The Relationship Between Functions and Their Reflections

Oct 13, 2025 by windailys.com 104 views

Introduction

In mathematics, functions are used to describe the relationship between variables. When dealing with functions, it's essential to understand the properties and behaviors of these functions, including their reflections. In this article, we will explore the relationship between any point on the graph of a function ${$ f(x) $}$ and its reflection ${$ -f(-x)$}$. We will examine the properties of these functions and determine the statement that best describes the true relationship between them.

What is a Function?

A function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the range. It is a way of describing a relationship between variables, where each input corresponds to exactly one output. Functions can be represented graphically, with the input variable on the x-axis and the output variable on the y-axis.

Reflections of Functions

A reflection of a function is a new function that is obtained by reflecting the original function about a specific line or axis. In this case, we are interested in the reflection of a function ${$ f(x) $}$ about the y-axis. This reflection is denoted as ${$ -f(-x)$}$.

Properties of Reflections

When a function is reflected about the y-axis, the x-coordinates of the points on the graph are negated, while the y-coordinates remain the same. This means that if ${$ (x, y)$}$ is a point on the graph of ${$ f(x) $}$, then ${$ (-x, y)$}$ is a point on the graph of ${$ -f(-x)$}$.

The Relationship Between Functions and Their Reflections

Now that we have a better understanding of functions and their reflections, let's examine the relationship between any point on the graph of a function ${$ f(x) $}$ and its reflection ${$ -f(-x)$}$. We can see that if ${$ (x, y)$}$ is a point on the graph of ${$ f(x) $}$, then ${$ (x, y)$}$ is also on the graph of ${$ -f(-x)$}$.

Why is This True?

This is true because the reflection of a function about the y-axis is simply a new function that is obtained by negating the x-coordinates of the points on the graph of the original function. Since the y-coordinates remain the same, the points on the graph of the original function and its reflection are the same.

Conclusion

In conclusion, the statement that best describes the true relationship between any point on the graph of a function ${$ f(x) $}$ and its reflection ${$ -f(-x)$}$ is:

If ${$ (x, y)$}$ is on the graph of ${$ f(x) $}$, then ${$ (x, y)$}$ is on the graph of ${$ -f(-x)$}$.

This is because the reflection of a function about the y-axis is simply a new function that is obtained by negating the x-coordinates of the points on the graph of the original function, while keeping the y-coordinates the same.

Example

Let's consider an example to illustrate this concept. Suppose we have a function ${$ f(x) = x^2 $}$. The graph of this function is a parabola that opens upwards. If we reflect this function about the y-axis, we get a new function ${$ -f(-x) = -(-x)^2 = -x^2 $}$. The graph of this new function is also a parabola, but it opens downwards.

Graphical Representation

To visualize this concept, let's consider the graph of the function ${$ f(x) = x^2 $}$. The graph of this function is a parabola that opens upwards. If we reflect this function about the y-axis, we get a new function ${$ -f(-x) = -(-x)^2 = -x^2 $}$. The graph of this new function is also a parabola, but it opens downwards.

Graphical Representation of the Relationship

To illustrate the relationship between the original function and its reflection, let's consider the graph of the function ${$ f(x) = x^2 $}$. The graph of this function is a parabola that opens upwards. If we reflect this function about the y-axis, we get a new function ${$ -f(-x) = -(-x)^2 = -x^2 $}$. The graph of this new function is also a parabola, but it opens downwards.

Graphical Representation of the Relationship Between Points

To illustrate the relationship between points on the graph of the original function and its reflection, let's consider the graph of the function ${$ f(x) = x^2 $}$. The graph of this function is a parabola that opens upwards. If we reflect this function about the y-axis, we get a new function ${$ -f(-x) = -(-x)^2 = -x^2 $}$. The graph of this new function is also a parabola, but it opens downwards.

Graphical Representation of the Relationship Between Points on the Graph of the Original Function and Its Reflection

Q: What is the relationship between a function and its reflection about the y-axis?

A: The reflection of a function about the y-axis is a new function that is obtained by negating the x-coordinates of the points on the graph of the original function, while keeping the y-coordinates the same.

Q: How do you determine if a point is on the graph of a function and its reflection?

A: To determine if a point is on the graph of a function and its reflection, you need to check if the x-coordinate of the point is negated when reflected about the y-axis. If the x-coordinate is negated, then the point is on the graph of the reflection.

Q: What is the relationship between the original function and its reflection about the y-axis?

A: The original function and its reflection about the y-axis are related in that they have the same y-coordinates, but the x-coordinates are negated in the reflection.

Q: How do you graph the reflection of a function about the y-axis?

A: To graph the reflection of a function about the y-axis, you need to negate the x-coordinates of the points on the graph of the original function, while keeping the y-coordinates the same.

Q: What is the difference between the original function and its reflection about the y-axis?

A: The original function and its reflection about the y-axis differ in that the x-coordinates are negated in the reflection, while the y-coordinates remain the same.

Q: Can you give an example of a function and its reflection about the y-axis?

A: Yes, consider the function ${$ f(x) = x^2 $}$. The reflection of this function about the y-axis is ${$ -f(-x) = -(-x)^2 = -x^2 $}$.

Q: How do you determine the equation of the reflection of a function about the y-axis?

A: To determine the equation of the reflection of a function about the y-axis, you need to negate the x-coordinates of the points on the graph of the original function, while keeping the y-coordinates the same.

Q: What is the significance of the reflection of a function about the y-axis?

A: The reflection of a function about the y-axis is significant because it helps to understand the properties and behaviors of the function, including its symmetry and periodicity.

Q: Can you give an example of a real-world application of the reflection of a function about the y-axis?

A: Yes, consider the motion of a particle under the influence of a force. The reflection of the position function about the y-axis can be used to model the motion of the particle in the opposite direction.

Q: How do you use the reflection of a function about the y-axis in calculus?

A: The reflection of a function about the y-axis is used in calculus to determine the derivative and integral of the function. The reflection can also be used to determine the symmetry and periodicity of the function.

Q: What are some common mistakes to avoid when working with the reflection of a function about the y-axis?

A: Some common mistakes to avoid when working with the reflection of a function about the y-axis include:

Negating the y-coordinates instead of the x-coordinates
Failing to keep the y-coordinates the same
Not checking for symmetry and periodicity

Q: How do you check for symmetry and periodicity in a function and its reflection about the y-axis?

A: To check for symmetry and periodicity in a function and its reflection about the y-axis, you need to examine the graph of the function and its reflection. If the graph of the function and its reflection are the same, then the function is symmetric about the y-axis. If the graph of the function and its reflection are the same after a certain period, then the function is periodic.

Q: Can you give an example of a function that is symmetric about the y-axis?

A: Yes, consider the function ${$ f(x) = x^2 $}$. The graph of this function is symmetric about the y-axis.

Q: Can you give an example of a function that is periodic?

A: Yes, consider the function ${$ f(x) = \sin(x) $}$. The graph of this function is periodic with a period of ${$ 2\pi $}$.

Q: How do you determine the period of a function?

A: To determine the period of a function, you need to examine the graph of the function and find the smallest interval after which the graph repeats itself.

Q: What is the significance of the period of a function?

A: The period of a function is significant because it helps to understand the behavior and properties of the function, including its symmetry and periodicity.

Q: Can you give an example of a real-world application of the period of a function?

A: Yes, consider the motion of a particle under the influence of a force. The period of the position function can be used to model the motion of the particle in a periodic manner.