The Function $f(x)=x^2+5x-6$ Is Shifted 4 Units To The Left To Create $g(x$\]. What Is $g(x$\]?A. $g(x)=\left(x^2+5x-6\right)+4$ B. $g(x)=(x-4)^2+5(x-4)-6$ C. $g(x)=(x+4)^2+5(x+4)-6$ D.

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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 left and apply it to the given function $f(x)=x^2+5x-6$. We will determine the new function $g(x)$ that results from shifting $f(x)$ 4 units to the left.

Understanding Function Shifts

When a function $f(x)$ is shifted $h$ units to the left, the new function $g(x)$ is obtained by replacing $x$ with $x+h$. This means that every occurrence of $x$ in the original function is replaced by $x+h$. In the case of a horizontal shift to the left, $h$ is a positive number.

Applying the Shift to $f(x)=x^2+5x-6$

To shift $f(x)=x^2+5x-6$ 4 units to the left, we need to replace $x$ with $x+4$. This means that every occurrence of $x$ in the original function is replaced by $x+4$. The new function $g(x)$ is obtained by applying this substitution to the original function.

Calculating $g(x)$

Let's calculate $g(x)$ by replacing $x$ with $x+4$ in the original function $f(x)=x^2+5x-6$.

$$g(x) = (x+4)^2 + 5(x+4) - 6$$

Simplifying $g(x)$

To simplify $g(x)$, we can expand the squared term and combine like terms.

$$g(x) = (x^2 + 8x + 16) + (5x + 20) - 6$$

$$g(x) = x^2 + 8x + 16 + 5x + 20 - 6$$

$$g(x) = x^2 + 13x + 30$$

Conclusion

In conclusion, the function $g(x)$ that results from shifting $f(x)=x^2+5x-6$ 4 units to the left is $g(x) = (x+4)^2 + 5(x+4) - 6$. This can be simplified to $g(x) = x^2 + 13x + 30$. Therefore, the correct answer is:

C. $g(x)=(x+4)^2+5(x+4)-6$

Discussion

The concept of function shifts is an essential part of mathematics, particularly in algebra and calculus. Understanding how functions change when they are shifted, stretched, or compressed is crucial for solving problems and modeling real-world phenomena. In this article, we applied the concept of horizontal shifts to the given function $f(x)=x^2+5x-6$ and determined the new function $g(x)$ that results from shifting $f(x)$ 4 units to the left.

Example Problems

1. Shift the function $f(x)=x^2-3x+2$ 3 units to the right. What is the new function $g(x)$?
2. Shift the function $f(x)=2x^2+4x-1$ 2 units to the left. What is the new function $g(x)$?
3. Shift the function $f(x)=x^2+2x-3$ 1 unit to the right. What is the new function $g(x)$?

Solutions

1. $g(x) = (x-3)^2 - 3(x-3) + 2$
2. $g(x) = (x+2)^2 + 4(x+2) - 1$
3. $g(x) = (x-1)^2 + 2(x-1) - 3$

Practice Problems

1. Shift the function $f(x)=x^2+4x-2$ 2 units to the left. What is the new function $g(x)$?
2. Shift the function $f(x)=2x^2-3x+1$ 1 unit to the right. What is the new function $g(x)$?
3. Shift the function $f(x)=x^2-2x-3$ 3 units to the left. What is the new function $g(x)$?

Solutions

1. $g(x) = (x-2)^2 + 4(x-2) - 2$
2. $g(x) = (x+1)^2 - 3(x+1) + 1$
3. $g(x) = (x-3)^2 - 2(x-3) - 3$

Conclusion

In conclusion, function shifts are an essential part of mathematics, and understanding how functions change when they are shifted, stretched, or compressed is crucial for solving problems and modeling real-world phenomena. In this article, we applied the concept of horizontal shifts to the given function $f(x)=x^2+5x-6$ and determined the new function $g(x)$ that results from shifting $f(x)$ 4 units to the left. We also provided example problems and solutions to help readers practice and understand the concept of function shifts.

Introduction

In our previous article, we explored the concept of function shifts and applied it to the given function $f(x)=x^2+5x-6$. We determined the new function $g(x)$ that results from shifting $f(x)$ 4 units to the left. In this article, we will provide a Q&A guide to help readers understand the concept of function shifts and how to apply it to different functions.

Q1: What is a function shift?

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 left, the new function is obtained by replacing $x$ with $x+h$, where $h$ is a positive number.

Q2: How do I shift a function to the left?

To shift a function to the left, you need to replace $x$ with $x+h$, where $h$ is a positive number. For example, if you want to shift the function $f(x)=x^2+5x-6$ 4 units to the left, you would replace $x$ with $x+4$.

Q3: How do I shift a function to the right?

To shift a function to the right, you need to replace $x$ with $x-h$, where $h$ is a positive number. For example, if you want to shift the function $f(x)=x^2+5x-6$ 4 units to the right, you would replace $x$ with $x-4$.

Q4: What is the difference between a horizontal shift and a vertical shift?

A horizontal shift moves a function to the left or right, while a vertical shift moves a function up or down. A horizontal shift is obtained by replacing $x$ with $x+h$ or $x-h$, while a vertical shift is obtained by multiplying the function by a constant.

Q5: How do I apply a function shift to a quadratic function?

To apply a function shift to a quadratic function, you need to replace $x$ with $x+h$ or $x-h$, where $h$ is a positive number. For example, if you want to shift the quadratic function $f(x)=x^2+5x-6$ 4 units to the left, you would replace $x$ with $x+4$.

Q6: How do I apply a function shift to a linear function?

To apply a function shift to a linear function, you need to replace $x$ with $x+h$ or $x-h$, where $h$ is a positive number. For example, if you want to shift the linear function $f(x)=2x-3$ 4 units to the left, you would replace $x$ with $x+4$.

Q7: How do I apply a function shift to a polynomial function?

To apply a function shift to a polynomial function, you need to replace $x$ with $x+h$ or $x-h$, where $h$ is a positive number. For example, if you want to shift the polynomial function $f(x)=x^3+2x^2-3x+1$ 4 units to the left, you would replace $x$ with $x+4$.

Q8: What are some common applications of function shifts?

Function shifts have many applications in mathematics, science, and engineering. Some common applications include:

• Modeling real-world phenomena, such as population growth or chemical reactions
• Solving optimization problems, such as finding the maximum or minimum of a function
• Analyzing the behavior of complex systems, such as electrical circuits or mechanical systems

Conclusion

In conclusion, function shifts are an essential part of mathematics, and understanding how functions change when they are shifted, stretched, or compressed is crucial for solving problems and modeling real-world phenomena. In this article, we provided a Q&A guide to help readers understand the concept of function shifts and how to apply it to different functions. We hope that this guide will be helpful in your studies and applications of function shifts.

Practice Problems

1. Shift the function $f(x)=x^2+4x-2$ 2 units to the left. What is the new function $g(x)$?
2. Shift the function $f(x)=2x^2-3x+1$ 1 unit to the right. What is the new function $g(x)$?
3. Shift the function $f(x)=x^2-2x-3$ 3 units to the left. What is the new function $g(x)$?

Solutions

1. $g(x) = (x-2)^2 + 4(x-2) - 2$
2. $g(x) = (x+1)^2 - 3(x+1) + 1$
3. $g(x) = (x-3)^2 - 2(x-3) - 3$

Discussion

The concept of function shifts is an essential part of mathematics, and understanding how functions change when they are shifted, stretched, or compressed is crucial for solving problems and modeling real-world phenomena. In this article, we provided a Q&A guide to help readers understand the concept of function shifts and how to apply it to different functions. We hope that this guide will be helpful in your studies and applications of function shifts.