Understanding the Basic Quadratic Function

Oct 13, 2025 by windailys.com 100 views

Introduction

In mathematics, transformations of functions are essential concepts that help us understand how functions can be manipulated and represented in various ways. The given function $f(x)=-\frac{1}{2}(x+3)^2+4$ is a transformation of the basic quadratic function $f(x)=x^2$ . In this article, we will explore the transformations involved in obtaining the given function from the basic quadratic function.

Understanding the Basic Quadratic Function

The basic quadratic function is given by $f(x)=x^2$ . This function represents a parabola that opens upwards, with its vertex at the origin (0, 0). The graph of this function is a simple U-shaped curve.

Transformations of the Basic Quadratic Function

To understand the transformations involved in obtaining the given function, let's analyze each component of the function $f(x)=-\frac{1}{2}(x+3)^2+4$ .

Vertical Compression by $\frac{1}{2}$

The coefficient of the squared term in the function $f(x)=-\frac{1}{2}(x+3)^2+4$ is $-\frac{1}{2}$ . This coefficient represents a vertical compression by a factor of $\frac{1}{2}$ . In other words, the graph of the function $f(x)=-\frac{1}{2}(x+3)^2+4$ is compressed vertically by a factor of $\frac{1}{2}$ compared to the graph of the basic quadratic function $f(x)=x^2$ .

Reflection over the $y$ -axis

The term $(x+3)$ in the function $f(x)=-\frac{1}{2}(x+3)^2+4$ indicates a reflection over the $y$ -axis. This is because the term $(x+3)$ represents a horizontal shift of 3 units to the left, which is equivalent to a reflection over the $y$ -axis.

Shift Left by 3

The term $(x+3)$ in the function $f(x)=-\frac{1}{2}(x+3)^2+4$ also indicates a shift of 3 units to the left. This is because the term $(x+3)$ represents a horizontal shift of 3 units to the left, which is equivalent to a shift of 3 units to the left.

Shift Up by 4

The constant term $+4$ in the function $f(x)=-\frac{1}{2}(x+3)^2+4$ indicates a shift of 4 units upwards. This is because the constant term $+4$ represents a vertical shift of 4 units upwards.

Conclusion

In conclusion, the function $f(x)=-\frac{1}{2}(x+3)^2+4$ represents a transformation of the basic quadratic function $f(x)=x^2$ . The transformation involves a vertical compression by $\frac{1}{2}$ , a reflection over the $y$ -axis, a shift of 3 units to the left, and a shift of 4 units upwards.

Final Answer

The final answer is A. Vertical compression by $\frac{1}{2}$ , reflection over the $y$ -axis, shift left by 3, shift up by 4.

Q&A: Understanding Transformations of the Basic Quadratic Function

In the previous article, we explored the transformations involved in obtaining the function $f(x)=-\frac{1}{2}(x+3)^2+4$ from the basic quadratic function $f(x)=x^2$ . In this article, we will answer some frequently asked questions related to transformations of the basic quadratic function.

Q: What is the effect of the coefficient $-\frac{1}{2}$ on the graph of the function $f(x)=-\frac{1}{2}(x+3)^2+4$ ?

A: The coefficient $-\frac{1}{2}$ represents a vertical compression by a factor of $\frac{1}{2}$ . This means that the graph of the function $f(x)=-\frac{1}{2}(x+3)^2+4$ is compressed vertically by a factor of $\frac{1}{2}$ compared to the graph of the basic quadratic function $f(x)=x^2$ .

Q: What is the effect of the term $(x+3)$ on the graph of the function $f(x)=-\frac{1}{2}(x+3)^2+4$ ?

A: The term $(x+3)$ represents a reflection over the $y$ -axis and a shift of 3 units to the left. This means that the graph of the function $f(x)=-\frac{1}{2}(x+3)^2+4$ is reflected over the $y$ -axis and shifted 3 units to the left compared to the graph of the basic quadratic function $f(x)=x^2$ .

Q: What is the effect of the constant term $+4$ on the graph of the function $f(x)=-\frac{1}{2}(x+3)^2+4$ ?

A: The constant term $+4$ represents a shift of 4 units upwards. This means that the graph of the function $f(x)=-\frac{1}{2}(x+3)^2+4$ is shifted 4 units upwards compared to the graph of the basic quadratic function $f(x)=x^2$ .

Q: How can we determine the type of transformation involved in obtaining the function $f(x)=-\frac{1}{2}(x+3)^2+4$ from the basic quadratic function $f(x)=x^2$ ?

A: To determine the type of transformation involved, we need to analyze each component of the function $f(x)=-\frac{1}{2}(x+3)^2+4$ . We can identify the coefficient, the term $(x+3)$ , and the constant term $+4$ as the components that represent the transformations.

Q: What is the final answer to the problem of determining the transformation involved in obtaining the function $f(x)=-\frac{1}{2}(x+3)^2+4$ from the basic quadratic function $f(x)=x^2$ ?

A: The final answer is A. Vertical compression by $\frac{1}{2}$ , reflection over the $y$ -axis, shift left by 3, shift up by 4.

Conclusion

Final Answer

The final answer is A. Vertical compression by $\frac{1}{2}$ , reflection over the $y$ -axis, shift left by 3, shift up by 4.

Additional Resources

For more information on transformations of functions, please refer to the following resources:

Math Open Reference: A comprehensive online reference for mathematics, including transformations of functions.
Khan Academy: A free online learning platform that includes video lessons and practice exercises on transformations of functions.
Wolfram Alpha: A computational knowledge engine that can be used to explore and visualize transformations of functions.