Question

Graph each system of inequalities. $$\left\\{\begin{array}{l}y \geq x^{2}-4 \\\y \leq x-2\end{array}\right.$$

Short answer

Shade the area where the regions \(y \geq x^{2}-4\) and \(y \leq x-2\) overlap.

Step-by-step solution

Understand the Inequalities

There are two inequalities to consider: 1. \(y \geq x^{2}-4\)2. \(y \leq x-2\).The goal is to graph the regions where these inequalities are satisfied.

Graph the Boundary of the First Inequality

Graph the equation \(y = x^{2} - 4\), which is a parabola opening upwards with its vertex at \((0, -4)\). Shade the region above the parabola because \(y\) is greater than or equal to \(x^{2} - 4\).

Graph the Boundary of the Second Inequality

Graph the equation \(y = x - 2\), which is a straight line with a slope of 1 and y-intercept of -2. Shade the region below this line because \(y\) is less than or equal to \(x - 2\).

Identify the Solution Region

The solution to the system of inequalities is the region where the shaded areas from both inequalities overlap. This represents the values that satisfy both \(y \geq x^{2}-4\) and \(y \leq x-2\).

Key concepts

Graphing Inequalities

Graphing inequalities involves representing areas of a graph that satisfy given conditions.
Rather than focusing solely on lines or curves as with equations, inequalities require you to shade regions.
For example, for an inequality like \(y \geq x^{2} - 4\), you first plot the related equality, \(y = x^{2} - 4\). Then, observe the direction of the inequality to determine which side of the parabola to shade.
This shaded area represents all the points \(x, y\) that make the inequality true.

Parabola

A parabola is a U-shaped curve represented by a quadratic equation such as \(y = x^{2} - 4\).
The vertex of this parabola is the point where it changes direction, here it's at (0, -4). To plot this, you can find a few specific points by substituting values for x and solving for y.
For instance, when x = \pm 2\, y = 0, giving you points (2, 0) and (-2, 0). These help shape the plot. Since the inequality is \(y \geq x^{2} - 4\), we shade the region above the parabola.
This shaded section indicates where y values are greater.

Linear Equations

Linear equations like \(y = x - 2\) create straight lines on a graph.
This specific line has a slope of 1 and y-intercept of -2, meaning it crosses the y-axis at (0, -2).
You can determine its slope by recognizing that for each unit increase in x, y increases by 1.
To graph this line, pick some x values to find corresponding y values, such as (0, -2) and (2, 0), and plot them.
Since \(y \leq x - 2\) is given, you need to shade the area below this line, indicating where the y values are smaller.

Solution Region

The solution region is the area of overlap where both inequalities are true simultaneously.
It requires graphing each inequality separately and noting where their shaded regions intersect. For \(y \geq x^{2} - 4\) and \(y \leq x - 2\), you first shade above the parabola (for the first inequality) and below the line (for the second).
The resulting intersection of these shaded areas is your solution region.
This region visually represents the set of all points \(x, y\) that satisfy both inequalities, helping you understand their combined constraints.