Problem 101 - College Algebra Essentials

Question

Exercises \(100-102\) will help you prepare for the material covered in the next section. Use a rectangular coordinate system to graph the circle with center \((1,-1)\) and radius 1

Step-by-Step Solution

Answer

The equation of the circle is \( (x-1)^2 + (y+1)^2 = 1 \), and it has been plotted using these parameters.

Step 1: Identify the given values

The given values are the coordinates of the center of the circle \((1,-1)\) and the radius of the circle which is 1.

Step 2: Form the Equation of the Circle

The general form for the equation of a circle is \( (x-a)^2 + (y-b)^2 = r^2 \). Substituting the center coordinates \((a,b)\) as \((1,-1)\) and \(r\) as 1 into the equation, results in \( (x-1)^2 + (y+1)^2 = 1 \)

Step 3: Graph the Circle

Using a plotting tool or a graph paper, label the x and y axes. Mark the centre of the circle at point (1,-1). Then, plot the circle with a radius of 1 unit.

Key Concepts

Rectangular Coordinate System

Imagine a grid where you can pinpoint any location with just two numbers—this is the essence of the rectangular coordinate system. It's made up of two perpendicular lines called axes. The horizontal axis is known as the X-axis, and the vertical one is called the Y-axis. Each point on the plane is defined by a unique pair of numbers, known as coordinates, which indicate its distance from these axes. For example, in the exercise, the center of the circle is at coordinates (1, -1), meaning it lies 1 unit to the right of the Y-axis (because of the positive '1') and 1 unit below the X-axis (due to the negative '-1').

This system is incredibly useful in graphing shapes like circles, as it allows us to visually represent the mathematical concepts and equations on paper or a digital screen with precision.

Equation of a Circle

The equation of a circle serves as a mathematical roadmap, telling us all the points (x,y) that sit along a circle's edge. It follows a straightforward pattern: (x-a)² + (y-b)² = r², where (a, b) is the center and r is the radius of the circle. In this equation, (x-a)² and (y-b)² are the squared distances from any point on the circle to the center point. Under this format, if you select any point that satisfies this equation, it will be a point on the circle. When we placed the given center (1, -1) and radius 1 into the formula, we got (x-1)² + (y+1)² = 1. This equation unlocks every point on the circle's boundary, which you can then plot on the grid to see the shape of the circle.

Radius of a Circle

The radius of a circle is not just any line—it's a straight shot from the circle's center to any point on the circle's edge. It's a constant distance, which in our exercise, is given as 1. Because it's consistent, the radius is a crucial piece of the puzzle that keeps the circle perfectly round. When graphing, if you measure out the length of the radius from the center in all directions and connect the dots, you’ll see the circle take shape. The uniformity of the radius length is what makes every point on this boundary equidistant from the center—creating the symmetry and balance that circles are famous for.

Center of a Circle

The center of a circle is, quite literally, at the heart of it all. It's the single point from which every edge of the circle is the same distance away—the radius. In coordinates, we represent the center with a pair of values (a, b), which in our graphing task are (1, -1). It's the control point from which we draw our radius to form the circle. When you're graphing, the center is the starting block. From there, you radiate outwards with your radius, ensuring that every edge is the same distance from this focal point, enabling you to sketch the perfect round figure of the circle on your rectangular coordinate grid.