Problem 5 - A Graphical Approach to College Algebra

Question

Skills Match each equation in Column I with the appropriate description in Column II. Do not use a calculator. A. Circle; center \((3,-4) ;\) radius 5 B. Parabola; opens left C. Parabola; opens upward D. Circle; center \((-3,4)\); radius 5 E. Parabola; opens right F. Circle; center \((0,0) ;\) radius \(\sqrt{5}\) G. No points on its graph H. Parabola; opens downward $$x^{2}+y^{2}=5$$

Step-by-Step Solution

Answer

Match with F: Circle; center \((0,0)\); radius \(\sqrt{5}\).

Step 1: Identify Equation Type

The given equation is \(x^2 + y^2 = 5\). This equation is of the form \(x^2 + y^2 = r^2\), which represents a circle.

Step 2: Determine Circle Parameters

For the circle equation \((x-h)^2 + (y-k)^2 = r^2\), the center is \((h, k)\) and the radius is \(r\). In our equation, there are no shifts in \(x\) or \(y\) (i.e., \(x - 0\) and \(y - 0\)), so the center is \((0, 0)\).

Step 3: Find Radius

Since \(r^2 = 5\), solving for \(r\) gives \(r = \sqrt{5}\). Therefore, the circle has a radius of \(\sqrt{5}\).

Step 4: Match with Column II

The equation \(x^2 + y^2 = 5\) represents a circle with center \((0, 0)\) and radius \(\sqrt{5}\). According to Column II, the correct match is F: Circle; center \((0,0)\); radius \(\sqrt{5}\).

Key Concepts

Center of a Circle

A circle on the coordinate plane is defined by its center and radius. The center of a circle is the point that is equidistant from all points on the circle. In the standard form of a circle equation, \[ (x - h)^2 + (y - k)^2 = r^2 \] - \((h, k)\) represents the center of the circle.- If no shifts are made to the values of \(x\) and \(y\), meaning there are no additional numbers subtracted in the equation, then the center is at the origin \((0, 0)\).
In our example, the equation is \(x^2 + y^2 = 5\), which simplifies to \((x - 0)^2 + (y - 0)^2 = 5\). This means the center is clearly \((0, 0)\). Understanding the location of the center helps in drawing and interpreting circles on a graph.

Radius of a Circle

The radius of a circle is the distance from the center to any point on the circle. This distance remains constant throughout the circle. In the circle's equation: - The equation form is \((x - h)^2 + (y - k)^2 = r^2\), where \(r\) is the radius.
To find the radius in an equation like \(x^2 + y^2 = 5\):1. Recognize the form is already simplified from \((x - 0)^2 + (y - 0)^2 = r^2\).2. Here, \(r^2 = 5\).
To find \(r\), simply take the square root of both sides, leading to \(r = \sqrt{5}\). It’s essential to remember that the radius describes the overall size of the circle, with larger radii leading to larger circles.
Stay attentive to whether \( r^2 \) is just a number like 5, which means the calculation requires finding the square root to determine the actual radius length.

Equation Matching

Equation matching involves relating a given equation to specific characteristics like the center and radius for circles. This exercise helps in identifying and verifying the geometric figures equations represent.To match a circle equation such as \(x^2 + y^2 = 5\) to its description:- Recognize the given form is \(x^2 + y^2 = r^2\), indicating it's a circle equation.- Identify parameters: - Center \((0, 0)\) - Radius \(\sqrt{5}\)
Referring to the problem statement, match these attributes:- **Identifier F in Column II** describes a Circle with center at \((0, 0)\) and radius \(\sqrt{5}\). This makes it the correct match.Matching equations to descriptions requires understanding the parameters defined by the standard equation's form. Through practice, this process becomes quick and intuitive, helping in effectively analyzing geometric figures.