The domain and range of a parabola depend on its orientation and position in the coordinate plane. In general, the standard form of a parabola is given by:
y = ax^2 + bx + c
where a, b, and c are constants, and x and y are the variables. The coefficient a determines the orientation of the parabola, and the constants b and c determine its position.
If a is positive, the parabola opens upward and its vertex is the minimum point. The domain of the parabola is all real numbers, and the range is the set of all real numbers greater than or equal to the y-coordinate of the vertex.
If a is negative, the parabola opens downward and its vertex is the maximum point. The domain of the parabola is all real numbers, and the range is the set of all real numbers less than or equal to the y-coordinate of the vertex.
The vertex of the parabola can be found using the formula:
x = -b/2a
y = f(x) = a(x - h)^2 + k
where h = -b/2a is the x-coordinate of the vertex, and k = f(h) is the y-coordinate of the vertex.
For example, consider the parabola y = 2x^2 - 4x + 1. The coefficient a is positive, so the parabola opens upward. The vertex of the parabola can be found by completing the square or by using the formula x = -b/2a:
h = -(-4)/(2*2) = 1
k = f(1) = 2(1)^2 - 4(1) + 1 = -1
So the vertex of the parabola is (1, -1). The domain of the parabola is all real numbers, and the range is the set of all real numbers greater than or equal to -1.
It's worth noting that the domain and range of a parabola can also be affected by any restrictions or conditions placed on the function, such as those arising from real-world applications or physical constraints.
