Question

Sketch the graph of the inequality. $$y \geq x^{2}-3$$

Short answer

The graph of the inequality \(y \geq x^{2}-3\) is a parabola opening upwards and situated 3 units below the y-axis. The solution region to the inequality, being \(y \geq x^{2}-3\), is the area above and on the curve of the parabola, including the parabola itself.

Step-by-step solution

Sketch the parabola

Draw the graph of the quadratic equation \(y = x^{2}-3\). This is a simple parabola shifted 3 units down the y-axis, with the vertex at (0, -3). Make sure the parabola opens upwards.

Identify the inequality

Notice that the inequality is \(y \geq x^{2}-3\). This means that we're interested in the areas at and above the curve. The parabola divides the plane into two regions, the region inside the curve 'below' and the region outside the curve 'above'.

Shade the solution region

As the given inequality is \(y \geq x^{2}-3\), this indicates the solution lies above or on the curve. So, shade the area above and on the parabola to represent the solution of the inequality.