Problem 22 - Precalculus: Mathematics for Calculus

Question

Graph the inequality. $$x^{2}+(y-2)^{2} \leq 4$$

Step-by-Step Solution

Answer

The graph is a circle centered at (0, 2) with radius 2, including its interior.

Step 1: Identify the Circle Equation

The given inequality is \(x^2 + (y - 2)^2 \leq 4\). Notice that this is the equation of a circle written as an inequality. The standard form of a circle's equation is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center of the circle and \(r\) is the radius.

Step 2: Determine the Center and Radius

From the inequality \(x^2 + (y - 2)^2 \leq 4\), we can identify \(h = 0\), \(k = 2\), and \(r^2 = 4\). Thus, the center of the circle is \((0, 2)\) and the radius is \(r = \sqrt{4} = 2\).

Step 3: Create the Graph Outline

Draw the outline of the circle with the center at \((0, 2)\) and a radius of 2 units. Since the inequality is \(\leq\), we will include the boundary of the circle in the graph. The boundary should be represented by a solid line.

Step 4: Shade the Interior

For the inequality \(x^2 + (y - 2)^2 \leq 4\), we need to shade the region that represents the interior of the circle. This indicates all the points \((x, y)\) that satisfy the inequality.

Key Concepts

Equation of a Circle

The equation of a circle is fundamental in understanding how to graph its geometric representation. A circle's equation in standard form is given by \((x - h)^2 + (y - k)^2 = r^2\). Here, \((h, k)\) represents the center of the circle, and \(r\) stands for the radius. This formula is vital as it helps pinpoint the exact location of the circle in a coordinate plane. When you encounter an inequality like \(x^2 + (y - 2)^2 \leq 4\), it resembles the standard circle equation but allows for additional elements, including shading. Graphing inequalities involves identifying how the standard equation transforms into a representation that includes shaded regions to show solutions satisfying the inequality.

Center of a Circle

The center of a circle is the point that is equidistant from all points on the circle's circumference. In the standard circle equation \((x - h)^2 + (y - k)^2 = r^2\), the center is represented by \((h, k)\). Understanding the center is crucial because it serves as the anchor point when graphing the circle. In the example inequality \(x^2 + (y - 2)^2 \leq 4\), our center is \((0, 2)\). This tells us that the circle is vertically displaced 2 units upward from the origin. Locating the center accurately ensures that the circle is graphed correctly on the coordinate plane, impacting any shading that illustrates the inequality's solutions.

Radius of a Circle

The radius is the distance from the center of the circle to any point on its edge. In the standard form, \((x - h)^2 + (y - k)^2 = r^2\), \(r\) represents the radius. Calculating the radius is simple once the equation is known. For the inequality \(x^2 + (y - 2)^2 \leq 4\), we see that \(r^2 = 4\). Taking the square root gives us a radius of \(r = 2\). The radius helps to determine how large the circle will appear on the graph. By ensuring the radius is plotted correctly, you can accurately indicate the bounds used for shading when graphing the inequality.

Shading Inequalities

Shading inequalities is an essential step when graphing an inequality involving a circle. It signifies which part of the plane includes all possible solutions to the inequality. For example, in the inequality \(x^2 + (y - 2)^2 \leq 4\), shading the circle's interior demonstrates all \((x, y)\) coordinate pairs that satisfy the inequality. Since the inequality symbol is \(\leq\), the circle's boundary is part of the solution set and is drawn with a solid line. Shading inside the circle shows that every point within the circle, including those on its border, meets the conditions set by the inequality. This visual representation assists in fully understanding the solutions that align with the graph's constraints.