Problem 42 - 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. $$ (x-3)^{2}+(y+4)^{2}=1 $$

Step-by-Step Solution

Answer

Equation represents a circle with center (3,-4) and radius 1; graph it by plotting the center and measuring a radius of 1.

Step 1: Identify the Equation Type

The given equation \[(x-3)^2 + (y+4)^2 = 1\] is in the standard form of a circle. A general circle equation is \((x-h)^2 + (y-k)^2 = r^2\).

Step 2: Re-write the Equation in Standard Form

The provided equation \[(x-3)^2 + (y+4)^2 = 1\] is already in the standard form of a circle which is \[(x-h)^2 + (y-k)^2 = r^2\]. Here, \(h=3\), \(k=-4\), and \(r=1\).

Step 3: Identify the Circle's Components

From the standard form \[(x-3)^2 + (y+4)^2 = 1\], the circle has a center at \((3, -4)\) and a radius of \(1\).

Step 4: Graph the Circle

To graph:- Start by plotting the center of the circle at the coordinate \((3, -4)\).- Measure a distance of \(1\) unit (the radius) from the center in all directions (up, down, left, right) to plot points that lie on the circle.- Connect these points smoothly to form the circle.

Key Concepts

Equations of Circles

In coordinate geometry, an equation of a circle tells us about its essential features, such as the center and the radius. A standard form equation for a circle is given as \[(x-h)^2 + (y-k)^2 = r^2\],where:

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

To identify and work with the equation of a circle, it’s crucial to recognize that the equation \[(x-3)^2 + (y+4)^2 = 1\] is already in the standard form with \(h=3\), \(k=-4\), and \(r=1\). This tells us the circle’s center is at point \((3, -4)\) and the radius is \(1\). By understanding the components represented in this equation, you gain insights into the circle's location and size on a graph.

Graphing Equations

Graphing equations involves plotting points on the coordinate plane based on the equation and visualizing the shape it forms. For circles, graphing starts with identifying specific features:

• Locate the center: For \[(x-3)^2 + (y+4)^2 = 1\], plot the center at \\((3, -4)\) on the graph.
• Determine the radius: Here, the radius is \(1\). From the center, you measure this distance in all cardinal directions—up, down, left, and right.

Once these points are plotted, smoothly connect them to form the outline of the circle. This process helps visualize how the circle sits relative to other shapes and points on the same graph. Graphing not only demonstrates the circle's placement but also its orientation within the plane.

Algebraic Equations

Algebraic equations form the foundation for various geometric shapes, including circles. These equations are expressions of algebra that hold an equality true under certain conditions. For a circle's equation, the algebra ensures that every \((x, y)\)pair satisfies the original circle equation as:

• The term \((x-h)^2\) ensures that the x-values are measured from the center (h), enhancing horizontal symmetry.
• The term \((y-k)^2\) ensures that the y-values are measured from the center (k), ensuring vertical symmetry.

Algebra helps in re-writing any circle's equation in its standard form so that centers and radii can be easily identified. Mastery of these algebraic manipulations allows for an accurate understanding of the circle's boundaries and dimensions from its equation.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, bridges the gap between algebra and geometry using coordinates to define shapes like circles. By positioning figures on a unique coordinate system, it enables precise analysis of geometric properties.For instance, the equation \[(x-3)^2 + (y+4)^2 = 1\] translates into geometric terms like the circle's radius and center when plotted.

• The coordinate system allows for exact plotting of the circle's center at \((3, -4)\), serving as a reference point for the shape's symmetry.
• Using unit measures, it clarifies how shapes relate with one another over the plane, revealing intersection points and more.

Thus, coordinate geometry provides the tools needed to connect algebraic equations to their graphical representations, ensuring that mathematical concepts are both visualized and understood in spatial terms.