How to Uncover the Range of a Quadratic Function: A Step-by-Step Adventure
Are you baffled by the enigma of finding the range of a quadratic function? Fear not, for this comprehensive guide will illuminate the path, guiding you through the maze of mathematical equations. Whether you’re a seasoned pro or just beginning your algebraic journey, get ready to conquer this challenge with newfound confidence!
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What is a Quadratic Function?
A quadratic function, often represented by the equation f(x) = ax² + bx + c, is a mathematical expression where the highest exponent of the variable x is 2. These functions take on a U-shape or an inverted U-shape in their graphical representation.
Unveiling the Range of a Quadratic Function
The range of a quadratic function refers to the set of all possible output values, also known as y-values. To determine the range, we need to delve into three possibilities:
1. Positive Leading Coefficient (a > 0)
- Case 1: Positive Discriminant (b² – 4ac > 0)
- The graph of the function forms a U-shaped curve.
- The range is the set of all y-values greater than or equal to the minimum value of the function.
- Case 2: Zero Discriminant (b² – 4ac = 0)
- The graph of the function is a parabola touching the x-axis at a single point.
- The range is the single y-value at the point of contact.
- Case 3: Negative Discriminant (b² – 4ac < 0)
- The graph of the function is an inverted U-shaped curve.
- The range is the set of all y-values less than or equal to the maximum value of the function.
2. Negative Leading Coefficient (a < 0)
- Case 1: Positive Discriminant (b² – 4ac > 0)
- The graph of the function forms an inverted U-shaped curve.
- The range is the set of all y-values less than or equal to the maximum value of the function.
- Case 2: Zero Discriminant (b² – 4ac = 0)
- The graph of the function is a parabola touching the x-axis at a single point.
- The range is the single y-value at the point of contact.
- Case 3: Negative Discriminant (b² – 4ac < 0)
- The graph of the function is a U-shaped curve.
- The range is the set of all y-values greater than or equal to the minimum value of the function.
3. Leading Coefficient Equal to Zero (a = 0)
In this case, the function is a linear function, not a quadratic function, and the range is all real numbers.
Real-World Example
Let’s imagine a rocket being launched from the Earth’s surface. The height of the rocket, h(t), can be modeled by the quadratic function h(t) = -4.9t² + 100t, where t represents the time in seconds.
To find the range of this function, we first determine the discriminant: b² – 4ac = 100² – 4(-4.9)(0) = 10000. The positive discriminant indicates a U-shaped curve.
Next, we find the vertex of the parabola using the formula (-b/2a, h(-b/2a)). Plugging in the given values, we get (-100/-9.8, h(-100/-9.8)) = (10.2, 510.2).
Therefore, the range of the function h(t) is all y-values greater than or equal to 510.2, representing the maximum height reached by the rocket before falling back to Earth.
A Handy Table for Reference
| Leading Coefficient (a) | Discriminant (b² – 4ac) | Range |
|---|---|---|
| Positive (a > 0) | Positive (b² – 4ac > 0) | y ≥ minimum value |
| Positive (a > 0) | Zero (b² – 4ac = 0) | Single y-value at the vertex |
| Positive (a > 0) | Negative (b² – 4ac < 0) | y ≤ maximum value |
| Negative (a < 0) | Positive (b² – 4ac > 0) | y ≤ maximum value |
| Negative (a < 0) | Zero (b² – 4ac = 0) | Single y-value at the vertex |
| Negative (a < 0) | Negative (b² – 4ac < 0) | y ≥ minimum value |
| Zero (a = 0) | Any value | All real numbers |
A Friendly Reminder
Just remember, the range of a quadratic function depends not only on the leading coefficient but also on the discriminant. By understanding the relationship between these factors, you’ll be able to find the range with ease and impress your math buddies.
Conclusion
Now, you possess the wisdom to unravel the enigmatic range of quadratic functions! Whether you’re tackling a high-school algebra problem or exploring real-world applications, this guide will empower you to conquer the world of quadratic adventures.
Don’t forget to check out our other articles to further enhance your mathematical prowess:
- The Art of Factoring Quadratic Expressions: A Beginner’s Guide
- The Secrets of Completing the Square: A Step-by-Step Revelation
- The Enchanted World of Solving Quadratic Equations: A Treasure Hunt for Solutions
FAQ about Finding the Range of a Quadratic Function
What is the range of a quadratic function?
Answer: The range is all possible output values from a function for all possible input values.
How do I find the range of a quadratic function in the form y = ax² + bx + c?
Answer: The range is determined by the sign of the coefficient "a". If a > 0, the range is [c, ∞), and if a < 0, the range is (-∞, c].
What is the difference between the domain and range of a quadratic function?
Answer: The domain is the set of all possible input values, while the range is the set of all possible output values.
How do I find the vertex of a quadratic function to determine the range?
Answer: The vertex is the point where the parabola changes direction. The x-coordinate of the vertex is -b/2a, and the y-coordinate is f(-b/2a).
What if the quadratic function has no real solutions (roots)?
Answer: In this case, the range is an empty set.
What if the quadratic function is a perfect square?
Answer: For a perfect square (y = (x – h)² + k), the range is [k, ∞).
How do I find the range of a quadratic function given its graph?
Answer: The range is the vertical interval spanned by the graph.
What is the relationship between the discriminant and the range of a quadratic function?
Answer: If the discriminant (b² – 4ac) is positive, the range is all real numbers. If the discriminant is zero, the range is a single point. If the discriminant is negative, the function has no real solutions (roots), and the range is an empty set.
Can a quadratic function have a negative range?
Answer: Yes, if the coefficient "a" is negative and the discriminant is negative (i.e., both the parabola opens downward and has no real roots).
Can the range of a quadratic function be a single value?
Answer: Yes, if the function is a perfect square (y = (x – h)² + k) or if the discriminant is zero.
