Problem 30 - Algebra 2

Question

Graph each equation. Describe the graph and its lines of symmetry. Then find the domain and range. $$ 11 x^{2}+11 y^{2}=44 $$

Step-by-Step Solution

Answer

The graph is a circle centered at (0,0) with a radius of 2. It has two lines of symmetry: \(y = 0\) and \(x = 0\). The domain and range are both \([-2, 2]\) inclusive.

Step 1: Rewrite the equation in standard form

The equation is given as \(11x^2 + 11y^2 = 44\). Divide the entire equation by 11 to obtain \(x^2 + y^2 = 4\), which is a standard form of circle equation with center at (0,0) and radius \(r = \sqrt{4} = 2\).

Step 2: Graph the circle

Plot the center point at (0,0) on a graph, then plot points at a distance of 2 units from the center in every direction. Sketch a circle passing through these points.

Step 3: Identify lines of symmetry

The circle will have two lines of symmetry. These are the horizontal and the vertical lines passing through the center. Here, they are \(y = 0\) and \(x = 0\).

Step 4: Determine the domain

The domain (set of possible x-values) for a circle with center (0,0) and radius 2 is \([-2, 2]\). It means x can take any value between -2 and 2 inclusive.

Step 5: Determine the range

The range (set of possible y-values) for the circle is also \([-2, 2]\). It means y can also take any value between -2 and 2 inclusive.

Key Concepts

Graphing Circles

Graphing a circle can be a fun and visually informative process. The equation provided, \(x^2 + y^2 = 4\), is the standard form of a circle equation. This tells us two important things: the center of the circle and its radius.

The center of this circle is at (0,0) \(\text{-- a location known as the origin}\).

• This makes it straightforward to identify where the circle is located on the graph.
• To find the radius, take the square root of 4, which gives a radius of 2.

Once you know the center and the radius, you'll position the center at (0,0) on a graph. From there, measure out the radius of 2 units in every direction. By plotting points at these distances, you can draw a round, symmetric shape that completes the circle.
This type of equation makes it clear that a circle is entirely symmetric centered around the origin, displaying the elegance of its shape in any direction.

Domain and Range

When examining circles, the domain and range can tell you a lot about the spread of the circle across the graph. The domain of a function refers to all the possible x-values. For a circle equation like \(x^2 + y^2 = 4\), the domain is determined by the width of the circle in the x-direction.

The domain is \([-2, 2]\).\

This means that the circle spans from -2 to 2 on the x-axis. Similarly, the range refers to the set of possible y-values or the height of the circle.

The range of this circle is also \([-2, 2]\), \

signaling its reach from the bottom to the top.

• Both domain and range values are inclusive, indicating that all values within -2 to 2 are part of the circle.
• This is because, geometrically, the points on the edge themselves contribute to the boundary of the circle.

Understanding domain and range helps in grasping how functions behave graphically, and circles are a perfect example of this balance along both axes.

Lines of Symmetry

Lines of symmetry make understanding geometric shapes much easier. For a circle, these lines show the mirror-like symmetry that this shape possesses.
A circle, such as \(x^2 + y^2 = 4\), has an infinite number of lines of symmetry. However, the most notable are the vertical and horizontal lines passing through the center.

• The vertical line is represented by \(x = 0\). This line cuts the circle into two equal halves vertically.
• Similarly, the horizontal line \(y = 0\) splits it into two equal halves horizontally.

These lines of symmetry highlight how a circle is a perfect example of rotational symmetry about the origin. Circles are symmetric in all directions, showing beautiful balance and harmony.By identifying these lines, you can easily predict how the circle reflects itself across axes.