Problem 62 - Intermediate Algebra

Question

Write each equation in standard form, if it is not already so, and graph it. If the graph is a circle, give the coordinates of its center and its radius. If the graph is a parabola, give the coordinates of its vertex. $$ x^{2}+(y-3)^{2}=25 $$

Step-by-Step Solution

Answer

The graph is a circle with center at (0, 3) and radius 5.

Step 1: Identify the Equation Type

The given equation is \(x^{2} + (y - 3)^{2} = 25\). This is a circle equation because it is in the form \((x - h)^{2} + (y - k)^{2} = r^{2}\), where \(h\) and \(k\) are the coordinates of the center and \(r\) is the radius.

Step 2: Determine the Center of the Circle

In the equation \((x - h)^{2} + (y - k)^{2} = r^{2}\), we compare and find \(h = 0\) and \(k = 3\). Thus, the center of the circle is \((0, 3)\).

Step 3: Find the Radius of the Circle

The right side of the equation \((x - 0)^{2} + (y - 3)^{2} = 25\) gives \(r^{2} = 25\). Solving for \(r\), we find \(r = \sqrt{25} = 5\). Therefore, the radius of the circle is \(5\).

Step 4: Graph the Circle

To graph the circle, plot the center at \((0, 3)\) on the coordinate grid. Then, draw a circle with a radius of \(5\) units around this center.

Key Concepts

Equations of Circles

The equation provided, \(x^2 + (y - 3)^2 = 25\), fits the standard form of a circle's equation. In general, the standard form of a circle's equation is represented as

• \((x-h)^2 + (y-k)^2 = r^2\),

where \((h, k)\) are the coordinates of the circle's center, and \(r\) is the radius of the circle.
To simplify:

• \(x^2 + (y - 3)^2 = 25\).

Here, the center \((h, k)\) is \((0, 3)\), since \(h = 0\) and \(k = 3\). The radius \(r\) is derived from \(r^2 = 25\), giving us \(r = 5\). For equations in this form, relaxation is key. Remember that understanding the equation itself tells you everything about the circle's position and size.

Graphing Circles

To graph a circle, you primarily need two pieces of information from the equation: the center \((h, k)\) and the radius \(r\). In our example, the center is at \((0, 3)\) and the radius is \(5\). Graphing a circle involves plotting these details on a coordinate grid.
First, mark the center at \((0, 3)\) on the graph.
Then, from this center point, count \(5\) units outward in all directions (up, down, left, right, and also towards each diagonal) to ensure you capture the circle's complete circumference.
Finally, draw a smooth curve, maintaining an equal distance from the center to connect these points around the center, forming a perfect circle. This visual representation is invaluable in understanding the spatial distribution defined by the equation.

Coordinate Geometry and Circles

In coordinate geometry, a circle's equation like \(x^2 + (y - 3)^2 = 25\) is instrumental in visualizing its properties on a grid. By analyzing the equation, you're engaging in a key aspect of coordinate geometry, which bridges algebra with spatial interpretation.

• With the circle's center set at \((0,3)\), you have a reference point.
• The radius \(5\) indicates every point on the circle's circumference is \(5\) units away from this center.

These aspects help build spatial reasoning as you see how particular values translate into distinct shapes and positions on the coordinate plane. This comprehension of circles is a pivotal concept in more complex geometric and real-world problem-solving scenarios where such spatial relationships play significant roles.