Question

Graph each system of inequalities. $$\left\\{\begin{array}{l}y+x^{2} \leq 1 \\\y \geq x^{2}-1\end{array}\right.$$

Short answer

The solution is the region between the curves y ≤ 1 - x^2 and y ≥ x^2 - 1.

Step-by-step solution

Graph the inequality y + x^2 ≤ 1

First, rewrite the inequality as y ≤ 1 - x^2. This represents a parabola that opens downwards. To graph, plot points for several values of x (e.g., -2, -1, 0, 1, 2) and calculate corresponding y values. Sketch the curve and shade the region below the parabola.

Graph the inequality y ≥ x^2 - 1

Rewrite the inequality as y ≥ x^2 - 1. This represents a parabola that opens upwards. Plot points by choosing several x values and finding their corresponding y values. Sketch the curve and shade the region above the parabola.

Identify the overlapping region

The solution to the system of inequalities is the region where the shaded areas from the two inequalities overlap. Examine your graphs and identify this region.

Key concepts

Parabolas

A parabola is a U-shaped curve that can open either upwards or downwards. Parabolas are defined by a quadratic equation of the form \(y = ax^2 + bx + c\). When \(a > 0\), the parabola opens upwards, and when \(a < 0\), it opens downwards.
The vertex is the highest or lowest point of the parabola. For the equations given in the exercise, let's identify their vertices:

• For \(y \le 1 - x^2\), we rewrite it to the form \(y = -x^2 + 1\). Here, the vertex is at \(0, 1\) and opens downwards since the coefficient of \(x^2\) is negative.
• For \(y \ge x^2 - 1\), the vertex is at \(0, -1\) and opens upwards because the coefficient of \(x^2\) is positive.

Understanding the direction in which the parabolas open helps in correctly shading the regions for the inequalities.

Inequalities

Graphing inequalities involves plotting a region, not just a line or curve. Here are the steps:

• Solve for \(y\) if necessary, making the inequality easier to graph.
• Graph the boundary line or curve. Use a solid line for \(\le\) or \(\ge\) and a dashed line for \(<\) or \(>\).
• Shade the correct side of the boundary line. For \(y \le\) or \(y \<\), shade below the line. For \(y \ge\) or \(y \>\), shade above the line.

In our exercise, we use these principles to graph \(y + x^2 \le 1\) and \(y \ge x^2 - 1\). The parabola from \(y + x^2 \le 1\) is shaded below the curve since it's \(\le\). The parabola from \(y \ge x^2 - 1\) is shaded above the curve as it's \(\ge\).

Graphing Systems

When graphing a system of inequalities, you need to find the region where all inequalities overlap. This overlapping region represents the solution to the system.
Follow these steps for each inequality:

• Graph each inequality separately on the same coordinate plane.
• Shade the solution region for each inequality.
• The solution to the system is where the shaded regions overlap.

In our exercise, we graph \(y \le 1 - x^2\) and \(y \ge x^2 - 1\). Then, we find the area where both shaded regions intersect. This intersected area is the solution to the system.
Clearly marking and shading the regions while being precise with the boundary lines helps in visualizing the correct solution.