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Chapter 5: Problem 87

In Exercises 79-88, sketch the graph of the equation. $$ x^{2}+(y-8)^{2}=25 $$

Short Answer

Expert verified
The graph is a circle centered at (0, 8) with radius 5 units.

Step by step solution

01

Identify the center and radius

The general equation of a circle is \((x-h)^{2}+(y-k)^{2}=r^{2}\). Compare this with the given equation \(x^{2}+(y-8)^{2}=25\), we get h=0, k=8 and \(r^{2}=25\). After square rooting r we get r=5.
02

Plot the center of the circle (h,k)

On a graph paper, plot the point (0,8) which signifies the center of the circle.
03

Draw the circle

From the center, measure 5 units in all directions: up, down, left and right to have four additional points on the graph. Using these points, sketch a complete circle on the plane.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Equation of a Circle
The equation of a circle is a key concept in geometry and provides a way to represent a circle algebraically. The standard form is \((x-h)^{2}+(y-k)^{2}=r^{2}\), where:
  • \(h\) and \(k\) are the x and y coordinates of the circle's center,
  • \(r\) is the radius of the circle.
In the given exercise, the equation is \(x^{2}+(y-8)^{2}=25\). By comparing this with the standard form, we identify the center at \((0,8)\) and the radius as \(5\), since \(r^{2} = 25\), so \(r = \sqrt{25} = 5\). This equation provides all necessary information to sketch the circle accurately on a graph.
Coordinate Plane
The coordinate plane is the foundation for graphing equations and geometrical shapes, such as circles. It is a two-dimensional surface where each point is defined by a pair of numerical coordinates:
  • The horizontal axis is known as the x-axis.
  • The vertical axis is called the y-axis.
In this exercise, you plot the circle based on its center \((0,8)\), meaning it's 8 units up from the origin along the y-axis. Understanding how to move along these axes is essential for accurately placing points and drawing shapes.
Graph Sketching
Graph sketching involves visualizing and drawing geometric figures based on algebraic equations. When sketching the graph of a circle, start with plotting the center, which in this case is \((0,8)\), and note it on the graph.
Once the center is marked, draw points that are \(5\) units away from the center in all four cardinal directions鈥攗p, down, left, and right. This step is crucial because it gives you guiding points that dictate the shape of the circle.
Finally, connect these points with a smooth, round line to complete the circle. This process helps reinforce the relationship between the equation and the graphically represented circle.
Geometry Concepts
Geometry concepts form the backbone of understanding shapes like circles in coordinate planes. Recognizing that a circle is a set of all points equidistant from a central point is fundamental.
  • The distance is measured as the radius, which dictates the size of the circle.
  • The center provided by the equation determines the circle's position on the coordinate plane.
  • In geometry, precision in distances and positioning is crucial for accurate drawing.
Visualizing these concepts while plotting ensures that the geometric interpretation remains coherent and accurate, linking the abstract algebraic equation to its concrete spatial representation on a plane.

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