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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 25
Chapter 3, Problem 25

Use each graph to determine an equation of the circle in (a) center-radius form and (b) general form.
Graph of a circle plotted on coordinate axes with labeled points at (-4,2), (-2,0), (-2,4), and (0,2).

Verified step by step guidance
1
Identify the center \((h, k)\) of the circle from the graph. This is the point where the circle is centered.
Determine the radius \(r\) of the circle by measuring the distance from the center to any point on the circle.
Write the equation of the circle in center-radius form using the formula: \[ (x - h)^2 + (y - k)^2 = r^2 \]
Expand the squared terms in the center-radius form to convert it into the general form. This involves expanding \((x - h)^2\) and \((y - k)^2\).
Simplify the expanded expression and rearrange all terms to one side to write the equation in general form: \[ x^2 + y^2 + Dx + Ey + F = 0 \] where \(D\), \(E\), and \(F\) are constants derived from the expansion.

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Key Concepts

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

Equation of a Circle in Center-Radius Form

The center-radius form of a circle's equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. This form directly shows the circle's center coordinates and radius, making it easy to write the equation when these values are known from the graph.
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Circles in Standard Form

Converting to General Form of a Circle

The general form of a circle's equation is x² + y² + Dx + Ey + F = 0. It is obtained by expanding the center-radius form and simplifying. Understanding how to expand and rearrange terms is essential to rewrite the equation in this standard polynomial form.
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Circles in General Form

Interpreting Graphs to Identify Circle Parameters

Analyzing a graph of a circle involves identifying the center point and measuring the radius, which is the distance from the center to any point on the circle. Accurate reading of these values from the graph is crucial for writing the correct equation in both forms.
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Circles in Standard Form
Related Practice
Textbook Question

Determine whether each equation defines y as a function of x. y = 6 -x2

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Determine whether the three points are the vertices of a right triangle. (-4,1),(1,4),(-6,-1)

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Determine whether each equation defines y as a function of x. x = (1/3)(y2)

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Textbook Question

Graph each function. See Examples 1 and 2. ƒ(x)=-(1/2)x2

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Textbook Question

Graph each piecewise-defined function.

f(x)={−3if x≤1−1if x>1f(x) =\(\begin{cases}\)-3 & \(\text{if }\) x \(\leq\) 1 \\-1 & \(\text{if }\) x > 1\(\end{cases}\)

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Textbook Question

Write an equation for each line described. Give answers in standard form for Exercises 11–20 and in slope-intercept form (if possible) for Exercises 21–32. horizontal, through (-7,4)