Problem 118 - Algebra and Trigonometry

Question

will help you prepare for the material covered in the next section. Use your graph of \(x^{2}+y^{2}=1\) from Exercise 117 to determine the relation's domain and range.

Step-by-Step Solution

Answer

The domain of the relation \(x^{2}+y^{2}=1\) is [-1, 1], and the range is also [-1, 1].

Step 1: Understanding the equation

Understand that the equation \(x^{2}+y^{2}=1\) represents a circle centered at the origin (0,0) with a radius of 1. Values of x and y are defined such that they satisfy this equation.

Step 2: Finding the domain

The domain of an equation is the set of all possible x-values. For this circle, x can take on any value between -1 and 1, inclusive, which can be mathematically represented as x: [-1, 1]. So the domain of this relation is [-1, 1].

Step 3: Finding the range

The range of an equation is the set of all possible y-values. Like x, y can also take on any value between -1 and 1, inclusive, which can be mathematically represented as y: [-1, 1]. So the range of this relation is [-1, 1].

Key Concepts

Domain and Range of Circle Equations

When dealing with circle equations like \(x^2 + y^2 = 1\), understanding the domain and range is crucial. First, recall that the equation \(x^2 + y^2 = 1\) represents a circle with a radius of 1 centered at the origin, (0,0). Since the circle is perfectly symmetric around its center, this offers insight into both the domain and range. The **domain** of an equation refers to all possible x-values that satisfy the equation. For the unit circle, the x-values can range from -1 to 1. This is because the furthest horizontal reach of the circle is 1 unit to the left and right from the origin. Thus, the domain of the circle \(x^2 + y^2 = 1\) is \[-1, 1\].

Similarly, the **range** of the equation is the set of y-values that make the equation true. Here, y-values also range from -1 to 1, as the circle extends 1 unit above and below its center. Therefore, the range of the circle \(x^2 + y^2 = 1\) is \[-1, 1\]. To summarize, for a circle equation such as \(x^2 + y^2 = 1\), both the domain and the range are the intervals \([-1, 1]\). Understanding these intervals helps in plotting the circle and analyzing how x and y interact within its boundary.

Coordinate Geometry and Circle Equations

Coordinate geometry, also called analytic geometry, is a branch of mathematics that connects algebra and geometry using a coordinate system. In the case of circle equations, it helps visualize and analyze the properties of circles in a two-dimensional plane.### Circle Basics
The equation \(x^2 + y^2 = 1\) is a standard form of a circle centered at the origin with a radius of 1. Such an equation tells us that any point \((x, y)\) on the circle is equidistant from the center (0,0).

• The term \(x^2 + y^2\) ensures the core concept of a circular shape is maintained.
• The number 1 on the right side represents the square of the radius.

### Applications in Coordinate Geometry
Coordinate geometry allows us to apply algebraic techniques to solve problems based on geometric figures. This includes:

• Calculating distances and midpoints.
• Understanding symmetries and transformations.
• Analyzing geometric properties like angles and tangents related to circles.

In simple terms, coordinate geometry makes it easier to work with abstract shapes, enabling a deeper understanding of geometric principles through an algebraic lens.

Graphing Circle Equations

Graphing a circle equation is a powerful way to visualize and deeply understand the relationship between x and y values. Starting with the equation \(x^2 + y^2 = 1\), let's explore how to graph it step by step. ### Plotting Steps
1. **Identify the center and radius:** The circle is centered at the origin (0,0) with a radius of 1.2. **Mark the radius on the axes:** From the origin, move 1 unit up, down, left, and right to plot these boundary points.3. **Draw the circle:** Use these points to sketch a circle around the origin, maintaining equal distance throughout.### Graphing Tips
- Ensure precision by using a compass if possible, as this maintains a consistent radius.- Label the axes and critical points like (-1,0), (1,0), (0,-1), and (0,1) to clarify the span of the circle.- Double-check that the circle touches but does not exceed these boundary points.Graphing not only allows us to see the full extent of the domain and range physically but also enhances the comprehension of the symmetrical nature of a circle. This visual approach reinforces the theoretical domain and range values derived algebraically from the equation.