Problem 41 - Intermediate Algebra

Question

Write each equation in standard form, if it is not already so, and graph it. The problems include equations that describe circles, parabolas, ellipses, and hyperbolas. $$ (x+1)^{2}+(y-2)^{2}=16 $$

Step-by-Step Solution

Answer

The equation is already in standard form. The circle's center is (-1, 2) with a radius 4.

Step 1: Identify the Type of Conic

The given equation \( (x+1)^{2}+(y-2)^{2}=16 \) is in the form \( (x-h)^2 + (y-k)^2 = r^2 \), which is the equation of a circle. Here, the center is at \( (h, k) = (-1, 2) \) and the radius \( r \) is \( 4 \) since \( 16 = r^2 \).

Step 2: Confirm the Standard Form for a Circle

The standard form for a circle is \( (x-h)^2 + (y-k)^2 = r^2 \). The given equation \( (x+1)^{2}+(y-2)^{2}=16 \) already matches this form, so it is already in standard form.

Step 3: Graph the Equation

To graph the circle: 1. Plot the center of the circle at \( (-1, 2) \).2. Since the radius is \( 4 \), mark points that are \( 4 \) units away from the center in all directions (up, down, left, right). 3. Draw the circle by connecting these points smoothly.

Key Concepts

Circle Equations

Circle equations are a fundamental part of conic sections in geometry. A circle, in its standard form equation, is expressed as \((x-h)^2 + (y-k)^2 = r^2\). This is the equation that defines a set of all points that are a fixed distance, known as the radius \(r\), from a central point \((h, k)\). Understanding the components of this equation is crucial for recognizing and graphing circles:

• Center \((h, k)\): This is the point from which distances (radius) are measured to determine the boundary of the circle.
• Radius \(r:\) This is a non-negative number representing the distance from the center to any point on the circle.

When working with circle equations, it is vital to identify these components correctly. In the example provided, \((x+1)^2 + (y-2)^2 = 16\), the center is \((-1, 2)\) and the radius is \(4\), since \(16\) is equal to \(r^2\). Writing equations in this form allows for easy interpretation and graphing of circles.

Graphing Conic Sections

Graphing conic sections involves plotting geometric shapes like circles, parabolas, ellipses, and hyperbolas on a coordinate plane. For a circle, once the equation is in standard form, graphing becomes straightforward:

• First, identify the center of the circle from the equation. In our example, the center is \((-1, 2)\).
• Next, determine the radius. Here, the radius is \(4\), since \(16 = 4^2\).
• Plot the center point on the graph.
• Using the radius, mark points that are 4 units away from the center in each direction (up, down, left, right).
• Finally, draw a smooth curve through these points to complete the circle.

By identifying the center and radius, you can easily visualize and graph the circle on a coordinate plane. This approach applies to other conic sections with their respective characteristics and equations.

Standard Form of Equations

The standard form of equations for conic sections provides a clear, simplified way to identify and graph them. Each conic section has its unique standard form:

• Circle: \((x-h)^2 + (y-k)^2 = r^2\)
• Parabola: For vertical axis \(y-k = a(x-h)^2\), for horizontal axis \(x-h = a(y-k)^2\)
• Ellipse: \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\)
• Hyperbola: \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\)

Converting to standard form often involves completing the square for quadratic expressions or simply rearranging terms. This step is crucial as it directly reveals key characteristics of the conic section, such as the center, vertices, focus, or asymptotes, depending on the type of conic. Accurately identifying these elements facilitates both the analysis and graphing of the equation. For instance, knowing the standard form of a circle's equation directly gives the center and radius, making the task of graphing routine and straightforward.