Stretching or Compressing the Function

Oct 13, 2025 by miodeya.com 107 views

Introduction

In this article, we will delve into the graph of a quadratic function with a square root, specifically the function q(x) = −x − 4‾‾‾‾‾√. We will analyze the original function and identify how it has been shifted, reflected, stretched, or compressed. This will provide us with a deeper understanding of the graph and its behavior.

Step 1: Identifying the Basic Function

The basic function in this case is a quadratic function, which is a polynomial function of degree two. The general form of a quadratic function is f(x) = ax^2 + bx + c, where a, b, and c are constants. In this case, the basic function is f(x) = x.

The Original Function

The original function is q(x) = −x − 4‾‾‾‾‾√. To understand the graph of this function, we need to identify how it has been modified from the basic function.

Shifting the Function

The original function has been shifted down by 4 units. This is because the constant term in the function is -4, which indicates that the graph has been shifted down by 4 units.

Reflecting the Function

The original function has been reflected across the y-axis. This is because the coefficient of the x-term is negative, which indicates that the graph has been reflected across the y-axis.

Stretching or Compressing the Function

The original function has been stretched vertically by a factor of 1. This is because the coefficient of the x-term is 1, which indicates that the graph has not been stretched or compressed horizontally.

Graphing the Original Function

To graph the original function, we need to combine the shifts, reflections, and stretches/compressions that we have identified. The graph of the original function will be a parabola that has been shifted down by 4 units and reflected across the y-axis.

Key Features of the Graph

The graph of the original function will have the following key features:

The vertex of the parabola will be at the point (0, -4).
The parabola will be reflected across the y-axis.
The parabola will be shifted down by 4 units.

Conclusion

In this article, we have analyzed the graph of a quadratic function with a square root, specifically the function q(x) = −x − 4‾‾‾‾‾√. We have identified how the original function has been shifted, reflected, stretched, or compressed from the basic function. This has provided us with a deeper understanding of the graph and its behavior.

Step 2: Graphing the Original Function

To graph the original function, we can use the following steps:

Start by graphing the basic function, which is f(x) = x.
Shift the graph down by 4 units to account for the constant term in the original function.
Reflect the graph across the y-axis to account for the negative coefficient of the x-term.
Stretch the graph vertically by a factor of 1 to account for the coefficient of the x-term.

By following these steps, we can graph the original function and identify its key features.

Key Takeaways

The original function q(x) = −x − 4‾‾‾‾‾√ has been shifted down by 4 units.
The original function has been reflected across the y-axis.
The original function has been stretched vertically by a factor of 1.

Final Thoughts

Introduction

In our previous article, we analyzed the graph of a quadratic function with a square root, specifically the function q(x) = −x − 4‾‾‾‾‾√. We identified how the original function has been shifted, reflected, stretched, or compressed from the basic function. In this article, we will answer some frequently asked questions about the graph of a quadratic function with a square root.

Q: What is the vertex of the parabola?

A: The vertex of the parabola is at the point (0, -4). This is because the constant term in the original function is -4, which indicates that the graph has been shifted down by 4 units.

Q: Is the parabola reflected across the y-axis?

A: Yes, the parabola is reflected across the y-axis. This is because the coefficient of the x-term in the original function is negative, which indicates that the graph has been reflected across the y-axis.

Q: Has the parabola been stretched or compressed?

A: The parabola has been stretched vertically by a factor of 1. This is because the coefficient of the x-term in the original function is 1, which indicates that the graph has not been stretched or compressed horizontally.

Q: What is the domain of the function?

A: The domain of the function is all real numbers, x ∈ (-∞, ∞). This is because the square root function is defined for all real numbers.

Q: What is the range of the function?

A: The range of the function is all real numbers, y ∈ (-∞, ∞). This is because the square root function can take on any real value.

Q: How can I graph the original function?

A: To graph the original function, you can follow these steps:

Start by graphing the basic function, which is f(x) = x.
Shift the graph down by 4 units to account for the constant term in the original function.
Reflect the graph across the y-axis to account for the negative coefficient of the x-term.
Stretch the graph vertically by a factor of 1 to account for the coefficient of the x-term.

Q: What are some key features of the graph?

A: Some key features of the graph include:

The vertex of the parabola is at the point (0, -4).
The parabola is reflected across the y-axis.
The parabola is shifted down by 4 units.
The parabola is stretched vertically by a factor of 1.

Q: Can I use this function in real-world applications?

A: Yes, this function can be used in real-world applications such as:

Modeling population growth or decline.
Analyzing the behavior of a physical system.
Predicting the outcome of a financial investment.

Conclusion

In this article, we have answered some frequently asked questions about the graph of a quadratic function with a square root. We have identified the vertex of the parabola, the reflection across the y-axis, and the stretching or compression of the graph. We have also provided some key features of the graph and discussed some real-world applications of the function.