Question

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

Short answer

Shade the region to the right of \(x = y^2\) and above \(y = x\).

Step-by-step solution

Understand the Inequalities

We are given two inequalities: 1) \(y^2 \leq x\) 2) \(y \geq x\). First, let's understand what each inequality represents graphically.

Graph the First Inequality

The inequality \(y^2 \leq x\) can be rewritten as \(x \geq y^2\). This represents the region to the right of the parabola \(x = y^2\). When graphing \(x = y^2\), draw a vertical parabola opening to the right.

Graph the Second Inequality

The inequality \(y \geq x\) represents the region above the line \(y = x\). When graphing \(y = x\), draw a 45-degree line passing through the origin.

Find the Intersection of the Regions

To find the solution to the system of inequalities, determine where the regions defined by \(x \geq y^2\) and \(y \geq x\) intersect. This is the overlapping area that satisfies both inequalities.

Graph the Solution Region

Shade the region that satisfies both inequalities. The final graph should highlight the area to the right of the parabola \(x = y^2\) and above the line \(y = x\). Make sure to include boundary lines appropriately.

Key concepts

graphing parabolas

Understanding how to graph parabolas is essential for solving many algebraic problems. A parabola is a U-shaped curve that can open either up, down, left, or right. In our problem, the parabola defined by the equation \( x = y^2 \) opens horizontally to the right.
For graphing, identify key points where the parabola intersects the axes. For instance, with \( x = y^2 \), when y = 0, then x = 0. Similarly, other points could be derived by plugging in values for y (both positive and negative) since \( y^2 \) will always yield non-negative results:

• For y = 1 or y = -1, x = 1
• For y = 2 or y = -2, x = 4
...
These points help in shaping the parabolic curve accurately.

inequalities

Inequalities show relationships where one side isn't strictly equal to the other. When graphing, shading helps illustrate these relationships. For \( y^2 \leq x\), we shade the region to the right of the parabola \( x = y^2 \) since all x-values in the shaded region are greater than or equal to y-values squared.
For the inequality \( y \geq x \), the shading is above and to the left of the line because y-values are greater than or equal to corresponding x-values. This approach helps visually capture where one variable is larger or smaller concerning the other.

graph intersection

Graph intersection points provide solutions to systems of inequalities. For a system, you graph each inequality and find their overlapping region.

For our example, draw the parabola \( x = y^2 \) and the line \( y = x \). Then determine where the shaded regions overlap. Start shading:

• Right of the parabola for \( y^2 \leq x \)
• Above the line for \( y \geq x \)

The common shaded area shows where both conditions hold true, providing the system's solution.

coordinate plane

The coordinate plane is essential for effectively graphing inequalities. It is a two-dimensional number line where any point is defined by an \( (x, y) \) pair.
Each axis has positive and negative directions, useful for graphing various functions, including lines, parabolas, and shaded regions.

When manually plotting points, use graph paper for precision in labeling axes and units. Ensure that:

• x-axis runs horizontally, and y-axis vertically
• points are accurately plotted (like (1, 1) for \( x = y \)

Accurate graphing on the coordinate plane illustrates relationships in their correct spatial context, making understanding and solving inequality systems straightforward.