To determine the graph of the equation [tex]\((x-2)^2 = -4(y-2)\)[/tex], let’s break down the properties of this equation step by step.
1. Form of the Equation:
The given equation [tex]\((x-2)^2 = -4(y-2)\)[/tex] is a type of quadratic equation representing a parabola. We can compare this with the standard form of a parabola that opens vertically, which is [tex]\((x - h)^2 = 4a(y - k)\)[/tex].
2. Vertex of the Parabola:
In the standard form [tex]\((x - h)^2 = 4a(y - k)\)[/tex], [tex]\((h, k)\)[/tex] represents the vertex of the parabola. For our equation, we have:
[tex]\[
(x - 2)^2 = -4(y - 2)
\][/tex]
By comparing, we determine that [tex]\(h = 2\)[/tex] and [tex]\(k = 2\)[/tex]. Therefore, the vertex of the parabola is [tex]\((2, 2)\)[/tex].
3. Direction of Opening:
The standard form [tex]\((x - h)^2 = 4a(y - k)\)[/tex] tells us the direction in which the parabola opens based on the coefficient [tex]\(4a\)[/tex]:
- If [tex]\(4a\)[/tex] is positive, the parabola opens upwards.
- If [tex]\(4a\)[/tex] is negative, the parabola opens downwards.
In our equation [tex]\((x - 2)^2 = -4(y - 2)\)[/tex], the coefficient on the right side is [tex]\(-4\)[/tex]. This indicates that [tex]\(4a\)[/tex] is negative ([tex]\(4a = -4\)[/tex]), hence the parabola opens downward.
4. Width of the Parabola:
The absolute value of [tex]\(4a\)[/tex] determines the width of the parabola. For [tex]\(4a = -4\)[/tex], the value of [tex]\(|4a| = 4\)[/tex], which is a typical coefficient size and suggests the standard width for a parabola without excessive narrowing or widening.
5. Graph Characteristics:
- Vertex: [tex]\((2, 2)\)[/tex]
- Direction: Opens downward
- Shape: Symmetrical about the vertical line through the vertex [tex]\(x = 2\)[/tex]
Putting all these together, the graph of the equation [tex]\((x-2)^2 = -4(y-2)\)[/tex] is a downward-opening parabola with its vertex at the point [tex]\((2, 2)\)[/tex].
Therefore, the correct graph displays these characteristics – a parabola opening downward with its vertex exactly at [tex]\((2, 2)\)[/tex].
