Problem 19 - Algebra A Combined Function

Question

The graph of each equation is a circle. Find the center and the radius, and then graph the circle. See Examples 5 through 7. $$ x^{2}+(y-2)^{2}=1 $$

Step-by-Step Solution

Answer

Center: (0, 2); Radius: 1.

Step 1: Identify the Standard Circle Equation

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

Step 2: Rewrite the Given Equation in Standard Form

Compare the given equation \(x^2 + (y - 2)^2 = 1\) with the standard form. Notice: \((x - 0)^2 + (y - 2)^2 = 1\).

Step 3: Determine the Center of the Circle

From the rewritten equation, identify \(h = 0\) and \(k = 2\), so the center of the circle is \((0, 2)\).

Step 4: Determine the Radius of the Circle

Identify \(r^2 = 1\) from the equation. Therefore, \(r = \sqrt{1} = 1\). The radius of the circle is 1.

Step 5: Graph the Circle

Plot the center of the circle at \((0, 2)\). Then, use a compass or draw freehand to sketch the circle with radius 1, ensuring it passes through points that are 1 unit away from the center.

Key Concepts

Center of a Circle

The center of a circle in a two-dimensional plane is a critical point that determines the position of the circle. To find the center, we look at the equation of the circle in its standard form, which is \((x - h)^2 + (y - k)^2 = r^2\).
Here, the center \(h, k\) represents the fixed point from which all points on the boundary of the circle are equidistant.
If the equation of a circle is already in the standard form, the values of \( h \) and \( k \) can be directly read off as the coordinates of the center.
For example, in the exercise equation \( x^2 + (y - 2)^2 = 1 \), it can be rewritten as \((x - 0)^2 + (y - 2)^2 = 1\), making the center \(0, 2\).
This means that all points on the boundary of the circle are at an equal distance from \(0, 2\), forming a perfect exponential metric around this central point.

Radius

The radius of a circle is the constant distance from the center to any point on the circle's boundary.
In the standard form equation, \((x - h)^2 + (y - k)^2 = r^2\), the variable \( r \) represents the radius.
This distance is crucial because it defines the size of the circle.
To determine the radius from the exercise \(x^2 + (y-2)^2 = 1\), you first identify \(r^2\).
In this case, \( r^2 = 1\), leading to \( r = \sqrt{1} = 1\).
The radius tells us that every point on the circle is exactly 1 unit from the center, in all directions, around circle.

Graphing Circles

Graphing a circle involves plotting its center and ensuring that all boundary points are equidistant from this center.
Using the center and radius, you can draw a circle representation on a graph.
Here's how you can do it:

• Plot the center of the circle at the given coordinates, such as \(0, 2\).
• Using a compass or a steady hand, draw a line from the center outward equal to the radius length.
• Sketch the circle around the center, maintaining a constant distance equal to the radius from start to end.

In the exercise, starting at the point \(0, 2\) and moving 1 unit outward in all directions will create the circle.
This method ensures the circle maintains its shape relative to standard metrics.

Standard Form of a Circle

The standard form of a circle's equation is a fundamental equation in geometry, given by:
\[(x - h)^2 + (y - k)^2 = r^2\].
This form is useful for quickly identifying significant properties like the center, \( (h, k) \), and the radius, \( r \).

The structure of the equation makes it easier to compare and convert a given circle equation into recognizable mathematical terms.
For instance, if given a generic form like \(x^2 + (y - b)^2 = c\), you can see it matches the standard form by identifying the square components.
This comparison reveals the center as \(h, k\), and the radius as \(r\), derived from \(r^2\).
The standard form showcases the symmetry and geometric properties of circles, assisting in various problem-solving scenarios.