Question

Write the equation of each circle in standard form. Graph. center at (0,0)\(;\) radius \(=4.82\)

Short answer

The standard form of the circle's equation is \(x^2 + y^2 = 23.2324\).

Step-by-step solution

Write the standard form of a circle's equation

The standard form of a circle's equation is \[ (x - h)^2 + (y - k)^2 = r^2 \] where \(h, k\) is the center of the circle and \(r\) is the radius.

Plug in the center coordinates and radius into the standard equation

Since the center is at \(0,0\) and the radius is \(4.82\), we plug these values into the standard form: \[ (x - 0)^2 + (y - 0)^2 = (4.82)^2 \].

Simplify the equation

Simplify the equation by squaring the radius and removing the zeros: \[ x^2 + y^2 = 23.2324 \].

Graph the circle

To graph the circle, locate the center at (0,0) and then measure the distance of 4.82 units in all directions to create the circumference of the circle.

Key concepts

Graphing Circles

Understanding how to graph circles on the coordinate plane starts with visualizing the circle as a set of points that are all at an equal distance from its center. When graphing a circle, that equal distance is called the radius. Here's how you turn that understanding into a graph:

• First, identify the center of the circle, which will be a point on your graph. In our example, the center is at the origin, (0,0).
• Next, use a compass or simply measure out the length of the radius from the center in all directions. In this exercise, since the radius is 4.82, you would measure 4.82 units from the center to multiple points that create an outline of the circle.
• Then connect these points smoothly to form the circumference of the circle. This gives you the visual representation of the circle on the graph.

To improve this exercise, a visual aid showing steps of drawing the circle on a graph, starting from plotting the center to measuring out the radius, and finally connecting the points, could be provided to help students better understand the process.

Radius and Center of a Circle

Every circle has a central point which is equidistant from every point on the edge of the circle. This point is known as the center. In mathematical notation, the center of a circle on a coordinate plane is often represented as a pair of coordinates \(h, k\). When you know the center, you can easily find any point on the circle by moving the length of the radius away from \(h, k\) in every direction. The radius is a straight line from the center to any point on the circle and is the same length for any point on the circle's edge. In our exercise, the center is \(0,0\) and the radius is 4.82. Knowing these two components is crucial for writing the equation of the circle in standard form and for graphing the circle, as outlined in the graphing section above.

For improvement purposes, students could benefit from interactive tools that allow them to adjust the radius and see how the size of the circle changes on the graph, enhancing their understanding of the relationship between radius and the size of the circle.

Equations of Circles in Standard Form

The equation of a circle in its standard form is a powerful expression that contains all the information you need to understand the circle's size and position. It is given by the formula \[ (x - h)^2 + (y - k)^2 = r^2 \], where \(h, k\) represents the coordinates of the circle's center, and \(r\) represents the radius. To convert the equation of a circle into standard form, you'll plug in the values of the center and radius into their respective places in the equation.

In the example given, we're dealing with a center at \(0,0\) and a radius of 4.82. By following the formula, we arrive at the standard form of \[ x^2 + y^2 = 23.2324 \]. This tells us that the circle is centered at the origin and has a radius such that any point on the circle is approximately 4.82 units from the center.

An improvement suggestion for the exercise could include breaking down the formula in a step-by-step manner with illustrations for each term to ensure clarity in understanding how each part of the formula corresponds to the physical properties of the circle.