To find the domain and range of a parabola, you first need to understand its general form and characteristics.
Domain
The domain of a parabola is the set of all possible x-values that the function can take. Since a parabola extends indefinitely in both the left and right directions, the domain is always all real numbers. In interval notation, this can be expressed as:
Domain: (-∞, ∞)
Range
Finding the range is a little more complicated and depends on whether the parabola opens upwards or downwards.
- If the parabola opens upwards (the coefficient of the x² term is positive), the vertex of the parabola represents the minimum point. The y-value of the vertex will be the lowest point on the graph, and the parabola extends infinitely upwards from that point.
- If the parabola opens downwards (the coefficient of the x² term is negative), the vertex represents the maximum point. The y-value of the vertex will be the highest point, and the parabola extends infinitely downwards from that point.
To find the vertex, you can use the formula:
x = -b/(2a)
Once you have the x-coordinate of the vertex, plug it back into the equation of the parabola to find the corresponding y-coordinate.
Examples
1. For the parabola given by y = x² – 4: This opens upwards. The vertex is at (0, -4). Thus, the range is:
Range: [-4, ∞)
2. For the parabola given by y = -x² + 6: This opens downwards. The vertex is at (0, 6). Thus, the range is:
Range: (-∞, 6]
In summary, the domain of any parabola is always all real numbers, while the range depends on whether the parabola opens upwards or downwards, determined by the vertex.