# Understanding the Domain and Range of a Quadratic Function

Published on: by Khan Academy

## Introduction

Explaining the concept of domain and range using a quadratic function as an example.

## Definition of domain and range

The domain of a function refers to the set of all valid inputs or x values for which the function is defined. In the case of f(x) = 3x^2 + 6x - 2, the domain is all real numbers since any real number can be squared, multiplied by 3, added 6 times, and then subtracted by 2 to yield a valid output. Imaginary and complex numbers aside, the domain encompasses a wide range of real numbers where the function holds true.

On the other hand, the range of a function represents the set of all possible output values or y values that the function can generate. In the case of the quadratic function f(x) = 3x^2 + 6x - 2, graphing the function reveals its parabolic shape with a vertex at (-1, -5). Due to this upward-opening parabola, the minimum value of the function at the vertex is -5, indicating that the range includes all real numbers greater than or equal to -5. As x values move away from the vertex, the function continues to increase indefinitely, showcasing the expansive range of the function.

## Exploration of the domain of the function f(x) = 3x^2 + 6x - 2

Exploring the domain of the function f(x) = 3x^2 + 6x - 2 involves understanding the set of all possible input values that can be substituted into the function. As mentioned earlier, any real number can serve as a valid input for this quadratic function. By squaring the input, multiplying it by 3, adding 6 times the input, and subtracting 2, the function is well-defined and yields a real number as its output. This comprehensive domain analysis showcases the versatility of the function with regards to the range of input values it can process.

Moreover, investigating the domain of f(x) = 3x^2 + 6x - 2 not only underscores the inclusiveness of real numbers but also highlights the fact that the function remains continuous and defined for all values within its domain. This exploration sheds light on the fundamental nature of the function and lays the groundwork for further analysis regarding its behavior, properties, and applications in various mathematical contexts.

## Graphing the function to visualize the range

Graphing the function f(x) = 3x^2 + 6x - 2 allows for a visual representation of the range of the function. By plotting various points on the graph, including the vertex at (-1, -5), the upward-opening parabolic shape of the function becomes evident. The symmetrical nature of the parabola around its vertex portrays a clear picture of how the function behaves as x values increase or decrease from the vertex.

Through graphing the function, one can observe that the range of the function f(x) = 3x^2 + 6x - 2 extends to all real numbers greater than or equal to -5. This visual depiction not only enhances the understanding of the function's behavior but also provides valuable insights into the maximum and minimum values that the function can attain. Graphing serves as a powerful tool for analyzing functions and exploring their properties in a tangible and intuitive manner.

## Calculation of the vertex of the function

Calculating the vertex of the function f(x) = 3x^2 + 6x - 2 involves determining the x-coordinate where the function reaches its minimum value. By applying the formula for finding the vertex of a quadratic function, namely -b/2a, one can identify the x-value at which the vertex occurs. In the case of f(x) = 3x^2 + 6x - 2, the vertex is located at x = -1, indicating the lowest point on the upward-opening parabolic curve.

Understanding and calculating the vertex of a function provides crucial information about its behavior and extremities. The vertex serves as a pivotal point on the graph where the function achieves its minimum or maximum value, depending on the nature of the parabolic curve. By pinpointing the vertex, one gains valuable insights into the overall shape and characteristics of the function, paving the way for a deeper analysis of its domain, range, and critical points.

## Determining the minimum value of the function at the vertex

Determining the minimum value of the function f(x) = 3x^2 + 6x - 2 at the vertex involves evaluating the function at the x-coordinate of the vertex. In the case of f(x) = 3x^2 + 6x - 2 and its vertex at x = -1, substituting this x-value into the function yields the corresponding minimum y-value, which is -5. This minimum value signifies the lowest point that the function can reach along its parabolic curve.

By determining the minimum value of the function at its vertex, one gains crucial information about the behavior and range of the function. Understanding that the function achieves its minimum value of -5 at the vertex sheds light on the limitations and extremities of the function within its domain. This critical analysis of the function's minimum value enhances the overall comprehension of its characteristics and the implications for mathematical applications.

## Highlights

- Explanation of domain as the set of valid inputs for a function
- Illustration of the parabolic shape of quadratic functions
- Calculation of specific points to understand the function's behavior
- Visualization of the function's graph to determine range
- Interpretation of the vertex as the minimum value of the function

## FAQ

**Q: What is the domain of a function?**

A: The domain of a function refers to the set of all valid input values for the function, in this case, all real numbers for the quadratic function f(x) = 3x^2 + 6x - 2.

**Q: How does graphing a function help in understanding its range?**

A: Graphing a function allows visualization of its behavior and helps determine the possible output values or range of the function.

**Q: What is the significance of the vertex in quadratic functions?**

A: The vertex of a quadratic function represents the minimum or maximum value the function can attain and provides valuable information about its behavior.

**Q: Why is the range of the function limited to values greater than or equal to -5?**

A: The quadratic function f(x) = 3x^2 + 6x - 2 has a vertex at y = -5, indicating that it cannot produce values lower than -5, leading to a range of all real numbers greater than or equal to -5.

**Q: How can understanding domain and range enhance mathematical analysis?**

A: Knowledge of domain and range helps in interpreting the behavior of functions, determining suitable input values, and identifying possible output ranges, contributing to a deeper understanding of mathematical concepts.