Equations Of Circles Common Core Algebra 2 Homework Practice

Unformatted text preview: Name: ___________________________________ Date: _________________ EQUATIONS OF CIRCLES COMMON CORE ALGEBRA II Various quadratic relationships can be placed into equations by knowing the locus definition of the relationship. We will explore this for parabolas in a future lesson. In this one, we will develop the equation of a circle by using the distance formula that you learned from Common Core Geometry. THE DISTANCE FORMULA The distance between two points x1 , y1 and x2 , y2 is given by: D x2 x1 y2 y1 2 2 Exercise #1: A circle is the collection of all points that are a set distance (the radius) away from a point (its center). The circle shown below has a radius of 5 and a center at the point 4, 2 . An arbitrary point on the circle, x, y , is shown marked. y (a) Using the distance formula show that the point 7, 2 x, y must lie on this circle (verify graphically). 4, 2 (b) Letting x2 , y2 x, y and x1 , y1 4, 2 , write the x distance formula for all points on this circle. (d) Show algebraically that the point 1, 2 must (c) Square both sides of the equation from (b) to create the standard form of a circle. also lie on the circle. THE EQUATION OF A CIRCLE A circle whose center is at h, k and whose radius is r is given by: x h 2 y k r2 2 Exercise #2: Which of the following equations would have a center of 3, 6 and a radius of 3? (1) x 3 y 6 9 (3) x 3 y 6 3 (2) x 3 y 6 9 (4) x 3 y 6 3 2 2 2 2 2 2 2 2 COMMON CORE ALGEBRA II, UNIT #6 – QUADRATIC FUNCTIONS AND THEIR ALGEBRA – LESSON #10 eMATHINSTRUCTION, RED HOOK, NY 12571, © 2015 Exercise #3: For each of the following equations of circles, determine both the circle’s center and its radius. If its radius is not an integer, express it in decimal form rounded to the nearest tenth. (a) x 2 y 7 100 (b) x 5 y 8 4 (c) x 2 y 2 121 (d) x 1 y 2 1 (e) x 2 y 3 49 (f) x 6 y 5 18 (g) x 2 y 2 64 (h) x 4 y 2 20 (i) x 2 y 2 57 2 2 2 2 2 2 2 2 2 2 2 Exercise #4: Write equations for circles A and B shown below. Show how you arrive at your answers. y 1, 8 2, 4 6, 4 B A 4, 1 x Exercise #5: By completing the square on both quadratic expressions in x and y determine the center and radius of a circle whose equation is x 2 10 x y 2 2 y 10 COMMON CORE ALGEBRA II, UNIT #6 – QUADRATIC FUNCTIONS AND THEIR ALGEBRA – LESSON #10 eMATHINSTRUCTION, RED HOOK, NY 12571, © 2015 Name: ___________________________________ Date: _________________ EQUATIONS OF CIRCLES COMMON CORE ALGEBRA II HOMEWORK FLUENCY 1. Each of the following is an equation of a circle. State the circle’s center and radius. In the cases where the radius is not an integer, give its value rounded to the nearest tenth. (b) x 3 y 7 36 (c) x 5 y 1 64 (d) x 2 y 9 100 (e) x 2 y 2 1 (f) x 2 y 5 25 (g) x 2 y 2 50 (h) x 3 y 2 200 (i) x 6 y 6 20 2 (a) x 2 y 2 144 2 2 2 2 2 2 2 2 2 2. Which of the following is true about a circle whose equation is x 5 y 3 36 ? 2 2 (1) It has a center of 5, 3 and an area of 12 . (2) It has a center of 5, 3 and a diameter of 6. (3) It has a center of 5, 3 and an area of 36 . (4) It has a center of 5, 3 and a circumference of 12 . 3. Which of the following is a correct equation for the circle shown graphed below? y (1) x 2 y 3 16 2 2 (2) x 2 y 3 4 2 2 x (3) x 2 y 3 4 2 2 (4) x 2 y 3 16 2 2 4. By completing the square on each of the quadratic expressions, determine the center and radius of a circle whose equation is shown below. x 2 6 x y 2 10 y 66 COMMON CORE ALGEBRA II, UNIT #6 – QUADRATIC FUNCTIONS AND THEIR ALGEBRA – LESSON #10 eMATHINSTRUCTION, RED HOOK, NY 12571, © 2015 5. Circles are described below by the coordinates of their centers, C, and one point on their circumference, A. Determine an equation for each circle in center-radius form. (a) C 5, 2 and A 11, 10 (b) C 2, 5 and A 3, 17 (c) C 5, 1 and A 2, 5 y 6. Solve the following system of equations graphically. x 2 y 2 25 x y 5 x2 7. Find the intersection of the circle x 2 y 2 29 and y x 3 algebraically. APPLICATIONS y 8. Jonas is designing a circular garden whose equation is x 2 y 2 49 . He wishes to place a walkway within the garden at all points within the circle that satisfy the inequality 2 y 2 . Graph the circle on the grid to the right and shade in all points that represent the walkway. COMMON CORE ALGEBRA II, UNIT #6 – QUADRATIC FUNCTIONS AND THEIR ALGEBRA – LESSON #10 eMATHINSTRUCTION, RED HOOK, NY 12571, © 2015 x ...
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Chapter 1

Chapter 1

1-1Expressions and FormulasSkills Practicep.1
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1-2Properties of Real NumbersSkills Practicep.3
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1-3Solving EquationsSkills Practicep.5
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1-4Solving Absolute Value EquationsSkills Practicep.7
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1-5Solving InequalitiesSkills Practicep.9
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1-6Solving Compound and Absolute Value InequalitiesSkills Practicep.11
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Chapter 2

Chapter 2

2-1Relations and FunctionsSkills Practicep.13
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2-2Linear Relations and FunctionsSkills Practicep.15
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2-3Rate of Change and SlopeSkills Practicep.17
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2-4Writing Linear EquationsSkills Practicep.19
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2-5Scatter Plots and Lines of RegressionsSkills Practicep.21
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2-6Special FunctionsSkills Practicep.23
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2-7Parent Functions and TransformationsSkills Practicep.25
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2-8Graphing Linear Absolute Value InequalitiesSkills Practicep.27
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Chapter 3

Chapter 3

3-1Solving Systems of Equations by GraphingSkills Practicep.29
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3-2Solving Systems of Equations AlgebraicallySkills Practicep.31
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3-3Solving Systems of Inequalities by GraphingSkills Practicep.33
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3-4Optimization with Linear ProgrammingSkills Practicep.35
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3-5Systems of Equations in Three VariablesSkills Practicep.37
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Chapter 4

Chapter 4

4-1Introduction to MatricesSkills Practicep.39
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4-2Operations with MatricesSkills Practicep.41
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4-3Multiplying MatricesSkills Practicep.43
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4-4Transformations with MatricesSkills Practicep.45
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4-5Determinants and Cramer's RuleSkills Practicep.47
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4-6Inverse Matrices and Systems of EquationsSkills Practicep.49
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Chapter 5

Chapter 5

5-1Graphing Quadratic EquationsSkills Practicep.51
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5-2Solving Quadratic Equations by GraphingSkills Practicep.53
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5-3Solving Quadratic Equations by FactoringSkills Practicep.55
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5-4Complex NumbersSkills Practicep.57
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5-5Completing the SquareSkills Practicep.59
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5-6The Quadratic Formula and the DiscriminantSkills Practicep.61
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5-7Transformations with Quadratic FunctionsSkills Practicep.63
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5-8Quadratic InequalitiesSkills Practicep.65
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Chapter 6

Chapter 6

6-1Operations with PolynomialsSkills Practicep.67
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6-2Dividing PolynomialsSkills Practicep.69
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6-3Polynomial FunctionsSkills Practicep.71
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6-4Analyzing Graphs of Polynomial FunctionsSkills Practicep.73
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6-5Solving Polynomial EquationsSkills Practicep.75
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6-6The Remainder and Factor TheoremsSkills Practicep.77
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6-7Roots and ZerosSkills Practicep.79
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6-8Rational Zero TheoremSkills Practicep.81
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Chapter 7

Chapter 7

7-1Operations on FunctionsSkills Practicep.83
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7-2Inverse Functions and RelationsSkills Practicep.85
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7-3Square Root Functions and InequalitiesSkills Practicep.87
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7-4nth RootsSkills Practicep.89
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7-5Operations with Radical ExpressionsSkills Practicep.91
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7-6Rational ExponentsSkills Practicep.93
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7-7Solving Radical Equations and InequalitiesSkills Practicep.95
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Chapter 8

Chapter 8

8-1Graphing Exponential FunctionsSkills Practicep.97
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8-2Solving Exponential Equations and InequalitiesSkills Practicep.99
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8-3Logarithms and Logarithmic FunctionsSkills Practicep.101
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8-4Solving Logarithmic Equations and InequalitiesSkills Practicep.103
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8-5Properties of LogarithmsSkills Practicep.105
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8-6Common LogarithmsSkills Practicep.107
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8-7Base e and Natural LogarithmsSkills Practicep.109
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8-8Using Exponential and Logarithmic FunctionsSkills Practicep.111
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Chapter 9

Chapter 9

9-1Multiplying and Dividing Rational ExpressionsSkills Practicep.113
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9-2Adding and Subtracting Rational ExpressionsSkills Practicep.115
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9-3Graphing Reciprocal FunctionsSkills Practicep.117
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9-4Graphing Rational FunctionsSkills Practicep.119
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9-5Variation FunctionsSkills Practicep.121
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9-6Solving Rational Equations and InequalitiesSkills Practicep.123
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Chapter 10

Chapter 10

10-1Midpoint and Distance FormulasSkills Practicep.125
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10-2ParabolasSkills Practicep.127
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10-3CirclesSkills Practicep.129
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10-4EllipsesSkills Practicep.131
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10-5HyperbolasSkills Practicep.133
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10-6Identifying Conic SectionsSkills Practicep.135
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10-7Solving Quadratic SystemsSkills Practicep.137
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Chapter 11

Chapter 11

11-1Sequences as FunctionsSkills Practicep.139
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11-2Arithmetic Sequences and SeriesSkills Practicep.141
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11-3Geometric Sequences and SeriesSkills Practicep.143
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11-4Infinite Geometric SeriesSkills Practicep.145
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11-5Recursion and IterationSkills Practicep.147
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11-6The Binomial TheoremSkills Practicep.149
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11-7Proof by Mathematical InductionSkills Practicep.151
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Chapter 12

Chapter 12

12-1Experiments, Surveys, and Observational StudiesSkills Practicep.153
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12-2Statistical AnalysisSkills Practicep.155
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12-3Conditional ProbabilitySkills Practicep.157
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12-4Probability DistributionsSkills Practicep.159
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12-5The Normal DistributionSkills Practicep.161
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12-6Hypothesis TestingSkills Practicep.163
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12-7Binomial DistributionsSkills Practicep.165
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Chapter 13

Chapter 13

13-1Trigonometric Functions in Right TrianglesSkills Practicep.167
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13-2Angles and Angle MeasureSkills Practicep.169
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13-3Trigonometric Functions of General AnglesSkills Practicep.171
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13-4Law of SinesSkills Practicep.173
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13-5Law of CosinesSkills Practicep.175
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13-6Circular FunctionsSkills Practicep.177
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13-7Graphing Trigonometric FunctionsSkills Practicep.179
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13-8Translations of Trigonometric GraphsSkills Practicep.181
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13-9Inverse Trigonometric FunctionsSkills Practicep.183
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Chapter 14

Chapter 14

14-1Trigonometric IdentitiesSkills Practicep.185
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14-2Verifying Trigonometric IdentitiesSkills Practicep.187
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14-3Sum and Difference of Angles FormulasSkills Practicep.189
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14-4Double-Angle and Half-Angle FormulasSkills Practicep.191
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14-5Solving Trigonometric EquationsSkills Practicep.193
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