Problem 102 - A Graphical Approach to College Algebra

Question

In Exercises \(97-108,\) graph by hand the equation of the circle or the parabola with a horizontal axis. $$(x+3)^{2}+(y-1)^{2}=1$$

Step-by-Step Solution

Answer

The circle is centered at \((-3, 1)\) with radius 1.

Step 1: Identify the Equation Type

The given equation is \((x+3)^2 + (y-1)^2 = 1\). This is the standard form of a circle, \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.

Step 2: Determine the Circle's Center

In the equation \((x+3)^2 + (y-1)^2 = 1\), compare with \((x-h)^2 + (y-k)^2 = r^2\):- \(h = -3\)- \(k = 1\). Thus, the center of the circle is \((-3, 1)\).

Step 3: Calculate the Radius

The right side of the equation is \(1\), which equals \(r^2\). Solve for \(r\) by taking the square root: \(r = \sqrt{1} = 1\). The circle has a radius of 1.

Step 4: Plot the Circle on a Graph

To graph the circle, you need the center and the radius:1. Plot the center of the circle at \((-3, 1)\).2. From the center, measure the radius of 1 in all directions (up, down, left, right) to locate points on the circle.3. Sketch the circle by connecting these points smoothly, keeping the center at \((-3, 1)\) and ensuring the distance to any point on the circle is 1.

Key Concepts

Equations of Circles

The equation of a circle in its standard form is written as \((x-h)^2 + (y-k)^2 = r^2\). Here, \(h\) and \(k\) represent the coordinates of the center of the circle, while \(r\) stands for the radius. The left-hand side of the equation involves squaring two expressions, which helps in describing the set of all points that are at a fixed distance (the radius) from a given point (the center).
Understanding this standardized equation is like having a blueprint for graphing the circle on a coordinate plane. All you need to find the locus of points that form the circle are the values of \(h\), \(k\), and \(r\). Simply plug these values into the equation to see how the circle is defined mathematically.

• Identify \(h\) and \(k\), which tell you where the center is located.
• Determine \(r\), the radius, to understand how far out the circle extends from the center.

Radius of a Circle

The radius of a circle is the distance from its center to any point on its boundary. In the standard circle equation \((x-h)^2 + (y-k)^2 = r^2\), the right side, \(r^2\), represents the square of the circle's radius. To find the actual radius, you simply take the square root of this value.
For the equation \((x+3)^2 + (y-1)^2 = 1\), the radius squared is \(1\), so the radius \(r\) is \(\sqrt{1} = 1\).
Knowing the radius is crucial for graphing purposes because it defines how large the circle will be on the graph.

• The radius tells you how far you need to measure out from the center in every direction to accurately draw the circle.
• Since the radius is always a positive number, it confirms the circle's size without ambiguity.

Center of a Circle

The center of a circle is a point on the coordinate plane from which all points on the circle's boundary are equidistant. In our standard form equation \((x-h)^2 + (y-k)^2 = r^2\), the center is given by the coordinates \((h,k)\).
Extracting these coordinates requires some simple comparison with the equation's form. From the example equation \((x+3)^2 + (y-1)^2 = 1\), you identify the center as \((-3, 1)\).
Locating the center on the graph provides a fixed point essential for drawing the circle.

• Start by plotting the center point, \((-3, 1)\) in this case, on the coordinate plane.
• This becomes the reference point from which the circle’s radius can be measured to plot additional points and complete the circle's shape.

Remember, a circle's center is a vital piece of information for not only graphing the circle accurately but also interpreting its position in a broader mathematical context.