Question

Sketch the graph of the inequality. $$x^{2}+y^{2} \leq 4$$

Short answer

The graph of the inequality \(x^{2}+y^{2} \leq 4\) is a filled circle with a radius of 2 centered at the origin (0,0). All points inside and on the circle are included in the solution.

Step-by-step solution

Analyzing the inequality

The inequality \(x^{2}+y^{2} \leq 4\) represents a circle with radius 2 centered at the origin of the coordinate system. The \(\leq\) sign means all the points on this circle and inside this circle are included in the solution set.

Sketching the circle

Draw a circle with radius of 2 units on a graph where the center is the origin (0,0). You can mark these points (2,0), (-2,0), (0,2) and (0,-2) - those are the points where the circle intersects the x-axis and y-axis.

Shading the area of solution

As the inequality is \(x^{2}+y^{2} \leq 4\), meaning it includes values equal to 4 (the points on the circle itself) and all values less than 4 (inside the circle). So, shade the entire circle including its boundary to visualize the solution set of the inequality.

Key concepts

Circle Graph

When you come across an equation like x^{2}+y^{2} \[\]\leq 4, you're dealing with what we call a circle graph in mathematics. A circle graph represents all the points that satisfy the equation of a circle. To understand it better, think of a flat surface, like a piece of paper, where you have two intersecting lines that are perpendicular to each other. These lines divide the paper into four equal parts, which are our reference points for plotting the circle.

The equation given is a special form that shows a circle centered at the origin – the point where these lines intersect – and its radius is the square root of the number after the ≤ sign. Here, the radius is the square root of 4, which is 2. To sketch the circle graph accurately, you would draw a round shape touching the points (2,0), (-2,0), (0,2), and (0,-2) on this flat surface or 'plane', as we like to call it in mathematics.

Coordinate System

The coordinate system, also known as the Cartesian plane, is fundamental when plotting graphs such as circles. It consists of a horizontal line called the x-axis and a vertical line called the y-axis. These axes meet at a point called the origin, which has coordinates (0,0).

When you're drawing a graph, you use these axes to find the location of points. For instance, in our circle equation x^{2}+y^{2} \[\]\leq 4, the points (2,0), (-2,0), (0,2), and (0,-2) lie on the axes. The first number in each pair, the x-coordinate, tells you how far to move right (positive) or left (negative) from the origin, and the second number, the y-coordinate, tells you how far to move up (positive) or down (negative) from the origin.

Solution Set of Inequalities

The solution set of inequalities can be thought of as a 'zone' where all the points meet the conditions set by an inequality. For a circle graph like the one described by x^{2}+y^{2} \[\]\leq 4, this zone includes not just points on the circle's boundary, but also all the points inside the circle.

This inequality uses a ≤ symbol, meaning 'less than or equal to', which indicates inclusion - in other words, every point whose distance from the origin is less than or equal to 2 is part of the solution set. If you were asked for a particular point, like (1,1), you'd plug these numbers into the equation to see if they satisfy the inequality, confirming if they're within the shaded area that represents our solution set.

Shading Techniques in Graphing

The art of shading techniques in graphing helps visually represent the solution set of an inequality. In the case of our circle inequality, x^{2}+y^{2} \[\]\leq 4, you should shade the entire area that represents the set of points included in the solution.

To do this effectively, first sketch the circle with a solid line to indicate that points on the circumference are included (since we have the ≤ sign). Next, use a pencil or a highlighter to shade the interior of the circle gently. This shaded region clearly shows anyone reading the graph that every point within the boundary is a solution to the inequality. Without shading, it would be less clear which side of the boundary line contains the solutions.