A Quadratic Function With The Vertex At \[$(5, -3)\$\] And Opening Downwards Has What Range?A. \[$[-3, \infty)\$\]B. \[$(-\infty, 5]\$\]C. \[$[5, \infty)\$\]D. \[$(-\infty, -3]\$\]
Introduction
Quadratic functions are a fundamental concept in mathematics, and understanding their properties is crucial for solving various mathematical problems. In this article, we will focus on quadratic functions with a vertex at a specific point and explore how to determine their range.
What is a Quadratic Function?
A quadratic function is a polynomial function of degree two, which means the highest power of the variable (usually x) is two. It can be written in the form:
f(x) = ax^2 + bx + c
where a, b, and c are constants, and a ≠0.
Vertex Form of a Quadratic Function
The vertex form of a quadratic function is given by:
f(x) = a(x - h)^2 + k
where (h, k) is the vertex of the parabola. In this form, the vertex is the point (h, k), and the parabola opens upwards if a > 0 and downwards if a < 0.
Given Information
We are given a quadratic function with the vertex at (5, -3) and opening downwards. This means that the parabola has a minimum point at (5, -3), and the value of the function decreases as we move away from this point.
Determining the Range
To determine the range of the quadratic function, we need to find the minimum and maximum values of the function. Since the parabola opens downwards, the minimum value of the function occurs at the vertex, which is (5, -3). The maximum value of the function occurs as x approaches infinity or negative infinity.
Analyzing the Options
Let's analyze the given options:
A. {[-3, \infty)$}$ B. {(-\infty, 5]$}$ C. {[5, \infty)$}$ D. {(-\infty, -3]$}$
Since the parabola opens downwards, the minimum value of the function is -3, which occurs at the vertex (5, -3). As x approaches infinity or negative infinity, the value of the function increases without bound. Therefore, the range of the quadratic function is all real numbers greater than or equal to -3.
Conclusion
Based on the analysis, the correct answer is:
A. {[-3, \infty)$}$
This is because the parabola opens downwards, and the minimum value of the function is -3, which occurs at the vertex (5, -3). The range of the quadratic function is all real numbers greater than or equal to -3.
Final Thoughts
Introduction
In our previous article, we discussed how to determine the range of a quadratic function with a vertex at a specific point. In this article, we will provide a Q&A section to help you better understand the concept and address any questions you may have.
Q: What is the vertex form of a quadratic function?
A: The vertex form of a quadratic function is given by:
f(x) = a(x - h)^2 + k
where (h, k) is the vertex of the parabola.
Q: What does the value of 'a' represent in the vertex form of a quadratic function?
A: The value of 'a' represents the direction and width of the parabola. If a > 0, the parabola opens upwards, and if a < 0, the parabola opens downwards.
Q: How do I determine the range of a quadratic function?
A: To determine the range of a quadratic function, you need to find the minimum and maximum values of the function. Since the parabola opens downwards, the minimum value of the function occurs at the vertex, and the maximum value of the function occurs as x approaches infinity or negative infinity.
Q: What is the minimum value of a quadratic function with a vertex at (5, -3) and opening downwards?
A: The minimum value of the function is -3, which occurs at the vertex (5, -3).
Q: What is the range of a quadratic function with a vertex at (5, -3) and opening downwards?
A: The range of the quadratic function is all real numbers greater than or equal to -3.
Q: Can a quadratic function have a range of (-∞, 5]?
A: No, a quadratic function with a vertex at (5, -3) and opening downwards cannot have a range of (-∞, 5]. The range of the function is all real numbers greater than or equal to -3.
Q: Can a quadratic function have a range of [5, ∞)?
A: No, a quadratic function with a vertex at (5, -3) and opening downwards cannot have a range of [5, ∞). The range of the function is all real numbers greater than or equal to -3.
Q: Can a quadratic function have a range of (-∞, -3]?
A: No, a quadratic function with a vertex at (5, -3) and opening downwards cannot have a range of (-∞, -3]. The range of the function is all real numbers greater than or equal to -3.
Conclusion
In conclusion, understanding the properties of quadratic functions is crucial for solving various mathematical problems. By analyzing the vertex form of a quadratic function and determining the range, we can solve problems involving quadratic functions. We hope this Q&A section has helped you better understand the concept and address any questions you may have.
Final Thoughts
If you have any more questions or need further clarification, please don't hesitate to ask. We are here to help you understand the concept of quadratic functions and their ranges.