For a quadratic function that graphs as a parabola, the domain, range, and vertex are fundamental properties. Specifically, for a quadratic function whose vertex is (-3, -8):
- Domain: All real numbers.
- Range: Depends on the parabola's opening direction; it is either y ≥ -8 (if opening upwards) or y ≤ -8 (if opening downwards).
- Vertex: The point (-3, -8).
Understanding Quadratic Function Properties
Quadratic functions are polynomial functions of the second degree, meaning the highest power of the variable (usually x) is 2. Their graphs are U-shaped curves called parabolas. Understanding the domain, range, and vertex is crucial for analyzing these functions.
The Vertex: The Turning Point
The vertex is the most critical point on a parabola. It represents the turning point where the function changes direction, reaching either its maximum or minimum y-value.
- Definition: The vertex of a parabola is the point (h, k) where the function reaches its peak (maximum) or lowest point (minimum).
- Value: Based on the given information, the vertex of this specific quadratic function is (−3, −8).
- Significance: The coordinates of the vertex, (h, k), are directly used to determine the range of the function.
Domain: All Possible Inputs
The domain of a function refers to all the possible x-values (inputs) for which the function is defined.
- Definition: For any standard quadratic function, regardless of its specific equation, there are no restrictions on the x-values that can be plugged in.
- Value: Therefore, the domain for this quadratic function (and all quadratic functions) is all real numbers. This can be expressed in interval notation as (-∞, ∞).
- Example: You can substitute any real number, positive, negative, or zero, for x into a quadratic equation, and it will always produce a valid y-value.
Range: Restricted Outputs
The range of a function refers to all the possible y-values (outputs) that the function can produce. Unlike the domain, the range of a quadratic function does have restrictions because the parabola has a turning point (the vertex).
- Definition: The range is determined by the y-coordinate of the vertex (k) and the direction in which the parabola opens.
- Determination:
- From the Vertex: The y-coordinate of the vertex is k = -8. This value serves as the boundary for the range.
- Direction of Opening:
- If the parabola opens upwards (when the leading coefficient 'a' in the quadratic equation
ax^2 + bx + cis positive), the vertex represents the minimum y-value. In this case, the range is all real numbers greater than or equal to -8, expressed as y ≥ -8 or [-8, ∞). - If the parabola opens downwards (when the leading coefficient 'a' is negative), the vertex represents the maximum y-value. In this case, the range is all real numbers less than or equal to -8, expressed as y ≤ -8 or (-∞, -8].
- If the parabola opens upwards (when the leading coefficient 'a' in the quadratic equation
- Conclusion: Without knowing the specific quadratic equation (e.g., whether its leading coefficient is positive or negative), we cannot definitively state if the range is
y ≥ -8ory ≤ -8. Both are possible interpretations depending on the parabola's orientation.
Summary Table of Properties
Here's a concise overview of the domain, range, and vertex for a quadratic function with a vertex at (-3, -8):
| Property | Description | Value for this Function |
|---|---|---|
| Vertex | The turning point of the parabola | (-3, -8) |
| Domain | All possible input (x) values | All real numbers or (-∞, ∞) |
| Range | All possible output (y) values | y ≥ -8 (if opening up) or y ≤ -8 (if opening down) |
For further exploration of quadratic functions and their properties, you can refer to resources on Domain and Range of Quadratic Functions.
