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 Instructional Focus DocumentPrecalculus
 TITLE : Unit 13: Conics SUGGESTED DURATION : 16 days

#### Unit Overview

Introduction
This unit bundles student expectations that address determining, analyzing, and writing equations for the conic sections formed when a plane intersects a double-napped cone. Connections are made between the locus definitions of the conic sections and their equations in rectangular coordinates. Additionally, converting between parametric equations and rectangular relations for conic sections and using these representations to model and solve problems are also included. 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 Geometry Unit 07, students identified the shapes of two-dimensional cross-sections of prisms, pyramids, cylinders, cones, and spheres. In Geometry Units 02 and 07, students derived and used the distance formula to verify geometric relationships. In Geometry Unit 06, students showed that the equation of a circle with center at the origin and radius r is x2 + y2 = r2 and wrote the equation for a circle centered at (h, k) with radius r using (xh)2 + (yk) = r2. In Algebra II Unit 05, students wrote the equation of a parabola using given attributes, including vertex, focus, directrix, axis of symmetry, and direction of opening. In Precalculus Unit 11, students graphed parametric equations, used parametric equations to model and solve problem situations, and converted between parametric equations and rectangular relations for various linear and parabolic situations.

During this Unit
Students explore the cross-sections of a double-napped cone in order to determine the four conic sections formed when a plane intersects a double-napped cone. Additionally, students explore the special cases of the intersection of a plane and a double-napped cone yielding the degenerate conic sections. Students determine the general characteristics of the four types of conic sections (circles, parabolas, ellipses, and hyperbolas), including shape, cross-section conditions, key attributes (vertex/vertices, asymptotes, etc.), equations (for circles and parabolas), and real-world applications. Students explore the locus definition of a circle and make connections to its equation in rectangular coordinates. Specifically, students connect the distance formula to the equation of a circle. Students write equations of circles in both mathematical and real-world problem situations. Students explore the locus definition of an ellipse and make connections to its equation in rectangular coordinates. Specifically, students use the distance formula to determine the general equation of an ellipse. Students write rectangular equations of ellipses using their various characteristics, including major axis, semi-major axis, minor axis, semi-minor axis, focal length, coordinates for the center, coordinates for the foci, and coordinates for the vertices. Students write equations of ellipses in both mathematical and real-world problem situations. Students explore the locus definition of a hyperbola and make connections to its equation in rectangular coordinates. Specifically, students explore the conditions that give rise to a hyperbola and then use the distance formula to determine the general equation of a hyperbola. Students write rectangular equations of hyperbolas using their various characteristics, including focal length, coordinates for the center, coordinates for the foci, coordinates for the vertices, distance from the center to the vertices, slopes of the linear asymptotes, and equations for the linear asymptotes. Students write equations of hyperbolas in both mathematical and real-world problem situations. Students explore the locus definition of a parabola and make connections to its equation in rectangular coordinates. Specifically, students explore how the distance formula yields the equation of a parabola. Students write equations of parabolas in both mathematical and real-world problem situations. Students convert rectangular relations for circles, parabolas, ellipses, and hyperbolas into their parametric equations, then convert parametric equations for these conics into their rectangular relations. Students model real-world situations using rectangular and parametric equations of ellipses and hyperbolas and use these equations to solve problems.

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

After this Unit
In subsequent mathematics courses, students will continue to apply conic sections as they arise in problem situations.

Algebraic and geometric reasoning serve an integral role in college readiness. Translating among multiple representations of equations 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; III. Geometric and Spatial Reasoning A1, C1; V. Statistical Reasoning A1, C2; VI. Functions B2; 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 “understand relations and functions and select, convert flexibly among, and use various representations for them” (p. 296). Additionally students should use visualization, spatial reasoning, and geometric modeling to solve problems including analyzing the cross-sections of three-dimensional objects (NCTM, 2000). Students should be provided with rich mathematical settings in which they can hone their visualization skills in order to use visualization as a problem-solving tool. Visualizing the shape of a conic section formed by the intersection of a plane and a cone can be facilitated by the use of physical models, drawings, and software capable of manipulating three-dimensional objects (NCTM, 2000). Herman (2012) describes how the modern study of conics is considered analytic geometry where the conic sections represent graphs of certain types of equations that are defined in terms of distance relationships between specific points and/or lines. Brown and Jones (2005) advocate for the use of two different coordinate systems which are well suited for graphing conic sections and exploring these distance relationships: a focus-directrix coordinate system and a focus-focus coordinate system. Using these alternative coordinate systems allows students to explore the definitions of the conic sections in a way that may be masked by using equations or technology to plot points alone (Brown & Jones, 2005). The study of conic sections continues into calculus, where students analyze conics and other implicitly defined curves using implicit differentiation (College Board, 2012).

Brown, E. M., & Jones, E. (2005). Understanding conic sections using alternate graph paper. Mathematics Teacher, 99(5), 322-327.
Herman, M. (2012). Exploring conics: Why does b2 – 4ac matter? Mathematics Teacher, 105(7), 526-532.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.

 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)
• Conic sections are curves obtained from the intersection of a plane and a double-napped cone, connect algebraic and geometric relations, and can be represented, transformed, and analyzed to model problem situations to make predictions and critical judgments.
• What kinds of mathematical and real-world situations can be modeled by …
• circles?
• parabolas?
• ellipses?
• hyperbolas?
• How can …
• circles
• parabolas
• ellipses
• hyperbolas
… be used to represent problem situations?
• How can conic sections be represented?
• What relationships exist between conic sections and the intersection of a plane with a double-napped cone?
• What connections can be made between multiple representations of conic sections?
• What graphs, key attributes, and characteristics are unique to the different types of conic sections?
• How are the locus definitions of conic sections connected to the equations of conic sections?
• How does the given information and/or representation influence the selection of an efficient method for writing equations of conic sections?
• How are properties and operational understandings used to transform equations that represent conic sections?
• How are properties and operational understandings used to transform …
• parametric equations into rectangular relations?
• rectangular relations into parametric equations?
• Relations and Geometric Reasoning
• Algebraic and Geometric Relations
• Parametric equations
• Rectangular relations
• Conic Sections
• Circles
• Parabolas
• Ellipses
• Hyperbolas
• 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 only four types of conic sections can be formed by the intersection of a plane with a double-napped cone: circles, parabolas, ellipses, and hyperbolas. Students may forget about the degenerate conic sections formed by a plane passing through the vertex of double-napped cones.
• Students may confuse the Pythagorean relationships between a, b, and c for ellipses (c2 =|a2b2|) and for hyperbolas (c2 = a2b2).
• Some students may believe that hyperbolas are made up of two symmetric parabolas. However, parabolas have different locus definitions than the branches of hyperbolas and therefore represent different geometric shapes.
• Some students may mistakenly graph a hyperbola with an incorrect axis. For example, students may mistakenly graph the hyperbola = 1 with a vertical axis rather than a horizontal axis.
• Some students may incorrectly graph parametric equations if they use a calculator in the wrong angle mode.
• When graphing parametric equations, some students may only graph the path of the parametric equations without noting specific t values along this path.

Underdeveloped Concepts:

• Some students may fail to take the square root of the r2 portion of the equation of a circle when determining the radius of the circle.
• Some students may fail to take the square root of the a2 or b2 portion of the equation of an ellipse or a hyperbola when determining the values of a and b.

#### Unit Vocabulary

Related Vocabulary:

 Asymptote Axis Axis of symmetry Center Circle Conic section Convert Coordinates Cross-section Degenerate conic section Directrix Distance Distance formula Double-napped cone Ellipse Equation Focal length Focus Horizontal Hyperbola Intersecting lines Intersection Line Locus Major axis Minor axis Parabola Parameter Parametric equations Parallel Path Perpendicular Point Radius Rectangular coordinates Rectangular relation Semi-major axis Semi-minor axis Slope Standard form Vertex Vertical
Unit Assessment Items System Resources Other Resources

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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.
• A Partial Specificity label indicates that a portion of the specificity not aligned to this unit has been removed.
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.3B Convert parametric equations into rectangular relations and convert rectangular relations into parametric equations.

Convert

PARAMETRIC EQUATIONS INTO RECTANGULAR RELATIONS

Including, but not limited to:

• Algebraic methods
• Solving one equation in a set of parametric equations (either x(t) or y(t)) for the parameter t
• Substituting the expression for t into the other equation (either y(t) or x(t))
• Applying trigonometric identities to eliminate the parameter t

Convert

RECTANGULAR RELATIONS INTO PARAMETRIC EQUATIONS

Including, but not limited to:

• Rectangular relations into general parametric equations
• Rectangular functions of the form y = f(x)
• Letting x = t
• Writing y as y = f(t)
• Rectangular relations of the form x = f(y)
• Letting y = t
• Writing x as x = f(t)
• Rectangular relations, given with specific information about x(t) or y(t)
• Write an expression to describe x(t) or y(t)
• Substitute the expression for x or y into the given rectangular relation
• Rectangular relations of circles, ellipses, and hyperbolas
• Circles
• Applying the trigonometric identity sin2t + cos2t = 1 to parameterize a circle
• Ellipses
• Applying the trigonometric identity sin2t + cos2t = 1 to parameterize an ellipse
• Hyperbolas
• Applying the trigonometric identity sec2t – tan2t = 1 to parameterize a hyperbola

Note(s):

• Algebra I solved literal equations for a specified variable.
• Algebra II used the composition of two functions.
• Precalculus extends these skills to convert between parametric and rectangular 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.
• VI.B. Functions – Analysis of functions
• VI.B.2. Algebraically construct and analyze new functions.
• 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.3C Use parametric equations to model and solve mathematical and real-world problems.

Use

PARAMETRIC EQUATIONS

Model, Solve

MATHEMATICAL AND REAL-WORLD PROBLEMS USING PARAMETRIC EQUATIONS

Including, but not limited to:

• Conics
• Circles
• General form:
• Variables and constants
• (hk) (center of circle)
• Ellipses
• General form:
• Variables and constants
• a (horizontal semi-axis)
• b (vertical semi-axis)
• (hk) (center of ellipse)
• Hyperbolas
• Hyperbola with a horizontal axis
• General form:
• Variables and constants
• a (distance from center to vertices)
• b (where ± represents the slopes of the linear asymptotes)
• (hk) (center of hyperbola)
• Hyperbola with a vertical axis
• General form:
• Variables and constants
• a (where ± represents the slopes of the linear asymptotes)
• b (distance from center to vertices)
• (hk) (center of hyperbola)
• Applications

Note(s):

• Geometry applied the trigonometric ratios (sine, cosine) to determine side lengths and angle measures in triangles.
• Algebra I analyzed the features of linear and quadratic functions.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• II. Algebraic Reasoning
• D2 – Translate among multiple representations of equations and relationships.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
P.3F Determine the conic section formed when a plane intersects a double-napped cone.

Determine

THE CONIC SECTION FORMED WHEN A PLANE INTERSECTS A DOUBLE-NAPPED CONE

Including, but not limited to:

• Circle
• Figure: a closed curve
• Intersection: The plane is perpendicular to the symmetry axis of the double-napped cone.
• Nappes: The plane intersects only one nappe of the double-napped cone.
• Parabola
• Figure: an unbounded curve
• Intersection: The plane is parallel to an edge of the double-napped cone.
• Nappes: The plane intersects only one nappe of the double-napped cone.
• Ellipse
• Figure: a closed curve
• Intersection: The plane is neither parallel to an edge, nor perpendicular to the symmetry axis, of the double-napped cone.
• Nappes: The plane intersects only one nappe of the double-napped cone.
• Hyperbola
• Figure: two unbounded curves
• Intersection: The plane is parallel to the symmetry axis of the double-napped cone.
• Nappes: The plane intersects both nappes of the double-napped cone.
• Degenerate conic sections
• A single point
• Intersection: The plane passes through the apex at an angle greater than the angle made by the symmetry axis and the edge of the cone.
• A single line
• Intersection: The plane passes through the apex and is tangent to the edges of the double-napped cone.
• A pair of lines
• Intersection: The plane passes through the apex at an angle less than the angle made by the symmetry axis and the edge of the cone.
• Applications
• Determine if objects in art and architecture can be formed using a cross-section of a double-napped cone.

Note(s):

• Geometry identified the shapes of two-dimensional cross-sections of prisms, pyramids, cylinders, spheres, and cones.
• Precalculus extends this knowledge to the double-napped cone and includes shapes such as the ellipse, hyperbola and parabola.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• III.A. Geometric and Spatial Reasoning – Figures and their properties
• III.A.1. Recognize characteristics and dimensional changes of two- and three-dimensional figures.
• 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.3G Make connections between the locus definition of conic sections and their equations in rectangular coordinates.

Make

CONNECTIONS BETWEEN THE LOCUS DEFINITION OF CONIC SECTIONS AND THEIR EQUATIONS IN RECTANGULAR COORDINATES

Including, but not limited to:

• Circles
• Locus definition of a circle
• A circle is the set of points (x, y) in a plane such that (x, y) is a distance of r units from a center point (h, k).
• Application of the distance formula and its transformation to the general rectangular equation for a circle
• Ellipses
• Locus definition of an ellipse
• For two points in a plane (a focus at F1 and another focus at F2), an ellipse is the set of points P at (x, y) such that the sum of the distances from P to each focus is constant (or PF1 + PF2 = 2a, where a is a constant).
• Application of the distance formula and its transformation to the general rectangular equation for an ellipse
• Parabolas
• Locus definition of a parabola
• For a point (the focus, F) and a line (called the directrix), a parabola is the set of all points (x, y) such that the distance from F to (x, y) is equivalent to the distance from (x, y) to the directrix.
• Application of the distance formula and its transformation to the general rectangular equation for a parabola
• Hyperbolas
• Locus definition of a hyperbola
• For two points (a focus at F1 and another focus at F2), a hyperbola is the set of points P at (x, y) such that the difference of the distances from P to each focus is constant (or |PF1 - PF2| = 2a, where a is a constant).
• Application of the distance formula and its transformation to the general rectangular equation for a hyperbola

Note(s):

• Geometry used the distance formula to verify geometric relationships.
• Geometry developed and used the equation of a circle, (xh)2 + (yk)2 = r2, centered at (h, k) with radius r.
• Algebra II developed the equation for a parabola using given attributes, including vertex, focus, directrix, axis of symmetry, and direction of opening.
• Precalculus extends this knowledge to include the other conic sections (ellipses and hyperbolas).
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• III.A. Geometric and Spatial Reasoning – Figures and their properties
• III.A.1. Recognize characteristics and dimensional changes of two- and three-dimensional figures.
• 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.3H Use the characteristics of an ellipse to write the equation of an ellipse with center (h, k).

Use

THE CHARACTERISTICS OF AN ELLIPSE

Including, but not limited to:

• Characteristics of an ellipse
• Major axis
• Semi-major axis
• Minor axis
• Semi-minor axis
• Focal length
• Coordinates for the center
• Coordinates for the foci
• Coordinates for the vertices

Write

THE EQUATION OF AN ELLIPSE WITH CENTER (hk)

Including, but not limited to:

• Using the general equation for an ellipse
• Ellipse with a horizontal major axis: = 1, with a > b
• Ellipse with a vertical major axis: = 1, with b > a
• Given a combination of two or more characteristics of an ellipse
• Major axis, 2a (horizontal) or 2b (vertical)
• Semi-major axis, a (horizontal) or b (vertical)
• Minor axis, 2a (horizontal) or 2b (vertical)
• Semi-minor axis, a(horizontal) or b (vertical)
• Focal length, c
• Equation relating focal length and the length of each semi-axis, |a2b2| = c2
• Coordinates for the center, (h, k)
• Coordinates for the foci
• Ellipse with a horizontal major axis: (h ± c, k)
• Ellipse with a vertical major axis: (h, k ± c)
• Coordinates for the vertices
• On the horizontal axis: (h ± a, k)
• On the vertical axis: (h, k ± b)
• Description of effects of changes in characteristics of an ellipse on the represetnative equation
• Use conic sections to solve problems involving modeling.

Note(s):

• Geometry wrote the equation for a circle centered at (h, k) with radius r using (xh)2 + (yk)2 = r2.
• Algebra II wrote the equation for a parabola using given attributes, including vertex, focus, directrix, axis of symmetry, and direction of opening.
• Precalculus extends this knowledge to write equations for the other conic sections (ellipses and hyperbolas).
• 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.
• III.A. Geometric and Spatial Reasoning – Figures and their properties
• III.A.1. Recognize characteristics and dimensional changes of two- and three-dimensional figures.
• III.C. Geometric and Spatial Reasoning  – Connections between geometry and other mathematical content strands
• III.C.1. Make connections between geometry and algebraic equations.
• 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.3I Use the characteristics of a hyperbola to write the equation of a hyperbola with center (h, k).

Use

THE CHARACTERISTICS OF A HYPERBOLA

Including, but not limited to:

• Characteristics of a hyperbola
• Coordinates for the center
• Distance from the center to the vertices
• Coordinates for the vertices
• Slopes of the linear asymptotes
• Equations for the linear asymptotes
• Focal length
• Coordinates for the foci

To Write

THE EQUATION OF A HYPERBOLA WITH CENTER (hk)

Including, but not limited to:

• Using the general equation for a hyperbola
• Hyperbola with a horizontal axis: = 1
• Hyperbola with a vertical axis: = 1
• Given a combination of two or more characteristics of a hyperbola
• Coordinates for the center, (h, k)
• Distance from the center to the vertices, a (horizontal) or b (vertical)
• Coordinates for the vertices
• Hyperbola with a horizontal axis: (h ± a, k)
• Hyperbola with a vertical axis: (h, k ± b)
• Slopes of the linear asymptotes, ±
• Equations for the linear asymptotes, yk = ± (xh)
• Focal length, c
• Equation relating focal length to a and b, a2 + b2 = c2
• Coordinates for the foci
• Hyperbolas with a horizontal axis: (h ± c, k)
• Hyperbolas with a vertical axis: (h, k ± c)
• Description of effects of changes in characteristics of a hyperbola on the representative equation
• Use conic sections to solve problems involving modeling.

Note(s):

• Geometry wrote equations for circles centered at (h, k) with radius r using (xh)2 + (yk)2 = r2.
• Algebra II wrote the equation for a parabola using given attributes, including vertex, focus, directrix, axis of symmetry, and direction of opening.
• Precalculus extends this knowledge to write equations for the other conic sections (ellipses and hyperbolas).
• 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.
• III.A. Geometric and Spatial Reasoning – Figures and their properties
• III.A.1. Recognize characteristics and dimensional changes of two- and three-dimensional figures.
• III.C. Geometric and Spatial Reasoning  – Connections between geometry and other mathematical content strands
• III.C.1. Make connections between geometry and algebraic equations.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.