Determine the Vertex and Axis of Symmetry for the parabola $x=-{(y-1)}^2$, and determine which direction the parabola will open.

• A. The parabola opens downwards; Vertex: $(0,1)$; Axis of Symmetry: $y=1$
• B. The parabola opens downwards; Vertex: $(0,-1)$; Axis of Symmetry: $x=1$
• C. The parabola opens to the left; Vertex: $(0,-1)$; Axis of Symmetry: $x=1$
• D. The parabola opens to the left; Vertex: $\(\left\)(0,1\(\right\))$; Axis of Symmetry: $y=1$

Answer: (D)

Verified step by step guidance

1. Rewrite the given equation \(x = -\left(y - 1\right)^2\) and recognize that it is in the form \(x = a(y - k)^2 + h\), which represents a parabola that opens either left or right, depending on the sign of \(a\).
2. Identify the vertex of the parabola from the equation. The vertex is at the point \((h, k)\), where \(h\) is the constant term added to the squared expression and \(k\) is the value subtracted from \(y\). In this case, \(h = 0\) and \(k = 1\), so the vertex is at \((0, 1)\).
3. Determine the direction the parabola opens by looking at the coefficient \(a\) in front of the squared term. Since \(a = -1\) (negative), the parabola opens to the left (because the parabola is expressed as \(x = a(y-k)^2 + h\)).
4. Find the axis of symmetry, which is the line that passes through the vertex and divides the parabola into two mirror images. For a parabola in the form \(x = a(y-k)^2 + h\), the axis of symmetry is the horizontal line \(y = k\). Here, it is \(y = 1\).
5. Summarize the results: the vertex is at \((0, 1)\), the axis of symmetry is the line \(y = 1\), and the parabola opens to the left.