Problem 3 - Algebra A Combined Function

Question

Graph each inequality. $$ x^{2}+y^{2} \geq 16 $$

Step-by-Step Solution

Answer

Shade the region outside the circle with radius 4 centered at the origin.

Step 1: Identify the Shape

The given inequality is based on the equation of a circle: \(x^2 + y^2 = r^2\). Here, \(r^2 = 16\), which implies \(r = 4\). Thus, the inequality \(x^2 + y^2 \geq 16\) represents all points on or outside a circle with a radius of 4 centered at the origin.

Step 2: Understand the Boundary Condition

The boundary of the region where the inequality holds is defined by the equation \(x^2 + y^2 = 16\). This is a circle centered at the origin with radius 4.

Step 3: Draw the Circle

Plot the circle with center at the origin (0,0) and a radius of 4. This means drawing a circle that crosses the x-axis at (-4, 0) and (4, 0), and the y-axis at (0, -4) and (0, 4).

Step 4: Determine and Shade the Region

Since the inequality is \(x^2 + y^2 \geq 16\), you need to shade the region outside and on the circle. This indicates the set of points that satisfy the inequality, meaning all points where the distance from the origin is greater than or equal to 4.

Key Concepts

Graphing Inequalities

Graphing inequalities is a step beyond simply graphing equations. Inequalities indicate a range of solutions, rather than just a line or curve. For instance, when graphing the inequality \(x^2 + y^2 \geq 16\), we are asked to show not just the circle described by \(x^2 + y^2 = 16\), but also the region where \(x^2 + y^2\) is greater than 16.

• The circle itself, defined by \(x^2 + y^2 = 16\), acts as a boundary. Any point on this circle means \(x^2 + y^2\) equals 16.
• The inequality symbol, \(\geq\), indicates that we also include points for which \(x^2 + y^2\) is greater than 16. This means shading the entire area outside the circle, including the circle itself.

Think of it as describing all the places a point could be that are at least 4 units away from the origin.

Circle Equations

Circle equations are frequently expressed as \(x^2 + y^2 = r^2\). This is fundamental in coordinate geometry. The equation gives a circle's shape, where \(r\) is the radius and the circle is centered at the origin (0,0).

• The equation \(x^2 + y^2 = 16\) implies a circle with radius 4, as the square root of 16 is 4.
• This form helps us understand not only the size of the circle but also its position in the coordinate plane.

To graph, pinpoint the center of the circle and plot points that are 4 units away in all directions. This will help create the boundary of the circle, aiding in visualizing and solving inequality solutions.

Boundary Condition

The concept of boundary conditions is essential when dealing with inequalities like \(x^2 + y^2 \geq 16\). It refers to the strict equal part of the inequality between two expressions.

• The boundary, described by \(x^2 + y^2 = 16\), is the dividing line where the inequality starts to apply. It separates points where \(x^2 + y^2\) can be equal to 16 from those where it is greater.
• In graphing terms, this boundary is the circle itself. Whether you include the boundary or not depends on whether the inequality is \(\geq\) or \(>\). Here, we include it as the inequality is \(\geq\).

Identify the boundary to recognize which parts of your graph will need shading, indicating the region that satisfies the inequality.

Coordinate Geometry

Coordinate geometry integrates algebraic equations with geometric visualizations. It’s key for understanding spatial relationships in graphs. When graphing the circle described by \(x^2 + y^2 = 16\):

• Place the center at the origin (0,0) on a Cartesian plane.
• Plot the circle with a fixed radius of 4 units.
• The axes help map out points, like (4,0) and (0,4), defining the circle's points of interception.

This discipline translates numerical data into geometric representation, making it easier to visualize solutions to equations and inequalities, like determining regions in graphs that satisfy specific conditions.