Problem 16 - Algebra and Trigonometry

Question

\(3-16=\) Graph the inequality. $$ x^{2}+(y-1)^{2} \leq 1 $$

Step-by-Step Solution

Answer

The inequality represents all points within and on a circle centered at (0, 1) with radius 1.

Step 1: Identify the Inequality

We start by examining the given inequality, \(x^2 + (y-1)^2 \leq 1\). This inequality describes a region that includes and is inside a circle with a center located at \((0, 1)\) and with a radius of 1.

Step 2: Understand the Components

The inequality \(x^2 + (y-1)^2 \leq 1\) is derived from the general equation of a circle \((x - h)^2 + (y - k)^2 = r^2\) where \(h\) and \(k\) are the coordinates of the center, and \(r\) is the radius. For our circle, \(h = 0\), \(k = 1\), and \(r = 1\).

Step 3: Draw the Circle

Graph a circle centered at \((0, 1)\) with radius 1. This circle includes all points at or within 1 unit distance from the center. Consider all edge points as part of the solution since \(\leq\) includes equality.

Step 4: Shade the Region Inside the Circle

Shade the inside of the circle to represent the solution set of the inequality. This shaded area includes all points \((x, y)\) such that the sum of the squares of \(x\) and \((y - 1)\) is less than or equal to 1.

Key Concepts

Circle Equation

A circle equation in coordinate geometry is a concise way to represent all points equidistant from a central point. The standard form of a circle equation is \[(x - h)^2 + (y - k)^2 = r^2\]where - \((h, k)\) represent the coordinates of the circle's center,- and \(r\) indicates the circle's radius.In the provided exercise, the equation \(x^2 + (y-1)^2 = 1\) shows us a circle centered at \((0, 1)\).The radius \(r = 1\) points out the distance from any point on the circle's boundary to its center. Essentially, this circle incorporates all points from the center up to one unit outward. This formula helps visualize and graph the circle accurately on a coordinate plane.

Inequality Solutions

An inequality involving a circle indicates a situation where solutions are not confined to the circle alone. In this problem, the inequality is \[x^2 + (y-1)^2 \leq 1\]Where the expression \( \leq \) implies that solutions exist within the circle and include points on the border.**Understanding Inequality Signs**- \( \leq \) or \( \geq \) means "less than or equal to" or "greater than or equal to".The inequality directs us to shade the circle and the area enclosed, indicating all points \((x, y)\) wherein the sum of the squares of \(x\) and \(y-1\) is one unit or less.This technique visually represents the solution set, providing a clear picture of feasible values satisfying the inequality condition.

Coordinate Geometry

Coordinate geometry, or analytic geometry, merges algebra and geometry to provide a unified approach to solving geometric problems. It involves the study of geometric figures using a coordinate system.In this exercise, using coordinate geometry helps translate the circle equation and associated inequality into a graphical representation on a coordinate grid.**Key Aspects of Coordinate Geometry**

• Coordinate Plane: Consists of a horizontal \(x\)-axis and a vertical \(y\)-axis, which intersect at the origin \((0, 0)\).
• Plotting Points: Any point on the plane is denoted \((x, y)\), where \(x\) is the horizontal distance from the origin and \(y\) is the vertical distance.
• Graphing Circle Equations: Employs the circle equation to accurately plot positions that maintain a certain distance from the center.

Utilizing these concepts, identifying the center and radius of the circle becomes straightforward through the equation's variables. Thus, enabling anyone to easily display such figures graphically and interpret inequalities visually.