituting the expression for $f(x)$ , we get:

g(x) = \sqrt{x} + 7 - 4

Simplifying the expression, we get:

g(x) = \sqrt{x} + 3

Properties of the New Function

The new function $g(x) = \sqrt{x} + 3$ has several properties that are different from the original function $f(x)$ . The graph of $g(x)$ is also a square root function, but it is shifted 3 units upwards instead of 7 units. This means that the vertex of the graph of $g(x)$ is at the point (0, 3).

Comparison with the Original Function

Let's compare the properties of the new function $g(x)$ with the original function $f(x)$ . Both functions are square root functions, but they have different shifts. The graph of $f(x)$ is shifted 7 units upwards, while the graph of $g(x)$ is shifted 3 units upwards.

Conclusion

In conclusion, we have seen how to transform the graph of a function $f(x) = \sqrt{x} + 7$ to create a new function $g(x)$ . The new function $g(x)$ is a translation 4 units down of the graph of $f(x)$ . We have also seen how to write the new function $g(x)$ in terms of the original function $f(x)$ . This demonstrates the importance of understanding transformations of graphs in mathematics.

Key Takeaways

The graph of $g(x)$ is a translation 4 units down of the graph of $f(x)$ .
The new function $g(x)$ can be written as $g(x) = f(x) - 4$ .
The graph of $g(x)$ is a square root function shifted 3 units upwards.
The vertex of the graph of $g(x)$ is at the point (0, 3).

Further Exploration

This article has provided a basic understanding of how to transform the graph of a function to create a new function. However, there are many other types of transformations that can be applied to graphs, such as reflections, rotations, and scaling. Exploring these transformations can help deepen our understanding of functions and their graphs.

References

[1] "Functions and Graphs" by Michael Sullivan
[2] "Mathematics for the Nonmathematician" by Morris Kline

Glossary

Transformation: A change in the position, size, or shape of a graph.
Translation: A transformation that moves a graph a certain distance in a specific direction.
Square Root Function: A function that takes the square root of a number as its input.
Vertex: The point on a graph where the function changes from increasing to decreasing or vice versa.
Q&A: Understanding Transformations of Graphs =============================================

Introduction

In our previous article, we explored the transformation of the graph of a function $f(x) = \sqrt{x} + 7$ to create a new function $g(x)$ . We saw how to write the new function $g(x)$ in terms of the original function $f(x)$ and how to identify the properties of the new function. In this article, we will answer some frequently asked questions about transformations of graphs.

Q: What is a transformation of a graph?

A transformation of a graph is a change in the position, size, or shape of the graph. This can include translations, reflections, rotations, and scaling.

Q: What is a translation of a graph?

A translation of a graph is a transformation that moves the graph a certain distance in a specific direction. For example, if we translate the graph of $f(x)$ 4 units down, the new function $g(x)$ will be $g(x) = f(x) - 4$ .

Q: How do I identify the properties of a new function after a transformation?

To identify the properties of a new function after a transformation, you need to analyze the transformation that was applied to the original function. For example, if the graph of $f(x)$ was translated 4 units down, the new function $g(x)$ will have the same shape as $f(x)$ but will be shifted 4 units down.

Q: Can I apply multiple transformations to a graph?

Yes, you can apply multiple transformations to a graph. For example, if you want to translate the graph of $f(x)$ 4 units down and then reflect it across the x-axis, you can apply the transformations in the following order:

Translate the graph of $f(x)$ 4 units down: $g(x) = f(x) - 4$
Reflect the graph of $g(x)$ across the x-axis: $h(x) = -g(x)$

Q: How do I write a new function after a transformation?

To write a new function after a transformation, you need to analyze the transformation that was applied to the original function. For example, if the graph of $f(x)$ was translated 4 units down, the new function $g(x)$ will be $g(x) = f(x) - 4$ .

Q: What are some common transformations of graphs?

Some common transformations of graphs include:

Translations: moving the graph a certain distance in a specific direction
Reflections: flipping the graph across a line or axis
Rotations: rotating the graph around a point or axis
Scaling: changing the size of the graph

Q: How do I determine the type of transformation that was applied to a graph?

To determine the type of transformation that was applied to a graph, you need to analyze the graph and identify the changes that were made. For example, if the graph was translated 4 units down, you can see that the vertex of the graph was shifted 4 units down.

Q: Can I apply transformations to graphs in a specific order?

Yes, you can apply transformations to graphs in a specific order. For example, if you want to translate the graph of $f(x)$ 4 units down and then reflect it across the x-axis, you can apply the transformations in the following order:

Translate the graph of $f(x)$ 4 units down: $g(x) = f(x) - 4$
Reflect the graph of $g(x)$ across the x-axis: $h(x) = -g(x)$

Conclusion

In conclusion, transformations of graphs are an essential concept in mathematics. By understanding how to apply transformations to graphs, you can create new functions and analyze the properties of those functions. We hope that this Q&A article has helped you understand transformations of graphs better.

Key Takeaways

A transformation of a graph is a change in the position, size, or shape of the graph.
Translations, reflections, rotations, and scaling are common transformations of graphs.
You can apply multiple transformations to a graph in a specific order.
To write a new function after a transformation, you need to analyze the transformation that was applied to the original function.

Further Exploration

This article has provided a basic understanding of transformations of graphs. However, there are many other types of transformations that can be applied to graphs, such as complex transformations and parametric transformations. Exploring these transformations can help deepen your understanding of functions and their graphs.

References

[1] "Functions and Graphs" by Michael Sullivan
[2] "Mathematics for the Nonmathematician" by Morris Kline

Glossary

Transformation: A change in the position, size, or shape of a graph.
Translation: A transformation that moves a graph a certain distance in a specific direction.
Reflection: A transformation that flips a graph across a line or axis.
Rotation: A transformation that rotates a graph around a point or axis.
Scaling: A transformation that changes the size of a graph.