Questions: Sketch a graph of y=-x^2-1

Transcript text: Sketch a graph of $y=-x^{2}-1$

Solution

Solution Steps

Step 1: Identify the Equation and its Components

The given equation is $ y = -x^2 - 1 $. This is a quadratic equation in the form $ y = ax^2 + bx + c $ where $ a = -1 $, $ b = 0 $, and $ c = -1 $.

Step 2: Determine the Vertex

For the equation $ y = -x^2 - 1 $, the vertex form is $ y = a(x-h)^2 + k $. Here, $ h = 0 $ and $ k = -1 $, so the vertex is at $ (0, -1) $.

Step 3: Plot the Vertex

Plot the vertex $ (0, -1) $ on the graph.

Step 4: Determine the Direction of the Parabola

Since $ a = -1 $ (which is negative), the parabola opens downwards.

Step 5: Calculate Additional Points

Choose values for $ x $ to find corresponding $ y $ values:

For $ x = 1 $, $ y = -(1)^2 - 1 = -1 - 1 = -2 $
For $ x = -1 $, $ y = -(-1)^2 - 1 = -1 - 1 = -2 $
For $ x = 2 $, $ y = -(2)^2 - 1 = -4 - 1 = -5 $
For $ x = -2 $, $ y = -(-2)^2 - 1 = -4 - 1 = -5 $

Step 6: Plot Additional Points

Plot the points $ (1, -2) $, $ (-1, -2) $, $ (2, -5) $, and $ (-2, -5) $ on the graph.

Step 7: Draw the Parabola

Connect the points with a smooth curve to form the parabola.

Final Answer

The graph of $ y = -x^2 - 1 $ is a downward-opening parabola with its vertex at $ (0, -1) $. The additional points $ (1, -2) $, $ (-1, -2) $, $ (2, -5) $, and $ (-2, -5) $ help in shaping the curve.

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