Problem 6 - Algebra 2

Question

Graph each equation. Identify the conic section and describe the graph and its lines of symmetry. Then find the domain and range. $$ x^{2}+y^{2}=49 $$

Step-by-Step Solution

Answer

The given equation represents a circle with a center at the origin (0,0) and a radius of 7 units. It has lines of symmetry along both x-axis and y-axis. Domain and Range of this function are both \[-7,7\].

Step 1: Graphing the Equation

Here, the equation given is \(x^{2} + y^{2} = 49\). This is the standard form of a circle centered at the origin (0,0). The radius of the circle can be found by taking the square root of 49 which gives 7. Thus it's a circle with radius 7 centered at the origin. Now, plot the circle using these values.

Step 2: Identifying the Conic Section

The equation given is of the form \(x^{2} + y^{2} = r^{2}\), which represents a circle in the Cartesian coordinate system.

Step 3: Describing the Graph and its Lines of Symmetry

The graph of \(x^{2} + y^{2} = 49\) is a circle centered at the origin (0,0) with a radius of 7 units. It has two lines of symmetry: one is the y-axis and the other is the x-axis.

Step 4: Finding the Domain and Range

Since the graph is a circle centered at the origin with radius 7, the domain (set of all x values) and the range (set of all y values) are both \[-7,7\].

Key Concepts

Circle Equation

A circle equation describes a geometric shape known as a circle on a coordinate plane. The standard form of a circle equation is \( x^2 + y^2 = r^2 \), where \( r \) represents the radius of the circle. This equation tells us that the circle is centered at the origin, which is the point \((0, 0)\). The radius \( r \) is the distance from the center to any point on the circle. To find the radius, you can take the square root of the number on the right side of the equation. In the equation \( x^2 + y^2 = 49 \), the number 49 is \( r^2 \), so the radius \( r \) is \( \sqrt{49} = 7 \). Understanding this basic circle equation helps us easily identify and graph circles.

Graphing Circles

Graphing circles involves plotting the circle on the coordinate plane based on its equation. To graph \( x^2 + y^2 = 49 \), follow these steps:

• Identify the center of the circle. For this equation, the center is the origin \((0, 0)\).
• Determine the radius, which in this case is 7, as calculated from \( r = \sqrt{49} \).
• From the center, mark points 7 units away in all directions: up, down, left, and right.
• Draw a smooth curve connecting these points to form a circle.

By following these steps, you create an accurate graph of the circle, visually representing the solutions to the circle equation. The circle should touch each point that is a radius length away from the center.

Lines of Symmetry

A circle is known for its perfect symmetry in the coordinate plane. It has an infinite number of lines of symmetry, but two primary axes stand out: the x-axis and the y-axis. These axes are special because they divide the circle into two identical halves:

• The x-axis is a horizontal line that mirrors the top half of the circle to the bottom half.
• The y-axis is a vertical line that reflects the left half to the right half.

These lines of symmetry highlight the balanced nature of a circle, making it one of the simplest and most symmetrical shapes in geometry. Each point on one side of the line finds a corresponding point directly across the line.

Domain and Range

The domain and range of a circle's graph dictate where the circle exists on the coordinate plane. For the equation \( x^2 + y^2 = 49 \), both values are limited by the circle's size and position:

• The **domain**, representing all possible x-values, is the interval from -7 to 7. This range is the horizontal spread of the circle, as it stretches 7 units left and 7 units right from the origin.
• The **range**, which represents all possible y-values, similarly spans from -7 to 7. This is the vertical spread, going 7 units above and 7 units below the center.

The domain and range are both \([-7, 7]\), encompassing the entirety of the circle's presence on the plane. This concept is crucial in understanding the limitation of circle equations within coordinate systems.