Problem 35 - Intermediate Algebra

Question

Identify whether equation, when graphed, will be a parabola, circle, ellipse, or hyperbola. Sketch the graph of equation. If a parabola, label the vertex. If a circle, label the center and note the radius. If an ellipse, label the center. If a hyperbola, label the \(x\) - or \(y\) -intercepts. \((x-7)^{2}+(y-2)^{2}=4\)

Step-by-Step Solution

Answer

The graph represents a circle with center at (7, 2) and radius 2.

Step 1: Identify the Type of Conic Section

First, look at the standard forms of conic sections. This equation \( (x-7)^{2} + (y-2)^{2} = 4 \) is already in the form \( (x-h)^{2} + (y-k)^{2} = r^{2} \), which is the standard form of a circle. Hence, the conic section described by this equation is a circle.

Step 2: Determine the Center and Radius

From the standard form \( (x-h)^{2} + (y-k)^{2} = r^{2} \), the center of the circle \( h, k \) is \( (7, 2) \). The radius \( r \) can be found by taking the square root of the right side of the equation. In this case, \( r = \sqrt{4} = 2\). Thus, the circle has a center at \( (7, 2) \) and a radius of \( 2 \).

Step 3: Sketch the Graph

To sketch the circle, plot the center at \( (7, 2) \) on the coordinate plane. From this center, measure a distance of \( 2 \) in all directions (up, down, left, right) to draw a circle. The equation describes all points that are exactly \( 2 \) units away from the center.

Step 4: Label the Center and Radius on the Graph

Clearly label the center of your circle \( (7, 2) \). Also, note on the graph that the circle has a radius of \( 2 \). This informs the viewer of the key characteristics of the circle.

Key Concepts

Circle Equations

In mathematics, the equation of a circle is used to represent all the points that are a fixed distance, known as the radius, from a given point called the center. The standard form of a circle's equation is \((x-h)^2 + (y-k)^2 = r^2\). Here, \( (h, k) \) represents the center of the circle, and \( r \) is the radius. This format is similar to how we might describe a circle in everyday terms: a set of all points equidistant from a center point.

• \(x\) and \(y\) are coordinates of any point on the circle.
• \(h\) and \(k\) are coordinates of the center of the circle.
• \(r\) represents the radius, or distance from the center to any point on the circle.

By identifying these components in a circle equation, like \((x-7)^2 + (y-2)^2 = 4\), we are able to determine the circle's specific location and size on a coordinate plane.

Graphing Circles

Graphing circles can be an enjoyable and visual way to understand their properties. To graph a circle from its equation, you simply locate its center and measure the radius in all directions. With the circle equation \((x-h)^2 + (y-k)^2 = r^2\), you start by plotting the center point \((h, k)\) on the graph.Next, you measure the radius \(r\) from the center point in four main directions: up, down, left, and right. These are called the cardinal directions on the graph, and the circle should touch these points, maintaining the same radius all around. It's like stretching a string of length \(r\), with one end fixed at the center point, and drawing a round curve as you move around.

• Plot the center \((h, k)\).
• Draw points at \(r\) units away in all cardinal directions.
• Connect the points smoothly for a perfect curve.

This practice clarifies how a circle looks mathematically and provides a foundation for sketching any circle from its equation.

Center and Radius of a Circle

Knowing how to find and interpret the center and radius of a circle is crucial when dealing with circle equations. Based on the standard circle equation \((x-h)^2 + (y-k)^2 = r^2\), the values \(h\) and \(k\) define the center, while \(r\) is the radius.For example, for the circle equation \((x-7)^2 + (y-2)^2 = 4\), you can directly extract that:

• The center is at \((h, k) = (7, 2)\).
• The radius \(r\) is \(\sqrt{4}\), which simplifies to 2.

This understanding allows you to visualize where the circle is positioned and how broad it is. Identifying the center tells you the exact point about which the circle is perfectly symmetrical, and knowing the radius informs you of the distance from that center to any point on the circle's edge.Finding these components helps not only in plotting graphs but also in solving practical problems related to geometry and spatial reasoning.