Problem 49 - Intermediate Algebra

Question

Identify whether equation, when graphed, will be a parabola, circle, ellipse, or hyperbola. Sketch the graph of equation. If a parabola, label the vertex. If a circle, label the center and note the radius. If an ellipse, label the center. If a hyperbola, label the \(x\) - or \(y\) -intercepts. \(\left(x+\frac{1}{2}\right)^{2}+\left(y-\frac{1}{2}\right)^{2}=1\)

Step-by-Step Solution

Answer

The equation is a circle with center \((-\frac{1}{2}, \frac{1}{2})\) and radius 1.

Step 1: Identify the Standard Form

The given equation is \( \left(x+\frac{1}{2}\right)^{2}+\left(y-\frac{1}{2}\right)^{2}=1 \). This resembles the standard form of a circle equation, \( \left(x-h\right)^2 + \left(y-k\right)^2 = r^2 \).

Step 2: Identify the Center and Radius

Comparing \( \left(x+\frac{1}{2}\right)^2 + \left(y-\frac{1}{2}\right)^2 = 1 \) to \( \left(x-h\right)^2 + \left(y-k\right)^2 = r^2 \), we find that \(h = -\frac{1}{2}\), \(k = \frac{1}{2}\), and \(r^2 = 1\). Thus, the center is \((-\frac{1}{2}, \frac{1}{2})\) and the radius is 1.

Step 3: Sketch the Graph

Draw a coordinate plane, mark the center of the circle at \((-\frac{1}{2}, \frac{1}{2})\). From this center, draw a circle with a radius 1 unit, ensuring the edge touches points 1 unit away in all directions.

Key Concepts

Circle Equation

In mathematics, the equation of a circle is fundamental in understanding how circles are represented in algebraic form. When we talk about the circle equation, we refer to the expression that defines all points equidistant from a specific point, known as the center. Typically, circles are discussed in relation to the Cartesian coordinate system. Here, a circle can be described using the equation:

• \[ (x - h)^2 + (y - k)^2 = r^2 \]

This formula represents a circle centered at the point \((h, k)\) with a radius \(r\). The terms \((x - h)^2\) and \((y - k)^2\) correspond to the square of the horizontal and vertical distances from the center, ensuring that these distances maintain the radius equal across all directions.

• \(h, k\): coordinates of the center
• \(r\): radius

Knowing how to derive and manipulate this equation is key to understanding various properties of circles.

Standard Form of a Circle

The term "standard form" refers to a traditional structured way of expressing the equation of a circle. In its standard form, a circle becomes easier to handle as its relevant characteristics, namely its center and radius, are explicitly stated. An equation like:

• \[ (x - h)^2 + (y - k)^2 = r^2 \]

quickly reveals the center \( (h, k) \) and the radius \(r\). By identifying portions of this formula with a given equation, you can easily determine these properties.

• If the equation requires transformation, complete the square or rearrange terms to fit this form.
• The standard form simplifies graphing since it directly indicates how the circle is placed on the coordinate plane.

The given problem's standard form makes analyzing and sketching the circle straightforward by directly providing the necessary information.

Graphing Circles

Graphing a circle accurately is all about understanding its equation and then translating this understanding onto a coordinate plane. To graph a circle:

• First, identify the center from the circle's equation. For instance, in the equation \((x - h)^2 + (y - k)^2 = r^2\), the center is at point \( (h, k) \).
• Next, note the radius, \(r\), which tells you the distance from the center to any point on the circle.
• Draw the center point, and then mark a point \(r\) units away in each direction: horizontally, vertically, and diagonally.
• Join these points with a smooth, circular curve to complete the graph.

Taking these steps helps in visualizing the circle in its correct position and size on the graph. Practice ensures precision in crafting circles accurately reflected by their equations.

Center and Radius of a Circle

The center and radius are fundamental aspects of circles, vital for both understanding and graphing them. The center, denoted \((h, k)\), is the fixed point around which the circle is uniformly distributed. The radius, \(r\), represents the constant distance from the center to any point on the circle's perimeter:

• Center \((h, k)\): Identified directly from the circle's equation in standard form, a shift in \(x\) and \(y\) coordinates indicates movement from the origin.
• Radius \(r\): Derived by taking the square root of the equation's right-hand side: \(r^2\). This value denotes how large or small the circle will appear.

For example, the equation \(((x+\frac{1}{2})^2 + (y-\frac{1}{2})^2 = 1)\) indicates a center at \((-\frac{1}{2}, \frac{1}{2})\) with a radius of 1. Understanding these components enables precise control over how a circle is analyzed and plotted, ensuring you capture its correct properties.