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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 36
Chapter 3, Problem 36

Describe the graph of each equation as a circle, a point, or nonexistent. If it is a circle, give the center and radius. If it is a point, give the coordinates. x2+y2+4x-8y+32=0

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Start by rewriting the given equation: \(x^2 + y^2 + 4x - 8y + 32 = 0\).
Group the \(x\) terms and \(y\) terms together: \((x^2 + 4x) + (y^2 - 8y) + 32 = 0\).
Complete the square for the \(x\) terms: take half of 4, which is 2, and square it to get 4. Add and subtract 4 inside the equation.
Complete the square for the \(y\) terms: take half of -8, which is -4, and square it to get 16. Add and subtract 16 inside the equation.
Rewrite the equation as \((x^2 + 4x + 4) + (y^2 - 8y + 16) + 32 - 4 - 16 = 0\), then simplify and express it in the form \((x + 2)^2 + (y - 4)^2 = r^2\) to identify the center and radius.

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Key Concepts

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

Standard Form of a Circle Equation

The standard form of a circle's equation is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius. Converting a general quadratic equation into this form helps identify the circle's properties clearly.
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Circles in Standard Form

Completing the Square

Completing the square is a method used to rewrite quadratic expressions in the form (x - h)^2 or (y - k)^2. This technique is essential to transform the given equation into the standard form of a circle.
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Solving Quadratic Equations by Completing the Square

Determining the Nature of the Graph

After rewriting the equation, the radius squared (r^2) determines the graph's nature: if r^2 > 0, it's a circle; if r^2 = 0, it's a point; if r^2 < 0, the graph does not exist in the real plane.
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The Natural Log
Related Practice
Textbook Question

Find the coordinates of the other endpoint of each line segment, given its midpoint and one endpoint. See Example 5(b).

midpoint (5,8)(5, 8), endpoint (13,10)(13, 10)

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Textbook Question

Find the coordinates of the other endpoint of each line segment, given its midpoint and one endpoint. See Example 5(b).

midpoint (12,6)(12, 6), endpoint (19,16)(19, 16)

Textbook Question

Give a rule for each piecewise-defined function. Also give the domain and range.

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Textbook Question

Use a graphing calculator to graph each equation in the standard viewing window. y = 3x + 4

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Textbook Question

Use a graphing calculator to graph each equation in the standard viewing window. 3x + 4y = 6

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Textbook Question

Determine whether each relation defines y as a function of x. Give the domain and range. See Example 5. x=y4

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