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

Answer: (D)

Verified step by step guidance

1. Identify the given equation of the parabola: \(x = -\left(y - 1\right)^2\). Notice that the variable \(x\) is expressed in terms of \(y\), which means the parabola opens horizontally (left or right), not vertically.
2. Recall the standard form for a horizontally opening parabola: \(x = a(y - k)^2 + h\), where \((h, k)\) is the vertex. Compare the given equation to this form to find the vertex coordinates.
3. From the equation \(x = -\left(y - 1\right)^2\), observe that \(h = 0\) and \(k = 1\), so the vertex is at the point \((0, 1)\).
4. Determine the axis of symmetry, which is the line that passes through the vertex and divides the parabola into two mirror images. For a horizontal parabola, the axis of symmetry is a horizontal line given by \(y = k\). Here, it is \(y = 1\).
5. Analyze the coefficient \(a\) in front of the squared term to find the direction the parabola opens. Since \(a = -1\) (negative), the parabola opens to the left.