Without actually graphing, identify the type of graph that each equation has.
$x^2+y^2=144$

Verified step by step guidance

1. Recognize the standard form of the equation: The given equation is \(x^2 + y^2 = 144\). This is a form of the equation \(x^2 + y^2 = r^2\), which is the standard form of a circle centered at the origin.
2. Identify the components of the equation: In the equation \(x^2 + y^2 = 144\), the terms \(x^2\) and \(y^2\) indicate that both variables are squared and have the same coefficient, which is 1 in this case.
3. Determine the radius of the circle: The equation \(x^2 + y^2 = r^2\) represents a circle with radius \(r\). Here, \(r^2 = 144\), so the radius \(r\) is the square root of 144.
4. Calculate the radius: The square root of 144 is 12, so the radius of the circle is 12.
5. Conclude the type of graph: Since the equation is in the form \(x^2 + y^2 = r^2\) and represents a circle with a radius of 12, the graph of this equation is a circle centered at the origin with a radius of 12.

Key Concepts

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double-napped cone. The main types include circles, ellipses, parabolas, and hyperbolas. Each type has a distinct equation and geometric properties. Understanding these shapes is crucial for identifying the type of graph represented by a given equation.

Standard Form of a Circle

The standard form of a circle's equation is given by (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. In the equation x² + y² = 144, it can be rewritten as (x - 0)² + (y - 0)² = 12², indicating a circle centered at the origin with a radius of 12. Recognizing this form is essential for identifying circular graphs.

Graphing Techniques

Graphing techniques involve understanding how to represent equations visually on a coordinate plane. This includes knowing how to plot points, identify key features like intercepts and vertices, and recognize symmetry. For equations like x² + y² = 144, these techniques help in visualizing the graph's shape and position without needing to plot every point.