Problem 97 - A Graphical Approach to Precalculus with Limits

Question

Graph by hand the equation of the circle or the parabola with a horizontal axis. $$x^{2}+y^{2}=1$$

Step-by-Step Solution

Answer

Graph a circle with center at (0, 0) and radius 1 on the coordinate plane.

Step 1: Identify the Type of Equation

The given equation is \(x^2 + y^2 = 1\). This is the general equation of a circle. A circle's equation is typically of the form \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center of the circle and \(r\) is the radius.

Step 2: Determine the Center and Radius

Comparing the given equation \(x^2 + y^2 = 1\) with the general circle equation \((x - h)^2 + (y - k)^2 = r^2\), it's clear that \(h = 0\), \(k = 0\), and \(r^2 = 1\). Thus, the center of the circle is \((0, 0)\) and the radius \(r\) is \(1\).

Step 3: Plot the Center

Start plotting by marking the center of the circle on the graph. In this case, the center is at the origin \((0, 0)\).

Step 4: Plot Points Using the Radius

From the center \((0, 0)\), measure a distance of \(1\) unit in all cardinal directions (up, down, left, right). Mark these points: \((1, 0)\), \((-1, 0)\), \((0, 1)\), and \((0, -1)\), since each represents a direction on the grid corresponding to the radius.

Step 5: Draw the Circle

Connect the points with a smooth, round curve. This forms the boundary of the circle. Ensure that it passes through each of the plotted points and looks symmetrical.

Key Concepts

Equation of a Circle

The equation of a circle is a fundamental concept in geometry, notable for its unique form and significant properties. The standard equation of a circle is \((x-h)^2 + (y-k)^2 = r^2\). Here,

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

Understanding components of this equation is vital. The expression \((x-h)^2 + (y-k)^2\) quantifies the sum of the squares of the distances from any point on the circle to the center. This ensures that each point on the circle is equidistant from the center. In the given exercise, \(x^2 + y^2 = 1\) indicates a circle centered at the origin, \((0,0)\), with a radius of 1 unit. Recognizing these features helps you comprehend the circle's shape and placement in a coordinate plane.

Plotting Graphs

Plotting graphs involves visualizing mathematical equations on a coordinate plane. To graph the equation \(x^2 + y^2 = 1\), begin by identifying key attributes, such as the center and radius. For this circle:

• Center: Located at origin \((0, 0)\).
• Radius: Measure of 1 unit.

Start by marking these coordinates directly on the graph. The next step requires translating the radius onto the graph. From the center, count 1 unit outward in all directions (up, down, left, right). This method ensures symmetry around the circle’s center, ensuring the accurate plotting of the points:

• \((1, 0)\)
• \((-1, 0)\)
• \((0, 1)\)
• \((0, -1)\)

These points are crucial as they will guide the smooth curve drawn to finish sketching the circle visually on the graph paper.

Geometric Shapes

Geometric shapes, like circles, are pervasive in math and nature. Understanding circles requires comprehensive knowledge of properties and the ability to represent them graphically. In geometry, a circle is defined as the set of all points in a plane that are equidistant from a given point, called the center. This circle perception opens doors to:

• Analyzing symmetry, since every point on the edge holds equal distance from the center.
• Calculating areas and circumferences which are pivotal in various applications, such as designing circular objects.
• Understanding transformations like rotations and translations, which preserve the circle’s properties.

Translating equations like \(x^2 + y^2 = 1\) into graphical forms assists in mastering these shapes. It applies both abstract mathematical concepts and practical skills, such as symmetry utilization and grid mapping, emphasizing the integral role of geometric shapes in mathematics.