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

#### Unit Overview

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)2 = 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.

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): II. Algebraic Reasoning B1, C1, D2; III. Geometric Reasoning C1; VIII. Problem Solving and Reasoning; IX. Communication and Representation; X. Connections.

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.

#### OVERARCHING UNDERSTANDINGS and QUESTIONS

Geometric and spatial reasoning are necessary to describe and analyze geometric relationships in mathematics and the real-world.

• Why are geometric and spatial reasoning necessary in the development of an understanding of geometric relationships?
• Why is it important to visualize and use diagrams to effectively communicate/illustrate geometric relationships?
• How do geometric and spatial reasoning allow for the understanding of different geometric systems as models for the world?

Relations are algebraic models that describe how two quantities relate to one another. Functions are a subset of relations.

• What are types of relations?
• How can relations be represented?
• Why do some relations not define a function?
• Why do some relations define a function?
• Why can function models describe how two variable quantities change in relation to one another?

Equations can be used to model and solve problem situations.

• Why are equations used to model problem situations?
• How are equations used to model problem situations?
• How are equations used to solve problem situations?
• Why is it essential to solve equations using various methods?
• How can solutions to equations be represented?
Performance Assessment(s) Overarching Concepts
Unit Concepts
Unit Understandings
 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.

Algebraic Reasoning

• Equations
• Multiple Representations
• Relations

Geometric Reasoning

• Conics

Associated Mathematical Processes

• Application
• Tools and Techniques
• Problem Solving Model
• Communication
• Representations
• Relationships
• Justification

Models and diagrams can be used to visualize and illustrate geometric relationships and aid in solving problems.

• Why are models and diagrams necessary for visualizing the geometric relationships found in the problem situation?
• How are models and diagrams used to organize information from the problem situation?
• How do models and diagrams aid in calculations when solving problems?
• Why is the coordinate plane used to diagram two-dimensional figures?

Conic sections have unique characteristics and can be represented in multiple ways, including as rectangular relations.

• What are the conic sections?
• How can conic sections be formed by the intersection of a plane with a double-napped cone?
• What are the degenerate conic sections, and how are they formed by the intersection of a plane with a double-napped cone?
• How can conic sections be represented using graphs?
• How can conic sections be represented algebraically as rectangular relations?

A circle is one of the four basic forms of conic sections.

• How is a circle defined as a locus of points? Explain.
• How does the locus definition of a circle connect to the equation of a circle in rectangular coordinates?

A parabola is one of the four basic forms of conic sections.

• How is a parabola defined as a locus of points? Explain.
• How does the locus definition of a parabola connect to the equation of a parabola in rectangular coordinates?

An ellipse is one of the four basic forms of conic sections and can be represented by the equation   = 1.

• How is an ellipse defined as a locus of points? Explain.
• How does the locus definition of an ellipse connect to the equation of an ellipse in rectangular coordinates?
• What are the characteristics of ellipses?
• How can the characteristics of an ellipse be determined from the graph of the ellipse?
• How can the characteristics of an ellipse be determined from the equation of the ellipse?
• How can the characteristics of an ellipse be used to write the equation of an ellipse?
• What situations can be modeled and solved by ellipses?

A hyperbola is one of the four basic forms of conic sections and can be represented by the equation  or  .

• How is a hyperbola defined as a locus of points? Explain.
• How does the locus definition of a hyperbola connect to the equation of a hyperbola in rectangular coordinates?
• What are the characteristics of hyperbolas?
• How can the characteristics of a hyperbola be determined from the graph of the hyperbola?
• How can the characteristics of a hyperbola be determined from the equation of the hyperbola?
• How can the characteristics of a hyperbola be used to write the equation of a hyperbola?
• What situations can be modeled and solved by hyperbolas?

Parametric equations have unique characteristics and can be represented in multiple ways, including as rectangular relations.

• What are the characteristics of parametric equations?
• How do the characteristics of parametric equations and their corresponding rectangular relations compare?
• How can parametric equations be used to represent functions and relations?
• How can parametric equations be represented using tables?
• How can parametric equations be represented using graphs?
• How can parametric equations be represented algebraically?
• How can parametric equations be converted into rectangular relations?
• How can rectangular relations be converted into parametric equations?

Parametric equations can be used to model and solve mathematical real-world problem situations.

• How are parametric equations used to model problem situations?
• What type of problem situations can be modeled by parametric equations?
• How can parametric equations be formulated to represent problem situations?
• What methods can be used to solve problem situations involving parametric equations?
• How is the reasonableness of solutions justified in problem situations?
• Why must the reasonableness of solutions be justified in problem situations?

#### 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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Texas Higher Education Coordinating Board – Texas College and Career Readiness Standards (select CCRS from Standard Set dropdown menu)

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

TEKS# SE# TEKS Unit Level Specificity

• Bold black text in italics: Knowledge and Skills Statement (TEKS)
• Bold black text: Student Expectation (TEKS)
• Strike-through: Indicates portions of the Student Expectation that are not included in this unit but are taught in previous or future unit(s)
• Blue text: Supporting information / Clarifications from TCMPC (Specificity)
• Blue text in italics: Unit-specific clarification
• Black text: Texas Education Agency (TEA); Texas College and Career Readiness Standards (TxCCRS)
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:
• X. Connections
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:
• VIII. Problem Solving and Reasoning
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:
• VIII. Problem Solving and Reasoning
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:
• IX. Communication and Representation
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:
• IX. Communication and Representation
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:
• X. Connections
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:
• IX. Communication and Representation
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. Algebraic Reasoning
• C1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations.
• D2 – Translate among multiple representations of equations and relationships.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
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. Geometric Reasoning
• C1 – Make connections between geometry and algebra.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
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 (xy) in a plane such that (xy) is a distance of r units from a center point (hk).
• 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 (xy) 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 (xy) such that the distance from F to (xy) is equivalent to the distance from (xy) 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 (xy) 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, (x – h)2 + (y – k)2 = r2, centered at (hk) 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:
• II. Algebraic Reasoning
• B1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to combine, transform, and evaluate expressions (e.g., polynomials, radicals, rational expressions).
• III. Geometric Reasoning
• C1 – Make connections between geometry and algebra.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
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, 2(horizontal) or 2b (vertical)
• Semi-major axis, (horizontal) or b (vertical)
• Minor axis, 2(horizontal) or 2b (vertical)
• Semi-minor axis, (horizontal) or b (vertical)
• Focal length, c
• Equation relating focal length and the length of each semi-axis, |a2 – b2| = c2
• Coordinates for the center, (hk)
• Coordinates for the foci
• Ellipse with a horizontal major axis: (h ± ck)
• Ellipse with a vertical major axis: (hk ± c)
• Coordinates for the vertices
• On the horizontal axis: (h ± ak)
• On the vertical axis: (hk ± 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 (hk) with radius r using (x – h)2 + (y – k)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. Algebraic Reasoning
• D2 – Translate among multiple representations of equations and relationships.
• III. Geometric Reasoning
• C1 – Make connections between geometry and algebra.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
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, (hk)
• Distance from the center to the vertices, a (horizontal) or b (vertical)
• Coordinates for the vertices
• Hyperbola with a horizontal axis: (h ± ak)
• Hyperbola with a vertical axis: (hk ± b)
• Slopes of the linear asymptotes, ±
• Equations for the linear asymptotes, y – k = ± (– h)
• Focal length, c
• Equation relating focal length to a and b, a2 + b2 = c2
• Coordinates for the foci
• Hyperbolas with a horizontal axis: (h ± ck)
• Hyperbolas with a vertical axis: (hk ± 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 (hk) with radius r using (x – h)2 + (y – k)2r2.
• 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. Algebraic Reasoning
• D2 – Translate among multiple representations of equations and relationships.
• III. Geometric Reasoning
• C1 – Make connections between geometry and algebra.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections