 Hello, Guest!
 Instructional Focus DocumentPrecalculus
 TITLE : Unit 12: Polar Equations SUGGESTED DURATION : 6 days

#### Unit Overview

Introduction
This unit bundles student expectations that address the polar coordinate system, including graphing points in polar coordinates, converting between polar and rectangular coordinates, and graphing polar equations by plotting points and using technology. Concepts are incorporated into both mathematical and real-world problem situations. According to the Texas Education Agency, mathematical process standards including application, tools and techniques, communication, representations, relationships, and justifications should be integrated (when applicable) with content knowledge and skills so that students are prepared to use mathematics in everyday life, society, and the workplace.

Prior to this Unit
In Grade 6 Unit 08, students plotted ordered pairs of rational numbers on a coordinate grid. In Geometry Unit 05, students used trigonometric ratios (including sine, cosine, and tangent) to determine the lengths of sides and the measures of angles in right triangles. In Precalculus Unit 07, students extended the use of the trigonometric ratios to solve real-world problems, including those involving navigational bearings. In Precalculus Unit 08, students represented rotation and reference angles on a coordinate grid. In Precalculus Unit 11, students graphed parametric equations and converted between parametric equations and rectangular relations. In Algebra I, Units 02 – 04, 08, and 09, students analyzed the graphs of linear, quadratic, and exponential functions. In Algebra II, Units 01, 02, and 05 – 11, students analyzed linear, quadratic, and exponential functions as well as absolute value, square root, cubic, cube root, rational, logarithmic functions. In Precalculus Units 01, 03 – 05, and 08, students analyzed the graphs of exponential, logarithmic, rational, polynomial, power, trigonometric, inverse trigonometric, and piecewise defined functions, including step functions.

In Algebra I Units 02 – 04, 08, and 09, Algebra II Units 01, 02, and 05 – 11, and Precalculus Units 01, 03 – 05, and 08, students analyzed the graphs of linear, quadratic, exponential, absolute value, square root, cubic, cube root, rational, logarithmic, polynomial, power, trigonometric, and inverse trigonometric functions.

During this Unit
Students graph points in the polar coordinate system using both radians and degrees. Students determine the polar coordinates for points and discover multiple ways to identify the same point in polar coordinates. Students convert polar coordinates into their rectangular coordinates and convert rectangular coordinates into their polar coordinates, while continuing to graph points in the polar coordinate system. Students graph polar equations by creating a table of values and plotting the polar points. Students graph polar equations with technology and analyze these graphs to explore various types of polar equations, including circles, cardioids, limacons, roses, and lemniscates.

Other considerations: Reference the Mathematics COVID-19 Gap Implementation Tool HS Precalculus

After this Unit
In Precalculus Unit 13, students will continue to study complex curves and relations that are not defined in terms of f(x). Specifically, students will explore the conic sections and their equations in rectangular coordinates. In subsequent mathematics courses, students will continue to apply polar coordinates and equations as they arise in problem situations.

Algebraic and geometric reasoning serve an integral role in college readiness. Translating among multiple representations of equations and relationships, analyzing the features of functions, and making connections between geometry and algebra are emphasized in the Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning B1; II. Algebraic Reasoning D1, D2; V. Statistical Reasoning A1, C2; VII. Problem Solving and Reasoning A1, A2, A3, A4, A5, B1, C1, D1, D2; VIII. Communication and Representation A1, A2, A3, B1, B2, C1, C2, C3; IX. Connections A1, A2, B1, B2, B3.

Research
According to the National Council of Teachers of Mathematics (2000), students in grades 9-12 should “specify locations and describe spatial relationships using coordinate geometry and other representational systems” (p. 308). Specifically, students should use Cartesian coordinates and other coordinate systems, including polar coordinates, to analyze geometric situations (NCTM, 2000). To this end, students should explore problem situations where using a different coordinate system (such as the polar coordinate system) provides a simpler representation of the geometric relation, while also being able to explain how both a polar representation and a rectangular representation describe the same geometric relation (NCTM, 2000). Cooney, Beckmann, & Lloyd (2010) contend that polar coordinates provide a criticism of the vertical line test, which is often used to determine the functionality of a relation graphed using Cartesian coordinates, but may deter students from exploring important aspects of graphical representations of other functional relationships. Specifically, the circle represented by the rectangular relation x2 + y2 = 16 does not represent a function for y in terms of x when using Cartesian coordinates; however, when this relation is graphed using the polar equation r(θ) = 4, then the set of ordered pairs of polar coordinates for this equation does represent a function of r in terms of θ (Cooney, Beckmann, & Lloyd, 2010). Lawes (2013) offers an alternative method for students to graph polar equations in addition to the traditional methods of plotting polar coordinates and using graphing technology. Specifically, Lawes recommends having students graph the polar function on the rectangular plane (where the horizontal axis represents and the vertical axis represents r) before graphing it on the polar plane. This method reinforces previous concepts related to graphing transformations of trigonometric functions, while also allowing students to visually determine the critical characteristics of polar curves and quickly sketch the graph of nearly any polar equation (Lawes, 2013). The study of polar equations continues into calculus, where students analyze planar curves (including those given in parametric form, polar form, and vector form) as an important part of functional analysis and derivative studies (College Board, 2012). These planar curves can represent a number of real-world applications, including problems incorporating velocity and acceleration. Additionally, students in calculus will determine the area of a region, including a region bounded by polar curves (College Board, 2012).

Cooney, T., Beckmann, S., & Lloyd, G. (2010). Developing essential understanding of functions for teaching mathematics in grades 9-12. Reston, VA: National Council of Teachers of Mathematics, Inc.
Lawes, J. F. (2013). Graphing polar curves. Mathematics Teacher, 106(9), 660-667.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.
Texas Education Agency & Texas Higher Education Coordinating Board. (2009). Texas college and career readiness standards. Retrieved from http://www.thecb.state.tx.us/index.cfm?objectid=E21AB9B0-2633-11E8-BC500050560100A9

 Quantitative relationships model problem situations efficiently and can be used to make generalizations, predictions, and critical judgements in everyday life. What patterns exist within different types of quantitative relationships and where are they found in everyday life? Why is the ability to model quantitative relationships in a variety of ways essential to solving problems in everyday life? Geometric, spatial, and measurement reasoning are foundational to visualizing, analyzing, and applying relationships within and between scale, shapes, quantities, and spatial relations in everyday life. Why is developing geometric, spatial, and measurement reasoning essential? How does geometric, spatial, and measurement reasoning affect how one sees and works in the world?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Polar equations are algebraic curves expressed in polar coordinates that connect algebraic and geometric relations and can be represented, transformed, and analyzed to model problem situations to make predictions and critical judgments.
• What connections exist between points represented in the rectangular and polar coordinate systems?
• How are properties and operational understandings used to convert between rectangular coordinates and polar coordinates?
• How are algebraic curves represented using polar equations?
• What kinds of mathematical and real-world situations can be modeled by polar equations?
• How can polar equations be used to represent problem situations?
• How can polar equations be represented?
• Relations and Geometric Reasoning
• Algebraic and Geometric Relations
• Rectangular coordinates
• Polar coordinates
• Polar equations
• Patterns, Operations, and Properties
• Associated Mathematical Processes
• Application
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

#### MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• Some students may think that inverse trigonometric functions are the same as the reciprocals of the trigonometric functions. For example, tan–1 is not the same as .
• Some students may incorrectly evaluate trigonometric functions or inverse trigonometric functions if they use a calculator in the wrong angle mode.
• Some students may incorrectly graph polar equations if they use a calculator in the wrong angle mode.
• Some students may believe that a point expressed in rectangular coordinates has a unique set of corresponding polar coordinates. Instead, each set of rectangular coordinates can be represented by an infinite set of polar coordinates.
• Some students may believe that polar equations are not functions since they do not pass the “vertical line test.” However, polar equations are functions where r is a function of θ.
• Some students may struggle to graph points in the polar coordinate system with negative r-values.

Underdeveloped Concepts:

• Some students may not understand how the θ-Step value affects the graph of parametric equations created using graphing technology.
• Some students may only graph polar equations for 0 ≤ θ < 2 . However, it may be necessary to extend these values in order to obtain a complete graph of a polar equation.

#### Unit Vocabulary

Related Vocabulary:

 Cardioid Circle Clockwise Convert Convex Cosine Counterclockwise Degrees Function Inverse functions Lemniscate Limaçon Polar coordinate system Polar coordinates Polar equation Pole Quadrant Radians Rectangular coordinates Relation Rose Rotation angle Sine Standard position Tangent xy-plane
Unit Assessment Items System Resources Other Resources

Show this message:

Unit Assessment Items that have been published by your district may be accessed through Search All Components in the District Resources tab. Assessment items may also be found using the Assessment Center if your district has granted access to that tool.

System Resources may be accessed through Search All Components in the District Resources Tab.

Texas Higher Education Coordinating Board – Texas College and Career Readiness Standards

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – STAAR Mathematics Resources

Texas Education Agency Texas Gateway – Revised Mathematics TEKS: Vertical Alignment Charts

Texas Education Agency Texas Gateway – Mathematics TEKS: Supporting Information

Texas Education Agency Texas Gateway – Interactive Mathematics Glossary

Texas Education Agency Texas Gateway – Resources Aligned to Precalculus Mathematics TEKS

Texas Instruments – Graphing Calculator Tutorials

TAUGHT DIRECTLY TEKS

TEKS intended to be explicitly taught in this unit.

TEKS/SE Legend:

• Knowledge and Skills Statements (TEKS) identified by TEA are in italicized, bolded, black text.
• Student Expectations (TEKS) identified by TEA are in bolded, black text.
• Portions of the Student Expectations (TEKS) that are not included in this unit but are taught in previous or future units are indicated by a strike-through.

Specificity Legend:

• Supporting information / clarifications (specificity) written by TEKS Resource System are in blue text.
• Unit-specific clarifications are in italicized, blue text.
• Information from Texas Education Agency (TEA), Texas College and Career Readiness Standards (TxCCRS), Texas Response to Curriculum Focal Points (TxRCFP) is labeled.
TEKS# SE# TEKS SPECIFICITY
P.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
P.1A Apply mathematics to problems arising in everyday life, society, and the workplace.

Apply

MATHEMATICS TO PROBLEMS ARISING IN EVERYDAY LIFE, SOCIETY, AND THE WORKPLACE
Including, but not limited to:

• Mathematical problem situations within and between disciplines
• Everyday life
• Society
• Workplace

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VII.D. Problem Solving and Reasoning – Real-world problem solving
• VII.D.1. Interpret results of the mathematical problem in terms of the original real-world situation.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
• IX.A.2. Connect mathematics to the study of other disciplines.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
• IX.B.2. Understand and use appropriate mathematical models in the natural, physical, and social sciences.
• IX.B.3. Know and understand the use of mathematics in a variety of careers and professions.
P.1B Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.

Use

A PROBLEM-SOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION, FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION, AND EVALUATING THE PROBLEM-SOLVING PROCESS AND THE REASONABLENESS OF THE SOLUTION
Including, but not limited to:

• Problem-solving model
• Analyze given information
• Formulate a plan or strategy
• Determine a solution
• Justify the solution
• Evaluate the problem-solving process and the reasonableness of the solution

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.1. Use estimation to check for errors and reasonableness of solutions.
• V.A. Statistical Reasoning – Design a study
• V.A.1. Formulate a statistical question, plan an investigation, and collect data.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VII.A.2. Formulate a plan or strategy.
• VII.A.3. Determine a solution.
• VII.A.4. Justify the solution.
• VII.A.5. Evaluate the problem-solving process.
• VII.D. Problem Solving and Reasoning – Real-world problem solving
• VII.D.2. Evaluate the problem-solving process.
P.1C Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.

Select

TOOLS, INCLUDING REAL OBJECTS, MANIPULATIVES, PAPER AND PENCIL, AND TECHNOLOGY AS APPROPRIATE, AND TECHNIQUES, INCLUDING MENTAL MATH, ESTIMATION, AND NUMBER SENSE AS APPROPRIATE, TO SOLVE PROBLEMS
Including, but not limited to:

• Appropriate selection of tool(s) and techniques to apply in order to solve problems
• Tools
• Real objects
• Manipulatives
• Paper and pencil
• Technology
• Techniques
• Mental math
• Estimation
• Number sense

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.1. Use estimation to check for errors and reasonableness of solutions.
• V.C. Statistical Reasoning – Analyze, interpret, and draw conclusions from data
• V.C.2. Analyze relationships between paired data using spreadsheets, graphing calculators, or statistical software.
P.1D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

Communicate

MATHEMATICAL IDEAS, REASONING, AND THEIR IMPLICATIONS USING MULTIPLE REPRESENTATIONS, INCLUDING SYMBOLS, DIAGRAMS, GRAPHS, AND LANGUAGE AS APPROPRIATE
Including, but not limited to:

• Mathematical ideas, reasoning, and their implications
• Multiple representations, as appropriate
• Symbols
• Diagrams
• Graphs
• Language

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• II.D. Algebraic Reasoning – Representing relationships
• II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
• II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
P.1E Create and use representations to organize, record, and communicate mathematical ideas.

Create, Use

REPRESENTATIONS TO ORGANIZE, RECORD, AND COMMUNICATE MATHEMATICAL IDEAS
Including, but not limited to:

• Representations of mathematical ideas
• Organize
• Record
• Communicate
• Evaluation of the effectiveness of representations to ensure clarity of mathematical ideas being communicated
• Appropriate mathematical vocabulary and phrasing when communicating mathematical ideas

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
P.1F Analyze mathematical relationships to connect and communicate mathematical ideas.

Analyze

MATHEMATICAL RELATIONSHIPS TO CONNECT AND COMMUNICATE MATHEMATICAL IDEAS
Including, but not limited to:

• Mathematical relationships
• Connect and communicate mathematical ideas
• Conjectures and generalizations from sets of examples and non-examples, patterns, etc.
• Current knowledge to new learning

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII. A.1. Analyze given information.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
•  VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
• IX.A.2. Connect mathematics to the study of other disciplines.
P.1G Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

Display, Explain, Justify

MATHEMATICAL IDEAS AND ARGUMENTS USING PRECISE MATHEMATICAL LANGUAGE IN WRITTEN OR ORAL COMMUNICATION

Including, but not limited to:

• Mathematical ideas and arguments
• Validation of conclusions
• Displays to make work visible to others
• Diagrams, visual aids, written work, etc.
• Explanations and justifications
• Precise mathematical language in written or oral communication

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.4. Justify the solution.
• VII.B. Problem Solving and Reasoning – Proportional reasoning
• VII.B.1. Use proportional reasoning to solve problems that require fractions, ratios, percentages, decimals, and proportions in a variety of contexts using multiple representations.
• VII.C. Problem Solving and Reasoning – Logical reasoning
• VII.C.1. Develop and evaluate convincing arguments.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
P.3 Relations and geometric reasoning. The student uses the process standards in mathematics to model and make connections between algebraic and geometric relations. The student is expected to:
P.3D Graph points in the polar coordinate system and convert between rectangular coordinates and polar coordinates.

Graph

POINTS IN THE POLAR COORDINATE SYSTEM

Including, but not limited to:

• Recognizing the meaning of the polar coordinates (r, θ)
• The “r” coordinate is the distance from the origin (or the “pole”).
• When r > 0, start at the origin (or pole) and move to the right a distance of r units.
• When r < 0, start at the origin (or pole) and move to the left a distance of r units.
• The “θ” coordinate is a rotation angle in standard position (in degrees or in radians).
• When θ > 0, rotate counter-clockwise.
• When θ < 0, rotate clockwise.
• Plotting points on a set of polar axes
• Naming coordinates for points in the polar coordinate system
• Multiple ways to name the same point
• If a point has coordinates (r, θ), then it can also be named in the following ways:
• In degrees
• (r, θ ± 360°n), where n ∈ Ζ
• (–r, θ ± 180°n), where n ∈ Ζ
• (r, θ ± 2πn), where n ∈ Ζ
• (–r, θ ± πn), where n ∈ Ζ

Convert

BETWEEN RECTANGULAR COORDINATES AND POLAR COORDINATES

Including, but not limited to:

• Using formulas to change polar coordinates (r, θ) into rectangular coordinates (x, y)
• x = rcosθ
• y<e/m> = rsinθ
• Using formulas to change rectangular coordinates (x, y) into polar coordinates (r, θ)
• r = • θ = tan–1 , Quadrant I
• θ = tan–1 + π, Quadrant II
• θ = tan–1 + π, Quadrant III
• θ = tan–1 + 2π, Quadrant IV

Note(s):

• 6th Grade Mathematics plotted ordered pairs of rational numbers on a coordinate grid.
• Geometry applied the trigonometric ratios (sine, cosine, tangent) to determine side lengths and angle measures in triangles.
• Precalculus combines these skills to plot points in the polar coordinate system and to convert between polar and rectangular coordinates.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
P.3E Graph polar equations by plotting points and using technology.

Graph

POLAR EQUATIONS

Including, but not limited to:

• Determine which graphing technique is most appropriate for a given situation
• Graphing calculator MODE to POLAR when applicable
• Plotting points
• Making a table
• Evaluating the polar equation
• Recognizing that the input (independent) variable is θ, on the right side of the table
• Plotting points on a polar graph
• Using both angle measurements
• Degrees
• Recognizing that different sets of coordinates may graph to the same location
• Using technology
• Mode settings
• Changing from “function” mode to “polar” mode
• Entering equations where r is a function of θ (or in the form r(θ))
• Window settings
• Minimum and maximum values of θ
• Interval between θ-values (“θ-Step”)
• Types of polar equations
• Circles
• r = a
• Center at the pole (or origin)
• r = acos θ, or r = asin θ
• Center on one of the coordinate axes
• Diameter = a
• Cardioids
• r = a + bcos θ, or r = a + bsin θ, where |a| = |b|
• Limacons
• r = a + bcos θ, or r = a + bsin θ, where |a| ≠ |b|
• If |a| < |b|, the limacon has an inner loop.
• If |b| < |a| < 2|b|, the limacon is called “dimpled.”
• If |a| ≥ 2|b|, the limacon is considered to be convex.
• Roses
• r = acos()
• r = asin()
• If n is odd, the rose has n petals.
• If n is even, the rose has 2n petals.
• The length of each petal is a (or a is the “radius” of the rose).
• Lemniscates
• r2 = a 2sin(2θ)
• r2 = a2cos(2θ)
• The length of each loop is a (or a is the "radius" of the lemniscate).

Note(s):

• Algebra II analyzed the graphs of quadratic, square root, exponential, logarithmic, cubic, cube root, absolute value, and rational functions.
• Precalculus extends these skills to include the graphing of polar equations.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• II.D. Algebraic Reasoning – Representing relationships
• II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems. 