explain how the rainge of a quadratic function is relted to the vertex of the parabola
The range of a quadratic function is directly related to the vertex of the parabola. To understand this relationship, let's start by reviewing the key properties of a quadratic function.
A quadratic function is a function of the form f(x) = ax^2 + bx + c, where a, b, and c are constants. The graph of a quadratic function is a parabola, which can open upwards (concave up) or downwards (concave down) depending on the value of the leading coefficient a.
The vertex of a parabola is the point on the graph where the parabola reaches its maximum or minimum value. If the parabola opens upwards (a > 0), the vertex represents the minimum point of the function, while if the parabola opens downwards (a < 0), the vertex represents the maximum point of the function.
Now, let's consider the range of a quadratic function. The range is the set of all possible y-values that the function can produce, or in other words, it represents the vertical extent of the graph.
If the quadratic function opens upwards (a > 0), the vertex is the lowest point on the parabola, and therefore the range of the function is all y-values greater than or equal to the y-coordinate of the vertex. In this case, the range extends infinitely upwards.
On the other hand, if the quadratic function opens downwards (a < 0), the vertex is the highest point on the parabola, and the range of the function is all y-values less than or equal to the y-coordinate of the vertex. In this case, the range extends infinitely downwards.
In summary, for a quadratic function, the range is determined by the orientation of the parabola and the y-coordinate of the vertex. If the parabola opens upwards, the range is all y-values greater than or equal to the y-coordinate of the vertex, and if the parabola opens downwards, the range is all y-values less than or equal to the y-coordinate of the vertex.