Problem 9 - 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. $$ 6 x^{2}+6 y^{2}=600 $$

Step-by-Step Solution

Answer

The conic section represented by the given equation is a circle with a radius of 10 units. It has lines of symmetry along the x-axis and y-axis. The domain and range are both from -10 to 10 (inclusive).

Step 1: Simplify the equation

First, look at the given equation \(6x^{2}+6y^{2}=600\). To simplify this, divide every term by 6 to get \(x^{2}+y^{2}=100\). This is the equation of a circle with radius \(\sqrt{100}\), which is 10.

Step 2: Identify the conic section

Since the equation is now in the form \(x^{2} + y^{2} = r^{2}\), it's clear that this represents a circle. The conic section is therefore, a circle.

Step 3: Describe the graph and the lines of symmetry

A circle is symmetric with respect to both the x-axis and y-axis. Hence, the lines of symmetry for this circle are x=0 (y-axis) and y=0 (x-axis).

Step 4: Determine the domain and range

Since the radius is 10, all points on the circle are within -10 and 10 along both x and y axes. Therefore, the domain (possible x-values) is: \(-10 \leq x \leq 10\) and the range (possible y-values) is \(-10 \leq y \leq 10\)

Key Concepts

circle

In geometry, a circle is a fundamental shape characterized by all points being equidistant from a central point. The standard equation of a circle centered at the origin is given by \(x^2 + y^2 = r^2\), where \(r\) is the radius. In our problem, by simplifying the equation \(6x^2 + 6y^2 = 600\) to \(x^2 + y^2 = 100\), we identify the circle's radius as 10, because \(\sqrt{100} = 10\).
Circles are unique among conic sections for their perfect symmetry and equidistance from the center, making them simple yet fascinating objects to study.
Understanding the equation of a circle helps in graphing and visualizing it accurately.

domain and range

The domain and range are key concepts in understanding the span of a graphed function. For a circle, these tell us where on the coordinate plane the circle is located. Since we've established the radius as 10, the circle stretches 10 units in every direction from its center at the origin \((0, 0)\).
Thus, the domain, or set of possible \(x\) values, ranges from -10 to 10, represented as

• Domain: \(-10 \leq x \leq 10\)

The range, or set of possible \(y\) values, is similarly from -10 to 10:

• Range: \(-10 \leq y \leq 10\)

Knowing the domain and range helps determine the circle's complete coverage in the plane.

lines of symmetry

Symmetry in geometric figures refers to a balanced and proportional similarity. For circles, this symmetry is particularly simple and extensive. A circle is symmetric across both the x-axis and the y-axis. This means that if you were to fold the circle along either axis, both halves would overlap perfectly. In mathematical terms, these lines of symmetry are:

• The y-axis, given by \(x = 0\)
• The x-axis, given by \(y = 0\)

This double symmetry is a defining characteristic of circles, offering them a unique balance and simplicity in their form. The concept of symmetry can make it easier to analyze and graph conic sections like circles.