Problem 43 - Algebra A Combined Function

Question

Identify whether each equation, when graphed, will be a parabola, circle,ellipse, or hyperbola. Sketch the graph of each 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

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

Step 1: Identify the Form

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

Step 2: Determine Circle Characteristics

Identify the characteristics of the circle from the standard circle form. The center \((h, k)\) of the circle is \((-\frac{1}{2}, \frac{1}{2})\) and the radius \(r\) is \(1\), since \(r^2 = 1\).

Step 3: Sketch the Circle

Sketch the circle on a graph, placing the center at \((-\frac{1}{2}, \frac{1}{2})\). From the center, draw a circle with a radius of 1, meaning all points are 1 unit away from the center.

Step 4: Label the Graph

On the sketched graph, label the center of the circle \((-\frac{1}{2}, \frac{1}{2})\) and the radius as 1 unit. Ensure these details are clearly marked to distinguish the graph as that of a circle.

Key Concepts

Circle Equation

The equation of a circle is foundational in understanding conic sections. In its standard form, a circle equation is expressed as \( (x-h)^2 + (y-k)^2 = r^2 \). Here, \((h, k)\) represents the center of the circle, and \(r\) denotes the radius. This formula is derived from the Pythagorean Theorem and helps describe all the points equidistant from the center, forming a perfect circle.

When working with a circle equation, identifying the components \((h, k)\) and \(r\) is essential as they provide the circle's position and size. For instance, if you have an equation like \( (x+\frac{1}{2})^2 + (y-\frac{1}{2})^2 = 1 \), you can immediately see that it's based on the pattern \( (x-h)^2 + (y-k)^2 = r^2 \).

Understanding this form is critical, as it allows you to directly visualize and graph the circle on a coordinate plane.

Graphing

Graphing a circle involves plotting all points that maintain a fixed distance, or radius, from the center point \((h, k)\). Start by identifying the center from the circle equation. For example, from \( (x+\frac{1}{2})^2 + (y-\frac{1}{2})^2 = 1 \), the center is \((-rac{1}{2}, rac{1}{2})\).

Once the center is located, use the radius \(r\) which, in this case, is 1 (since \( r^2 = 1 \)). Then, begin at the center and measure out 1 unit in all directions—left, right, up, and down. Connecting these points forms a perfect circle.

Graphing accurately requires careful marking of the center and understanding how far the radius stretches from this point. A clear graph should resemble a circle rather than an oval or ellipse, emphasizing its equal radii in all directions.

Radius

In a circle equation, the radius \(r\) is a critical quantity representing the constant distance from the center of the circle to any point on its border. The radius is extracted from the equation form \( (x-h)^2 + (y-k)^2 = r^2 \).

• If \(r^2 = 1\), as in the example equation, the radius \(r\) is 1.
• The larger the radius, the bigger the circle, indicating a wider spread of points from the center.

The radius is always a positive value and cannot be negative since distance cannot be negative.

Practically, knowing the radius allows you to graph and scale the circle appropriately on paper, ensuring each plotted point is equidistant from the circle's center, creating a symmetric and accurate circle depiction.

Center of Circle

The center of a circle is the fixed point \((h, k)\) from which all points on the circle are equidistant. It's a crucial feature that determines the circle's position in a plane. This center is directly spotted from the standard equation form \( (x-h)^2 + (y-k)^2 = r^2 \).

\(h\) and \(k\) are the coordinates that define the horizontal and vertical position of the circle's center. For instance, in the equation \( (x+\frac{1}{2})^2 + (y-\frac{1}{2})^2 = 1 \), subtracting inside the expressions reveals \( (h, k) = (-\frac{1}{2}, \frac{1}{2}) \).

Accurately identifying the center is a primary step in graphing or analyzing a circle, as it serves as the anchor for the entire shape, ensuring symmetry and correct placement of the radius around it.