Problem 62 - Intermediate Algebra

Question

Sketch the graph of each equation. If the graph is a parabola, find its vertex. If the graph is a circle, find its center and radius. $$x^{2}+(y+5)^{2}=5$$

Step-by-Step Solution

Answer

The center of the circle is (0,-5) and the radius is \(\sqrt{5}\).

Step 1: Identify the Equation Type

The equation given is \[ x^{2} + (y+5)^{2} = 5 \]This is the standard form of a circle's equation: \[ (x - h)^2 + (y - k)^2 = r^2 \] Comparing, we see this equation represents a circle.

Step 2: Determine the Center

Using the circle equation \( (x - h)^2 + (y - k)^2 = r^2 \), we identify \( h = 0 \) and \( k = -5 \). Thus, the center of the circle is at the point \((0, -5)\).

Step 3: Determine the Radius

In the circle's equation, the value on the right side \( r^2 = 5 \). Therefore, the radius is \[ r = \sqrt{5} \].

Step 4: Sketch the Graph

Plot the center of the circle at \((0, -5)\) on the coordinate plane. Using the radius \(\sqrt{5}\), draw the circle around the center. Ensure the circle has a radius of approximately \(2.24\), since \(\sqrt{5} \approx 2.24\).

Key Concepts

Standard Form Equation

Understanding that the equation \(x^{2}+(y+5)^{2}=5\) fits the mold of a standard circle equation is crucial. The standard form of a circle's equation is expressed as \((x - h)^2 + (y - k)^2 = r^2\).
Here, \(h\) and \(k\) represent the x and y coordinates of the circle's center, and \(r\) is the radius.
By comparing equations, we can identify the values of \(h\), \(k\), and \(r^2\) directly from the equation. With this form, understanding other parts of the circle becomes simple.

Center of a Circle

To find the center of the circle from its equation \((x - h)^2 + (y - k)^2 = r^2\), you need to extract the values of \(h\) and \(k\).
These values are the coordinates of the circle's center. In this problem:

• The term \((x - h)^2\) suggests \(h = 0\), since \(x\) is not adjusted by any number.


• The term \((y - k)^2\) corresponds to \((y + 5)^2\), indicating that \(k = -5\).

Therefore, the circle's center is located at the point \((0, -5)\) on the coordinate plane.

Radius Calculation

Finding the radius of a circle involves recognizing the format of the equation \(r^2\).
In the equation \(x^{2}+(y+5)^{2}=5\), the number \(5\) is equivalent to \(r^2\).
For the radius \(r\), take the square root of \(5\), which results in \(r = \sqrt{5}\).
This evaluates to approximately \(2.24\). It's essential to perform this calculation since the radius is the distance from the center to any point on the circle.

Graph Sketching

Sketching a graph of a circle involves a few clear steps to visualize the mathematical properties.
Start by plotting the center point at \((0, -5)\) on a coordinate plane. This is the fixed point from which the circle extends outward.

• With the center plotted, use the radius \(\sqrt{5}\) (approximately \(2.24\)) to draw the circle.


• Measuring from the center, mark several points exactly \(2.24\) units away.


• Ensure that these points create a smooth round shape as you connect them back to the starting point.

Following these steps will help accurately sketch the circle's graph, giving a clear view of its boundary.