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Ch. 2 - Graphs and Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 2 - Graphs and FunctionsProblem 9
Chapter 3, Problem 9

Without actually graphing, identify the type of graph that each equation has.
x2+y2=144x^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.
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.
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.
Calculate the radius: The square root of 144 is 12, so the radius of the circle is 12.
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.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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Geometries from Conic Sections

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.
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Circles in Standard Form

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.
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Graphs and Coordinates - Example
Related Practice
Textbook Question

Determine whether each statement is true or false. If false, explain why. The graph of y = x2 + 2 has no x-intercepts.

Textbook Question

In the following exercises, (a) find the center-radius form of the equation of each circle described, and (b) graph it. center (0, 0), radius 6

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Textbook Question

Let ƒ(x)=x2+3 and g(x)=-2x+6. Find each of the following. (ƒ+g)(3)

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Textbook Question

To answer each question, refer to the following basic graphs. Which one is the graph of ƒ(x)=√x? What is its domain?

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Textbook Question

Determine whether each statement is true or false. If false, explain why. The midpoint of the segment joining (0, 0) and (4, 4) is 2.

Textbook Question

Determine whether each relation defines a function. {(5,1),(3,2),(4,9),(7,8)}