Problem 61 - 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. $$(x-4)^{2}+y^{2}=7$$

Step-by-Step Solution

Answer

The graph is a circle with center (4, 0) and radius \(\sqrt{7}\).

Step 1: Identify the Type of Graph

The equation given is \( (x-4)^2 + y^2 = 7 \). This equation is in the form \((x-h)^2 + (y-k)^2 = r^2\), which represents a circle equation. Here, \(h = 4\), \(k = 0\), and \(r^2 = 7\).

Step 2: Determine the Center of the Circle

From the equation \((x-4)^2 + y^2 = 7\), we can see that the center \((h, k)\) of the circle is \((4, 0)\).

Step 3: Calculate the Radius of the Circle

In the equation, \(r^2 = 7\), so to find the radius \(r\), we take the square root of 7. Thus, \(r = \sqrt{7}\).

Step 4: Sketch the Graph

To sketch the graph:1. Plot the center of the circle at \((4, 0)\).2. Draw a circle with radius \(\sqrt{7}\), approximately 2.645, using the center as the reference point.

Key Concepts

Circle Equation

The circle equation is a fundamental concept in geometry and algebra that helps define a circle on a coordinate plane. An important aspect to understand is that the standard form of a circle equation 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 our exercise, the given equation is \((x-4)^2 + y^2 = 7\).

• Comparing it to the standard form, it's clear that the circle has its center at \((h, k) = (4, 0)\).
• The equation also shows us that \(r^2 = 7\).

This format is very helpful as it enables you to quickly identify the key features of the circle, the center and the radius square, directly from the equation.

Graphing Techniques

Graphing a circle involves several straightforward steps that help ensure you accurately depict the shape on a coordinate plane. Here’s a simple method you can follow:

• First, plot the center of the circle. Using coordinates \((h, k)\), locate this point on the graph. In our exercise, this is the point \((4, 0)\).
• Next, identify the radius of the circle from the radius squared term in the equation. If the equation is of the form \((x-h)^2 + (y-k)^2 = r^2\), you need to find \(r\) by taking the square root of the number on the right side.
• Once you know the radius, use a compass or a similar tool to draw a circle around the center point. Ensure that every point on the circle is exactly a radius length from the center.

This process accurately depicts a circle. Make sure to keep your graph neat, helping you maintain precision in sketching.

Radius Calculation

Understanding how to calculate the radius from a circle's equation is crucial in graphing and determining the circle's size.
Within a circle equation, such as \((x-h)^2 + (y-k)^2 = r^2\), the term \(r^2\) represents the square of the radius. To find \(r\):

• Identify \(r^2\) in the equation. In our exercise, \(r^2 = 7\).
• To solve for \(r\), take the square root of \(r^2\). This gives \(r = \sqrt{7}\).

This square root method works universally for circle equations and is a simple yet powerful tool to master. Despite seeming complex, calculations involving square roots become straightforward once you are familiar with the process. Always approximate square roots for decimal values as necessary, such as using \(\sqrt{7} \approx 2.645\) in practical graphing.