In a quadratic equation of the form [tex]\( a(x - h)^2 + k \)[/tex], the expression is written in vertex form. Here, [tex]\( (h, k) \)[/tex] represents the vertex of the parabola.
To understand why [tex]\( (h, k) \)[/tex] represents the vertex, let's consider the general steps and reasoning:
1. Starting with Vertex Form:
The vertex form of a quadratic function is [tex]\( f(x) = a(x - h)^2 + k \)[/tex].
2. Understanding the Terms:
- [tex]\( a \)[/tex] is a coefficient that determines the direction and the width of the parabola. If [tex]\( a > 0 \)[/tex], the parabola opens upwards. If [tex]\( a < 0 \)[/tex], it opens downwards.
- [tex]\( (x - h) \)[/tex] indicates that the parabola is shifted [tex]\( h \)[/tex] units horizontally. If [tex]\( h \)[/tex] is positive, the shift is to the right. If [tex]\( h \)[/tex] is negative, the shift is to the left.
- [tex]\( k \)[/tex] represents a vertical shift. If [tex]\( k \)[/tex] is positive, the parabola moves [tex]\( k \)[/tex] units up. If [tex]\( k \)[/tex] is negative, it moves [tex]\( k \)[/tex] units down.
3. Finding the Vertex:
- The term [tex]\( (x - h)^2 \)[/tex] reaches its minimum value when [tex]\( (x - h) = 0 \)[/tex]—that is, when [tex]\( x = h \)[/tex].
- At [tex]\( x = h \)[/tex], the value of the quadratic function [tex]\( f(x) \)[/tex] is [tex]\( f(h) = a(h - h)^2 + k = a \cdot 0 + k = k \)[/tex].
4. Conclusion:
- Therefore, the coordinate [tex]\( (h, k) \)[/tex] is the point where the quadratic function reaches its minimum (if [tex]\( a > 0 \)[/tex]) or maximum (if [tex]\( a < 0 \)[/tex]) value.
Thus, in the quadratic equation [tex]\( a(x - h)^2 + k \)[/tex], the point [tex]\( (h, k) \)[/tex] represents the vertex of the parabola. This is the highest or lowest point on the graph, depending on the sign of [tex]\( a \)[/tex].
