Problem 54 - 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. $$x^{2}+y^{2}=49$$

Step-by-Step Solution

Answer

The graph is a circle with center at (0, 0) and radius 7.

Step 1: Identify the Equation Type

The given equation is in the form of \( x^2 + y^2 = 49 \). This is the standard equation of a circle, which has no \(xy\) term and no linear terms such as \(x\) or \(y\).

Step 2: Compare with Standard Form

In the standard form of a circle \( (x - h)^2 + (y - k)^2 = r^2 \), we identify: \(h = 0\), \(k = 0\), and \(r^2 = 49\). Thus, the circle is centered at \((0, 0)\) and has a radius \(r\).

Step 3: Calculate the Radius

Since \(r^2 = 49\), we determine the radius \(r\) by taking the square root: \(r = \sqrt{49} = 7\).

Step 4: Graph the Circle

To sketch the graph, plot the center of the circle at \((0, 0)\) and draw a circle with radius 7. It extends 7 units in all directions from the center.

Key Concepts

Standard Form of a Circle

The standard form of a circle is a way of representing the equation of a circle in the Cartesian coordinate system. It is given as \((x - h)^2 + (y - k)^2 = r^2\). In this formula:

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

This equation tells us everything we need to know about the circle's size and position on the graph. By comparing any given circle's equation with the standard form, you can quickly identify the circle's center and radius.
For example, if your equation is \(x^2 + y^2 = 49\), you notice it doesn't have any \(h\) or \(k\). That means the center is at the origin \((0, 0)\), and we equate \(r^2 = 49\). The radius is found by taking the square root of 49, which is 7.

Graphing Circles

Graphing circles may seem a bit challenging at first, but it's straightforward once you grasp how the equation relates to the graph. In a coordinate plane, a circle is perfectly round and symmetrical.
To graph a circle:

• Start by plotting the center of the circle as identified by \((h, k)\).
• From this central point, measure out the circle's radius \(r\) in all four cardinal directions: up, down, left, and right.

Use these points as guides to draw a smooth and even circle. The goal is to make sure every part of the circle is equidistant from the center, and it should extend evenly away from this point.
Graphing on paper might involve drawing a guide with a compass if you're working manually, but software tools often automate this process, making it easy to visualize circles precisely.

Radius and Center of a Circle

Understanding the radius and center of a circle is crucial in working with circle equations. The center, given by \((h, k)\) in the equation \((x-h)^2 + (y-k)^2 = r^2\), is the point from which every part of the circle is equally distant.
The radius \(r\) is the constant distance from the center to any point on the circle's edge. This distance determines the size of the circle:

• A larger radius means a bigger circle.
• A smaller radius results in a smaller circle.

To find these values in an equation like \(x^2 + y^2 = 49\), you identify that \((h, k)\) is \((0, 0)\), making the origin the center. The radius \(r\) is determined by solving \(r^2 = 49\), where you find \(r = 7\). Knowing these concepts allows you to interpret and sketch any circle equation you encounter effectively.