Problem 28 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 23-34: Sketch the graph of the circle or semicircle. $$ x^{2}+(y-2)^{2}=25 $$

Step-by-Step Solution

Answer

The circle is centered at (0, 2) with a radius of 5.

Step 1: Identify the Standard Equation

The equation given is in the form \( x^2 + (y-k)^2 = r^2 \), which is the standard form of a circle equation. Here, \( h = 0 \), \( k = 2 \), and \( r^2 = 25 \).

Step 2: Determine the Center and Radius

From the standard form \( (x-h)^2 + (y-k)^2 = r^2 \), the center of the circle is \((h, k)\). Here it is \((0, 2)\). The radius \( r \) is the square root of 25, which is 5.

Step 3: Plot the Center

Plot the center of the circle on the coordinate plane at the point \((0, 2)\). This is the starting point for drawing the circle.

Step 4: Sketch the Circle

With the center at \((0, 2)\) and radius 5, draw a circle by marking points that are a distance of 5 units away from the center in all directions.

Key Concepts

Standard Form of a Circle

Understanding the standard form of a circle is crucial in graphing and analyzing circles in math. The standard form equation of a circle is \[ (x-h)^2 + (y-k)^2 = r^2 \]where:

• \((h, k)\) represents the center of the circle.
• \(r\) is the radius of the circle.

In any given circle equation, identifying these components makes it infinitely easier to graph the circle. For instance, in the equation \[ x^2 + (y-2)^2 = 25 \], the center of the circle, \((h, k)\), is found by comparing the given equation with the standard form. Here, the equation tells us the center is at \((0, 2)\) and the radius is \[ \sqrt{25} = 5 \]. Notice how the parts of the circle equation provide immediate values for these crucial circle characteristics. This structure allows quick conversion of equations into visual representations on a graph.

Graphing Circles

Graphing a circle involves using its equation to lay it out visually on a coordinate plane. Follow these steps:

First, identify the center and radius from the standard form of the equation as outlined earlier. The center \((h, k)\) is your starting point.

Next, plot the center on the coordinate plane. For the equation \[ x^2 + (y-2)^2 = 25 \], you would place a point at \((0, 2)\).

After placing the center, use the radius to find points around the center. From \((0, 2)\), measure 5 units in all directions (up, down, left, right, and diagonals) to locate points on the circle.

Finally, draw a smooth curve through these points connecting them to form a circle. Graphing tools or careful hand plotting may be helpful to achieve an accurate circle.

Radius and Center of a Circle

The radius and center of a circle are the two key components that define the circle's position and size. The center is simply the middle point of the circle from which all boundary points (the edge of the circle) are equidistant.

To find the center from an equation like \[ (x-h)^2 + (y-k)^2 = r^2 \], extract \((h, k)\). For \[ x^2 + (y-2)^2 = 25 \], the center is \((0, 2)\).

The radius, \(r\), is the distance from the center to any point on the circle. Mathematically, it's the square root of the number on the right-hand side of the equation. Again, using our example equation, the radius is \[ \sqrt{25} = 5 \].

These elements help not only in drawing the circle but also in understanding its spatial characteristics on any given plane, allowing us to visualize the circle fully.