Problem 50 - Algebra 2

Question

The graph of which equation of a circle contains all the points in the table below? $$\begin{array}{|c|c|c|c|}\hline x & {-3} & {0} & {3} \\ \hline y & {0} & { \pm 3} & {0} \\ \hline\end{array}$$

Step-by-Step Solution

Answer

The equation of the circle which contains all the points in the given table is \(x^2 + y^2 = 9\).

Step 1: Interpreting the Table

From the given table, one can note that the corresponding 'y' values to 'x' are all zero or +- some number. This suggests that the x values indicate the possible max and min radius, thus the origin of the circle will be 'x = 0', which is the middle value of 'x'.

Step 2: Determining the center of the circle

As concluded previously, the 'x' value of the center of the circle is '0'. The 'y' value of '0' in the table corresponds to this 'x' value. So, the origin of the circle is at (0,0). In terms of the standard equation of a circle, this means h = 0 and k = 0.

Step 3: Calculating the radius of the circle

From the table, the maximum 'y' value is +-3, given when 'x' is 0. The center of the circle has been determined to be at (0,0). Hence, the radius of the circle will be the distance from (0,0) to (0,3). By applying the distance formula, this radius value becomes '3'.

Step 4: Obtaining the equation

With the values of 'h', 'k', and 'r' obtained from the previous steps, the equation of the circle can now be established using the standard equation, \( (x-h)^2 + (y-k)^2 = r^2 \). Substituting the determined values, the equation becomes, \( x^2 + y^2 = 9 \).

Key Concepts

Graphing Circles

Understanding how to graph a circle on a coordinate plane is a fundamental part of geometry. A circle is defined by its center point and its radius. The standard equation for a circle is:

• \((x-h)^2 + (y-k)^2 = r^2\)

where \((h,k)\) is the center of the circle and \(r\) is the radius.
To graph a circle, begin at the center point \((h,k)\) and make sure the radius \(r\) is accurate by plotting points that are \(r\) units away from the center in all directions: up, down, left, right, and diagonally.
By connecting these boundary points smoothly, you form the circle. This visual representation helps reinforce the understanding of the circle's geometric properties.
It's crucial to remember that every point on the circle is equidistant from the center, making a perfect round shape.

Geometry

In geometry, circles have unique and useful properties. These properties form the basis for complex geometric reasoning and problem-solving. Understanding the circle's elements such as radius, diameter, and circumference helps in solving geometric problems.

• The radius is the distance from the center of the circle to any point on the edge. It is constant throughout the circle.
• The diameter, which is twice the length of the radius, crosses through the center representing the longest distance across the circle.
• The circumference is the perimeter of the circle calculated by \(2\pi r\), where \(\pi\) is approximately 3.14159.

In the context of the exercise, using these properties helps us identify the circle's equation from given points. Each property works together to give a complete picture of the circle's breadth and scope.

Coordinate Plane

The coordinate plane is a crucial tool for visualizing and solving problems involving geometric shapes like circles. It consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical), which intersect at the origin \((0,0)\).
The coordinate plane allows for precise positioning of geometric figures by using coordinates, which are ordered pairs \((x, y)\).
In the exercise, the origin of a circle was identified as \((0,0)\), and points on the circle were identified using coordinates from the table. This table showed how x and y values relate to each other, helping us derive the circle equation.
Understanding how coordinates work makes it easier to calculate distances, such as the radius, and formulate equations that describe geometric entities accurately.