Shifting the Function to the Right

Oct 13, 2025 by windailys.com 124 views

Introduction

In mathematics, function transformations are essential concepts that help us understand how functions change when they are shifted, stretched, or compressed. One of the most common transformations is the horizontal shift, where a function is moved to the left or right by a certain number of units. In this article, we will explore the concept of shifting a function to the right and apply it to the function $f(x)=x^2-6x+9$ . We will learn how to create a new function $g(x)$ by shifting $f(x)$ 5 units to the right.

Understanding the Function $f(x)=x^2-6x+9$

Before we dive into the transformation, let's take a closer look at the function $f(x)=x^2-6x+9$ . This is a quadratic function, which means it has a parabolic shape. The function can be written in the form $f(x)=a(x-h)^2+k$ , where $(h,k)$ is the vertex of the parabola. In this case, the vertex is $(3,0)$ , which means the function opens upwards and has a minimum value of 0 at $x=3$ .

Shifting the Function to the Right

To shift the function $f(x)$ 5 units to the right, we need to replace $x$ with $(x-5)$ . This is because when we shift a function to the right, we are essentially moving the graph of the function to the right by 5 units. By replacing $x$ with $(x-5)$ , we are effectively moving the graph of the function 5 units to the right.

Creating the New Function $g(x)$

Now that we have replaced $x$ with $(x-5)$ , we can create the new function $g(x)$ by plugging this expression into the original function $f(x)$ . This gives us:

g(x)=(x-5)^2-6(x-5)+9

Simplifying the Expression

To simplify the expression, we can expand the squared term and combine like terms:

g(x)=(x^2-10x+25)-6(x-5)+9

g(x)=x^2-10x+25-6x+30+9

g(x)=x^2-16x+64

Conclusion

In this article, we have learned how to shift a function to the right by replacing $x$ with $(x-5)$ . We have applied this concept to the function $f(x)=x^2-6x+9$ and created a new function $g(x)$ by shifting $f(x)$ 5 units to the right. We have also simplified the expression for $g(x)$ to get the final answer.

Answer

The correct answer is:

A. $g(x)=(x-5)^2-6(x-5)+9$

This is the correct answer because we have replaced $x$ with $(x-5)$ in the original function $f(x)$ to create the new function $g(x)$ .

Discussion

The concept of shifting a function to the right is an essential concept in mathematics, particularly in algebra and calculus. It is used to create new functions by transforming existing functions. In this article, we have learned how to shift a function to the right by replacing $x$ with $(x-5)$ . We have also applied this concept to the function $f(x)=x^2-6x+9$ and created a new function $g(x)$ by shifting $f(x)$ 5 units to the right.

Example Problems

Here are some example problems that you can try to practice your skills:

Shift the function $f(x)=x^2+4x+4$ 3 units to the right.
Shift the function $f(x)=x^2-2x+1$ 2 units to the left.
Shift the function $f(x)=x^2+5x+6$ 4 units to the right.

Tips and Tricks

Here are some tips and tricks that you can use to help you with function transformations:

Always replace $x$ with $(x-h)$ when shifting a function to the right.
Use the formula $g(x)=(x-h)^2+k$ to create a new function by shifting an existing function.
Simplify the expression for the new function by combining like terms.

Conclusion

In conclusion, shifting a function to the right is an essential concept in mathematics that helps us create new functions by transforming existing functions. We have learned how to shift a function to the right by replacing $x$ with $(x-5)$ and applied this concept to the function $f(x)=x^2-6x+9$ . We have also simplified the expression for the new function $g(x)$ to get the final answer.

Introduction

In our previous article, we explored the concept of shifting a function to the right and applied it to the function $f(x)=x^2-6x+9$ . We learned how to create a new function $g(x)$ by shifting $f(x)$ 5 units to the right. In this article, we will answer some frequently asked questions about function shifts and provide additional examples to help you practice your skills.

Q&A

Q: What is a function shift?

A: A function shift is a transformation that moves a function to the left or right by a certain number of units. When a function is shifted to the right, we replace $x$ with $(x-h)$ , where $h$ is the number of units we want to shift the function.

Q: How do I shift a function to the right?

A: To shift a function to the right, we replace $x$ with $(x-h)$ , where $h$ is the number of units we want to shift the function. For example, to shift the function $f(x)=x^2-6x+9$ 5 units to the right, we replace $x$ with $(x-5)$ .

Q: How do I shift a function to the left?

A: To shift a function to the left, we replace $x$ with $(x+h)$ , where $h$ is the number of units we want to shift the function. For example, to shift the function $f(x)=x^2-6x+9$ 3 units to the left, we replace $x$ with $(x+3)$ .

Q: What is the difference between a horizontal shift and a vertical shift?

A: A horizontal shift moves a function to the left or right, while a vertical shift moves a function up or down. In this article, we are focusing on horizontal shifts.

Q: How do I simplify the expression for a new function after a shift?

A: To simplify the expression for a new function after a shift, we can combine like terms and use the distributive property to expand the expression.

Q: Can I shift a function more than once?

A: Yes, you can shift a function more than once. For example, you can shift the function $f(x)=x^2-6x+9$ 3 units to the right and then 2 units to the left.

Example Problems

Here are some example problems that you can try to practice your skills:

Shift the function $f(x)=x^2+4x+4$ 2 units to the right.
Shift the function $f(x)=x^2-2x+1$ 4 units to the left.
Shift the function $f(x)=x^2+5x+6$ 3 units to the right and then 2 units to the left.

Tips and Tricks

Here are some tips and tricks that you can use to help you with function transformations:

Always replace $x$ with $(x-h)$ when shifting a function to the right.
Use the formula $g(x)=(x-h)^2+k$ to create a new function by shifting an existing function.
Simplify the expression for the new function by combining like terms.
Practice, practice, practice! The more you practice, the more comfortable you will become with function transformations.

Conclusion

In conclusion, function shifts are an essential concept in mathematics that helps us create new functions by transforming existing functions. We have answered some frequently asked questions about function shifts and provided additional examples to help you practice your skills. Remember to always replace $x$ with $(x-h)$ when shifting a function to the right, and simplify the expression for the new function by combining like terms. With practice, you will become more comfortable with function transformations and be able to apply them to a wide range of problems.