Let's analyze the quadratic function [tex]\( f(x) = a(x-h)^2 + k \)[/tex] step-by-step:
### (a) Vertex Form
The given function [tex]\( f(x) = a(x-h)^2 + k \)[/tex] is already in the vertex form of a quadratic equation. In this form, the vertex of the parabola is clearly indicated by the coordinates [tex]\((h, k)\)[/tex]. Therefore, the graph of [tex]\( f \)[/tex] is a parabola with the vertex at the point [tex]\((h, k)\)[/tex].
### (b) Determining the Direction and Nature of the Vertex When [tex]\( a > 0 \)[/tex]
When [tex]\( a > 0 \)[/tex]:
- The coefficient [tex]\( a \)[/tex] is positive, which means the parabola opens upwards.
- The vertex [tex]\( (h, k) \)[/tex] represents the minimum point of the parabola.
- This is because the parabola opens upwards and, consequently, the lowest point on the graph is at the vertex.
### (c) Determining the Direction and Nature of the Vertex When [tex]\( a < 0 \)[/tex]
When [tex]\( a < 0 \)[/tex]:
- The coefficient [tex]\( a \)[/tex] is negative, which means the parabola opens downwards.
- The vertex [tex]\( (h, k) \)[/tex] represents the maximum point of the parabola.
- This is because the parabola opens downwards and, consequently, the highest point on the graph is at the vertex.
### Summary
Combining the analysis:
- (a) The graph of [tex]\( f \)[/tex] is a parabola with vertex [tex]\( (h, k) \)[/tex].
- (b) If [tex]\( a > 0 \)[/tex], the graph opens upwards, and [tex]\( f(h) = k \)[/tex] is the minimum value of [tex]\( f \)[/tex].
- (c) If [tex]\( a < 0 \)[/tex], the graph opens downwards, and [tex]\( f(h) = k \)[/tex] is the maximum value of [tex]\( f \)[/tex].
Thus, the completed sentences are:
(b) If [tex]\( a > 0 \)[/tex], the graph of [tex]\( f \)[/tex] opens upwards. In this case [tex]\( f(h) = k \)[/tex] is the minimum value of [tex]\( f \)[/tex].
(c) If [tex]\( a < 0 \)[/tex], the graph of [tex]\( f \)[/tex] opens downwards. In this case [tex]\( f(h) = k \)[/tex] is the maximum value of [tex]\( f \)[/tex].
