What Are the Domain and Range of a Quadratic Function?

TL;DR

The domain of the quadratic function is all real numbers, meaning any real x-value can be used. The range is all real numbers greater than or equal to -5, since the vertex at (-1, -5) is the minimum point, and the function increases indefinitely in both directions.

Transcript

Determine the domain and range of the function f of x is equal to 3x squared plus 6x minus 2. So, the domain of the function is: what is a set of all of the valid inputs, or all of the valid x values for this function? And, I can take any real number, square it, multiply it by 3, then add 6 times that real number and then subtract 2 from it. So ess... Read More

Key Insights

• 🔢 The domain of a function refers to the set of valid inputs or x values, which for this quadratic function is all real numbers.
• 🌆 The range of a function refers to the set of possible outputs or y values, which for this quadratic function is all real numbers greater than or equal to -5.
• 😥 The vertex of a quadratic function is the minimum or maximum point of the parabolic graph.
• 📈 The vertex of this quadratic function is (-1, -5), and the graph is upward opening.
• ❣️ The parabolic shape of the function allows the y-values to increase indefinitely as x values get larger or smaller.
• 🙃 The graph of the function is symmetrical around the vertex, with mirror images on both sides.
• 💛 The vertex represents the minimum value of the function, and the function cannot take on y-values lower than the y-value of the vertex.

Questions & Answers

Q: What is the domain and range of the quadratic function f(x) = 3x^2 + 6x - 2?

The domain is all real numbers, as any real number can be squared, multiplied by 3, added to 6 times the number, and then subtracted by 2. The range is all real numbers greater than or equal to -5, as the parabola has a minimum point at (-1, -5) and increases indefinitely in both directions.

Q: Are there any complex numbers in the domain or range of the function?

No, the function only includes real numbers in its domain and range. Complex numbers are not included, as they are separate from the set of real numbers.

Q: How can the vertex of a quadratic function be calculated?

The vertex of a quadratic function can be calculated using the formula -b/2a, where a and b are the coefficients of the quadratic equation. In this case, the vertex is (-1, -5).

Q: Can the function have y-values lower than -5?

No, the function cannot have y-values lower than -5. The graph of the function is upward opening, and the vertex is the minimum point. As x values increase or decrease from the vertex, the y-values increase indefinitely.

Summary & Key Takeaways

• The domain of the function is the set of all valid inputs or x values, which is all real numbers.

• The range of the function is the set of all possible outputs or y values, which is all real numbers greater than or equal to -5.

• By graphing the function, it can be observed that the parabolic shape of the function has a minimum point (vertex) at (-1, -5), and as x values increase or decrease, the function increases indefinitely.