Question

1. $y = -2(x - 1)^2 - 3$

Explanation:

Step1: Recall Vertex Form of Parabola

The vertex form of a parabola is \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex and \(a\) determines the direction and width. For \( y = -2(x - 1)^2 - 3 \), compare with \( y = a(x - h)^2 + k \). Here, \(h = 1\), \(k = -3\), and \(a = -2\).

Step2: Determine Direction of Opening

Since \(a = -2 < 0\), the parabola opens downward (because a negative \(a\) in vertex form means the parabola opens down).

Step3: Identify the Vertex

From the vertex form, the vertex \((h, k)\) is \((1, -3)\).

Step4: Analyze the Graph Sketch (Optional, but for Visual)

To sketch, plot the vertex \((1, -3)\). Since \(a = -2\) (vertical stretch by 2 and reflection over x - axis), the parabola opens downward with vertex at \((1, -3)\).

Answer:

The parabola \( y = -2(x - 1)^2 - 3 \) has vertex \((1, -3)\) and opens downward. (If the question was about identifying vertex/direction, this is the key info. If it was to sketch, follow the steps above for plotting.)