Problem 38 - Algebra 2

Question

Graph each circle so that the center is at the origin. Then write the equation. diameter 2.5

Step-by-Step Solution

Answer

The equation of the circle is \( x^2 + y^2 = 1.5625 \). Note: The plotted circle could vary depending on the type of graphing tools used.

Step 1: Determine radius

The radius of a circle is half of its diameter. Given the diameter is 2.5, the radius of the circle is \( \frac{2.5}{2} = 1.25 \).

Step 2: Write circle equation

Use the general equation of a circle \( x^2 + y^2 = r^2 \) . Substituting the radius value calculated above, the equation of the circle becomes \( x^2 + y^2 = (1.25)^2 = 1.5625 \).

Step 3: Plot the circle

On a graph, plot the circle with its center at the origin (0,0). The circle should pass through the points at which x and y coordinates are 1.25 or -1.25, due to its radius being 1.25. Hence, four points where circle can pass through are (1.25,0), (-1.25,0), (0,1.25) and (0,-1.25).

Key Concepts

Radius and Diameter

The radius and diameter are fundamental concepts when working with circles. The diameter is the distance across the circle, passing through the center. The radius, on the other hand, is half the diameter. It extends from the center to any point on the circle's boundary. In mathematical terms:

• Diameter: The full width of the circle, which is always twice the radius.
• Radius: Half the diameter, calculated by dividing the diameter by two.

For example, if a circle's diameter is 2.5 units, then its radius is computed as follows:\(\text{Radius} = \frac{2.5}{2} = 1.25 \text{ units}.\)Understanding these definitions is crucial because the radius is used to write the equation of a circle, such as \(x^2 + y^2 = r^2\). This equation is essential for plotting and analyzing circles.

Plotting Graphs

Plotting graphs is a way to visualize mathematical concepts, like circles. Start by finding key points that lie on the circle. This begins with understanding that the center is at the origin, which is the point (0,0). From there, use the radius to identify boundary points. For a circle with a radius of 1.25, it will touch the graph at several distinct locations:

• (1.25, 0)
• (-1.25, 0)
• (0, 1.25)
• (0, -1.25)

These points mark where the circle touches the x and y axes. Connect these points smoothly to form a circle. Creating a graph offers a visual perspective, making it easier to understand abstract concepts like equations of circles.

Center at the Origin

When a circle's center is at the origin, its mathematical equation becomes simplified. The origin is the point (0,0) on a graph where the x-axis and the y-axis intersect. For circles, this point is often used as the center because it simplifies calculations and helps in understanding symmetrical properties. The general equation for a circle with its center at the origin is given by:\[ x^2 + y^2 = r^2 \]Here, \(r\) is the radius. So, for a radius of 1.25, we plug this value into the equation as:\[ x^2 + y^2 = (1.25)^2 = 1.5625 \]This equation shows that every point \((x, y)\) on the circle is equidistant from the origin, with that distance being the circle's radius. Setting the center at the origin also assists with plotting since it aligns the circle symmetrically across the axes.