Unit Overview. Content Area: Math Unit Title: Functions and Their Graphs Target Course/Grade Level: Advanced Math Duration: 4 Weeks


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1 Content Area: Math Unit Title: Functions and Their Graphs Target Course/Grade Level: Advanced Math Duration: 4 Weeks Unit Overview Description In this unit the students will examine groups of common functions such as linear, quadratic, square root, and cubic, and the characteristic that allow them to be identified and translated between algebraic and graphical formats. Transformations of these functions will also be addressed, including shifting and reflecting. Additional topics include combinations of functions and inverse functions. Students are evaluated by a unit test, quizzes, homework, class participation, along with other alternate assessments throughout the unit. Concepts Graphs of equations. Linear equations in two variables. Functions. Analyzing graphs of functions. A library of functions Shifting, reflecting, and stretching graphs. Combinations of functions Inverse functions Mathematical modeling CED.HS.02 CED.HS.04 REI.HS.01 REI.HS.10 F BF.HS.01 F BF.HS.03 F BF.HS.04 F IF.HS.01 F IF.HS.02 F IF.HS.04 F IF.HS.05 F IF.HS.07 Concepts & Understandings Learning Targets Understandings Sketch the graphs of equations Find and use the slopes of lines to write and graph linear equations in two variables Evaluate functions and find their domains Analyze graphs of functions Identify and graph rigid and nonrigid transformations of functions. Find arithmetic combinations and compositions of functions Find inverse functions graphically and algebraically Write algebraic models for direct, inverse, and joint variation.
2 F IF.HS.08 F IF.HS.09 F LE.HS.02 CO.HS.02 GPE.HS.05 See Addendum Guiding Questions How do we use a tchart or inputoutput table to graph a function? How many points do we need to graph a line?...a parabola?...a circle? If we have too few points could they represent more than one possible graph? What characteristic(s) do all lines share? What component of the equation of any line does this represent? What information do we need to graph any line? Is there more than one combination of pieces of information that will allow us to graph a line?...to write the equation of a line? Do all relations have a unique value of y for each value of x? How would the graphs of those relations look? What does the graph of a function look like? Can there be any 2 points in the same vertical line? What graphical characteristics might make graphing the function easier? What characteristics do all lines have in common? Parabolas? How do these common characteristics help us identify their graphs? Help us graph their equations? What does adding/subtracting a number "inside" the quadratic function do to its graph? "Outside" the function? What does a negative "outside the quadratic function do to its graph? Inside" the function? What does a value outside the graph other than 1 or 1 do to the quadratic graph? What would be easier, evaluating a function for a given value and taking the output as input into a second function, or finding a function that combines both of those functional processes into one function? What if we need to do this for many given values? What kind of data points and their graph (as in section 1.2) does a linear equation model or best represent? What shaped graph might be best modeled with a quadratic equation? Unit Results Students will... Sketch the graphs of equations Find and use the slopes of lines to write and graph linear equations in two variables Evaluate functions and find their domains Analyze graphs of functions Identify and graph rigid and nonrigid transformations of functions. Find arithmetic combinations and compositions of functions Find inverse functions graphically and algebraically Write algebraic models for direct, inverse, and joint variation. Suggested Activities The following activities can be incorporated into the daily lessons: Graphing familiar functions using tcharts. Using graph information to write an equation (specifically of a circle). Using tcharting to graph data in reallife application. Finding the x and y intercepts of a function.
3 Determining if a function is symmetric w/ respect to either axis or the origin, and use these techniques to sketch a function. Using tcharting to graph a function not symmetric w/ respect to either axis or the origin. Applying slope to solve reallife problems. Writing equations of parallel and perpendicular lines. Using tcharting to graph data in reallife application. Writing linear equations in two variables. Finding the slope of a line through two points. Identifying which equation is a function. Evaluating a function for a given value of x. Finding the domain of a function given a list of its ordered pairs. Applying the evaluation of a function to a reallife application. Testing for functions. Finding the zeros of a function, both algebraically and graphically. Testing a function as to whether it is odd, even or neither. Finding the domain and range of a function using its graph. Using the graph of the function, identify on which intervals it is increasing or decreasing. Using a graphing calculator, approximate the relative minimum or maximum of a function. Writing a linear function given input values and their function values. Evaluating a step function. Graphing a piecewise function. Sketching the graph of a function as the shift of a common function. Identifying the graph/equation of a function as the reflection of a common function. Sketching the graph of a function as the stretching (widening or narrowing) of a common function. Combining two functions arithmetically (adding, subtracting, multiplying, dividing). Decomposing a function into two separate functions. Appling the composition of functions to a reallife problem. Finding the inverse of a function graphically. Graphing inverses on the same set of coordinate axes and verify that they are, in fact, inverses graphically. Finding a function's inverse through opposite operations, and verify that they are, in fact, inverses. Finding the inverse of a function algebraically. Using the Horizontal Line Test to graphically verify that a function has an inverse. Plotting data points of a given sample set of points, and a math model of these points on the same graph and compare. Finding a math model for direct variation reallife situations. Finding a math model for inverse variation reallife situations. Finding a math model for joint variation reallife situations. Content Area: Math Unit Title: Polynomial and Rational Functions Target Course/Grade Level Advanced Math Duration: 5 weeks Unit Overview Description This unit introduces the use of a graphing utility as a tool to investigate mathematical ideas, to support analytic work, and to solve problems with numerical and graphical methods. The emphasis is on functions and graphs, the man building blocks of calculus. Students will learn how to sketch and analyze graphs of polynomial functions and how to use long division and synthetic division to divide polynomials by other polynomials. Then they will learn how to determine the numbers of rational and real zeros of polynomial functions, and find the zeros. More over they will learn how to sketch the graphs of rational functions and how to recognize and find partial fraction decompositions of rational expressions. The student will get evaluated by quizzes and by tests after completion of sections and the chapter. Concepts & Understandings
4 Concepts Quadratic functions Polynomial functions of higher degree Polynomial and synthetic division Zeros of polynomial functions Rational functions Partial fractions CPI Codes APR.HS.05 CED.HS.01 CED.HS.03 SSE.HS.01 SSE.HS.03 Understandings Analyze graphs of quadratic functions Write quadratic functions in standard form and use the results to sketch graphs of functions Use quadratic functions to model and solve reallife problems Use transformations to sketch graphs of polynomial functions Use the leading coefficient test to determine the end behavior of graphs of polynomial functions Use zeros of polynomial functions as sketching aids Use the intermediate value theorem to help locate zeros of polynomial functions Use long division to divide polynomials by other polynomials Use synthetic division to divide polynomials by binomials of the form (x k) Use the remainder theorem and factor theorem Use the fundamental theorem of algebra to determine the number of zeros of polynomial functions Find rational zeros of polynomial functions find zeros of polynomials by factoring find the domains of rational functions find the horizontal and vertical asymptotes of graphs of rational functions analyze and sketch graphs of rational functions sketch graphs of rational functions that have slat asymptotes use rational functions to model and solve reallife problems recognize partial fraction decompositions of rational expressions find partial fraction decompositions of rational expression Learning Targets F
5 BF.HS.01 F BF.HS.05 F IF.HS.02 F IF.HS.04 F IF.HS.06 APR.HS.01 APR.HS.02 APR.HS.03 APR.HS.04 APR.HS.05 APR.HS.06 APR.HS.07 See Addendum 21 st Century Themes and Skills Guiding Questions How do you analyze graphs of quadratic functions? How do you write quadratic functions in standard form and use the results to sketch graphs of functions? How do you use quadratic functions to model and solve reallife problems? How do you use transformations to sketch graphs of polynomial functions? How do you use the leading coefficient test to determine the end behavior of graphs of polynomial functions? How do you use zeros of polynomial functions as sketching aids? How do you use the intermediate value theorem to help locate zeros of polynomial functions? How do you use long division to divide polynomials by other polynomials? How do you use synthetic division to divide polynomials by binomials of the form (x k)? How do you use the remainder theorem and factor theorem? How do you use the fundamental theorem of algebra to determine the number of zeros of polynomial functions? How do you find rational zeros of polynomial functions? How do you find zeros of polynomials by factoring? How do you find the domains of rational functions? How do you find the horizontal and vertical asymptotes of graphs of rational functions? How do you analyze and sketch graphs of rational functions? How do you sketch graphs of rational functions that have slat asymptotes? How do you use rational functions to model and solve reallife problems? How do you recognize partial fraction decompositions of rational expressions? How do you find partial fraction decompositions of rational expression?
6 Unit Results Students will... Analyze graphs of quadratic functions Write quadratic functions in standard form and use the results to sketch graphs of functions Use quadratic functions to model and solve reallife problems Use transformations to sketch graphs of polynomial functions Use the leading coefficient test to determine the end behavior of graphs of polynomial functions Use zeros of polynomial functions as sketching aids Use the intermediate value theorem to help locate zeros of polynomial functions Use long division to divide polynomials by other polynomials Use synthetic division to divide polynomials by binomials of the form (x k) Use the remainder theorem and factor theorem Use the fundamental theorem of algebra to determine the number of zeros of polynomial functions Find rational zeros of polynomial functions Find zeros of polynomials by factoring Find the domains of rational functions Find the horizontal and vertical asymptotes of graphs of rational functions Analyze and sketch graphs of rational functions Sketch graphs of rational functions that have slat asymptotes Use rational functions to model and solve reallife problems Recognize partial fraction decompositions of rational expressions Find partial fraction decompositions of rational expression Suggested Activities The following activities can be incorporated into the daily lessons: Sketching graphs of quadratic functions Graphing a parabola in standard form Finding the vertex and xintercepts of a parabola Writing the equation of a parabola Finding all the real zeros of the function Sketching the graph of the rational function Writing the partial fraction decomposition for the rational expression Using the rational zero test to list all possible rational zeros of the function Identifying any horizontal or vertical asymptotes of the function Using synthetic division to find the value of the function Using long division to divide the function Using the intermediate value theorem to help locate zeros of polynomial functions Using the remainder theorem and factor theorem Finding zeros of polynomials by factoring Finding the domains of rational functions Using rational functions to model and solve reallife problems Recognizing partial fraction decompositions of rational expressions Finding partial fraction decompositions of rational expression Unit Overview Content Area: Math Advanced Math Unit Title: Exponential and Logarithmic Functions Target Course/Grade Level: 12th grade/ Advanced Math Duration: 4 weeks Description In this unit the students will examine exponential and logarithmic functions, their associated equations
7 and graphs. Properties of logarithms will be introduced, as one element in solving logarithmic and exponential equations. Related mathematical models and applications are addressed, as well. Students are evaluated by a unit test, quizzes, homework, class participation, along with other alternate assessments throughout the unit. Concepts Exponential Functions and their graphs Logarithmic functions and their graphs Properties of logarithms Exponential and logarithmic equations Exponential and logarithmic models CPI Codes FIF.HS.07 FLE.HS.01 FLE.HS.02 Concepts & Understandings Learning Targets Understandings Identifying a function as exponential allows us to more easily graph it or write its equation. Identifying a function as logarithmic allows us to more easily graph it or write its equation. Properties & formulas relating to logarithms allow us to transform, simplify & solve logarithmic expressions, and logarithmic & exponential equations. FLE.HS.03 FLE.HS.04 FLE.HS st Century Themes and Skills See Addendum Guiding Questions What function represents a constant rate of change? What if the dependant variable increase at more than a constant rate, for example, doubling each time the dependant variable increased by one...what would that graph look like? What if the independent increased by 1/2 each time the dependant increased by one? What test does a function need to pass to have an inverse? Does the exponential function pass this test? Will it have an inverse function? If it does have an inverse, what would the graph look like? Were there properties to make exponential expressions simpler? What were some of those properties? What properties or rules do we use to solve linear equations? Quadratic equations? How do we use inverse operations to solve equations? If an exponential equation has two expressions, with the same base, set equal, what can we say about the exponents? Do you think we would benefit from similar properties for logarithms? How would you know that a situation could be modeled using a logarithmic equation? An exponential equation? Why do we try to approximate the data of a given reallife situation with a particular equation? Unit Results
8 Students will... Recognize, evaluate and graph exponential functions with various bases, including e Recognize, evaluate and graph logarithmic functions with various bases, including e Use logarithmic properties and formulas to evaluate, rewrite, condense, and expand logarithmic expressions, as well as model & solve reallife Solve exponential and logarithmic equations of various types, and use these equations to model and solve reallife applications Recognize and Use the Most Common Models of Exponential & Logarithmic Functions Recognize and use the five most common types of models involving exponential and logarithmic functions; exponential growth & decay, logistic growth, & logarithmic models. Suggested Activities The following activities can be incorporated into the daily lessons: Evaluate exponential functions Graph exponential functions, with positive or negative exponents Transform (shift; reflect)graphs of exponential equations Evaluate and graph natural exponential functions Apply exponential functions to reallife problems Evaluate logarithmic functions, w/ or w/out a calculator Manipulate logarithmic expressions and solve log Graph logarithmic functions, comparing them to exponential graphs Shift graphs of logarithmic functions Content Area: Math Unit Title: Trigonometry Target Course/Grade Level Advanced Math Duration: 5 weeks Unit Overview Description The study of trigonometry is extended to circular functions. Radian measure is defined and the relationship between radians and degrees reinforced. The unit circle is introduced, circular functions defined and the similarity to triangular trigonometry noted. Students graphically represent the sine and cosine curves. Graphs are analyzed and amplitude, vertical and horizontal shifts, stretches and shifts studied by changing the a, b and c in the general equations y = a sin bx + c and y = a cos bx + c. Trigonometric identities are presented. Trigonometric proofs are introduced. Students will how to use the fundamental trigonometric identities and also they will know how to evaluate the compositions of trigonometric functions and inverse trigonometric functions. These skills are extended to be applied to reallife situations. Evaluation occurs via homework, quizzes and test upon completion of the chapter. Concepts & Understandings Concepts Radian and degree measure Trigonometric functions; The Unit circle Right triangle trigonometry Trigonometric functions of any angle Graphs of sine and Cosine functions Graphs of other trigonometric functions Inverse trigonometric functions Applications and models Learning Targets Understandings Describe an angle and convert between radian and degree measure Identify a unit circle and its relationship to real numbers Evaluate trigonometric functions of any angle Use the fundamental trigonometric identities Sketch the graphs of trigonometric functions and translations of graphs of sine and cosine functions Evaluate the inverse trigonometric functions Evaluate the compositions of trigonometric functions and inverse trigonometric functions.
9 CPI Codes SRT.HS.06 SRT.HS.07 SRT.HS.08 SRT.HS.09 SRT.HS.10 SRT.HS.11 See Addendum 21 st Century Themes and Skills Guiding Questions How do you describe an angle and convert between radian and degree measure? How do you identify a unit circle and its relationship to real numbers? How do you evaluate trigonometric functions of any angle? How do you use the fundamental trigonometric identities? How do you sketch the graphs of trigonometric functions and translations of graphs of sine and cosine functions? How do you evaluate the inverse trigonometric functions? How do you evaluate the compositions of trigonometric functions and inverse trigonometric functions? Unit Results Students will... Describe an angle and convert between radian and degree measure Identify a unit circle and its relationship to real numbers Evaluate trigonometric functions of any angle Use the fundamental trigonometric identities Sketch the graphs of trigonometric functions and translations of graphs of sine and cosine functions Evaluate the inverse trigonometric functions Evaluate the compositions of trigonometric functions and inverse trigonometric functions. Suggested Activities The following activities can be incorporated into the daily lessons: Converting from degrees to radians Converting from radians to degrees Finding angular and linear speed Identifying a unit circle and its relationship to real numbers Evaluating trigonometric functions using the unit circle Using the domain and period to evaluate sine and cosine function Using a calculator to evaluate trigonometric functions Evaluating trigonometric functions Using the fundamental trigonometric identities Using a calculator to evaluate trigonometric functions Using reference angles to evaluate trigonometric function Evaluate trigonometric functions of real numbers
10 Sketching the graphs of basic sine and cosine functions Using amplitude and period to help sketch the graphs of sine and cosine functions Sketching translations of the graphs of sine and cosine functions Using sine and cosine functions to model reallife data Sketching the graphs of tangent functions Sketching the graphs of cotangent functions Sketching the graphs of secant and cosecant functions. Sketch the graphs of damped trigonometric functions Evaluating the inverse trigonometric functions Evaluating the compositions of trigonometric functions Solving reallife problems involving right triangles Solving reallife problems involving harmonic motion Content Area: Math Unit Title Analytic Trigonometry Target Course/Grade Level Advanced Math Duration: 5 weeks Unit Overview Description In this unit the students will learn many of the trigonometric identities and conversion formulas. These will be used to simplify trigonometric expressions, solve trigonometric equations, and prove trigonometric identities. Students are evaluated by a unit test, quizzes, homework, class participation, along with other alternate assessments throughout the unit. Concepts Using fundamental identities Verifying trigonometric identities Solving trigonometric equations Sum and difference formulas Multipleangle and productto sum formulas Concepts & Understandings Understandings Recognize and write the fundamental trigonometric identities Use the fundamental trigonometric identities to evaluate trigonometric functions, simplify trigonometric expressions, and rewrite trigonometric expressions. Verify trigonometric identities. Use standard algebraic techniques to solve trigonometric equations Solve trigonometric equations of quadratic type Solve trigonometric equations involving multiple angles Use inverse trigonometric functions to solve trigonometric equations Use sum and difference formulas to evaluate trigonometric functions, verify identities, and solve trigonometric equations Use multipleangle formulas to rewrite and evaluate trigonometric functions Use powerreducing formulas to rewrite and evaluate trigonometric functions Use halfangle formulas to rewrite and evaluate trigonometric functions Use productto sumand sumto product formulas to rewrite and evaluate trigonometric functions
11 CPI Codes SRT.HS.06 SRT.HS.07 SRT.HS.08 SRT.HS.09 SRT.HS.10 SRT.HS.11 Learning Targets FTF.HS.07 FTF.HS.08 FTF.HS st Century Themes and Skills See Addendum Guiding Questions What is a trigonometric identity? Do you remember the trigonometric identities we studied in Chapter 4? Could you write them by category? What are the identities that show the functions as reciprocals? Which ones were derived using the Pythagorean Theorem? Which ones show that "co functions of complementary angles are congruent"? Which ones describe the functions as either even or odd? Whatever our technique may be, how do we know that we are done; that we have proven the identity? If we are unsure of what our first step should be, should we do anything, as long as it's algebraically correct and uses the identities we already know? Given all of the many identities we already know, does it seem like there will always be only one way to prove an identity? What are some of the techniques we use to solve linear equations?.quadratic equations? Have we looked at finding (w/out a calculator) the values of trigonometric functions of all types of angles? or just the conveniently "familiar" ones, such as 30, 45, 60, and 90? Unit Results Students will... Recognize and write the fundamental trigonometric identities Use the fundamental trigonometric identities to evaluate trigonometric functions, simplify trigonometric expressions, and rewrite trigonometric expressions. Verify trigonometric identities.
12 Use standard algebraic techniques to solve trigonometric equations Solve trigonometric equations of quadratic type Solve trigonometric equations involving multiple angles Use inverse trigonometric functions to solve trigonometric equations Use sum and difference formulas to evaluate trigonometric functions, verify identities, and solve trigonometric equations Use multipleangle formulas to rewrite and evaluate trigonometric functions Use powerreducing formulas to rewrite and evaluate trigonometric functions Use halfangle formulas to rewrite and evaluate trigonometric functions Use productto sumand sumto product formulas to rewrite and evaluate trigonometric functions Suggested Activities The following activities can be incorporated into the daily lessons: Using identities to evaluate a function Simplifying a trigonometric expression Factoring and simplifying a trigonometric expressions Adding and rewriting a trigonometric expression Trigonometric substitution Verifying a trigonometric identity Combining fractions before using identities Converting to sines and cosines Factoring an equation of quadratic type Using inverse functions Evaluating trigonometric expression Evaluating trigonometric function Proving a co function identity Deriving reduction formulas Solving a trigonometric function Solving multipleangle equation Using doubleangle formulas to analyze graphs Evaluating functions involving double angles Deriving a tripleangle formula Using a halfangle formula Using a sumto product formula Content Area: Math Unit TitleAdditional Topics in Trigonometry Target Course/Grade Level Advanced Math Duration: 3 weeks Unit Overview Description In this unit the students will learn how to use the Laws of Sines and Cosines to solve oblique triangles, and to model & solve reallife problems; how to represent vectors graphically, perform basic & more complex operations with vectors, & use them to model and solve reallife problems; and graph complex #s, write them in trigonometric form, and perform basic operations with them. examine groups of common functions such as linear, quadratic, square root, and cubic, and the characteristic that allow them to be identified and translated between algebraic and graphical formats. Transformations of these functions will also be addressed, including shifting and reflecting. Additional topics include combinations of functions and inverse functions. Students are evaluated by a unit test, quizzes, homework, class participation, along with other alternate assessments throughout the unit. Concepts Concepts & Understandings Understandings
13 CPI Codes Garfield High School Laws of Sines & Cosines. Learning Targets The Laws of Sines & Cosines can be used to find missing sides or angles of oblique triangles. Find the areas of oblique triangles Use the Law of sines to model and solve reallife problems. SRT.HS.10 SRT.HS st Century Themes and Skills See Addendum Guiding Questions How do we find missing sides or angles of nonright (oblique) triangles if we don't have a right angle and don't have a hypotenuse? What if we dropped a perpendicular to one of the sides & created two right triangles...could we then find our missing pieces by using our right triangle trigonometric and performing some equation substitution? Unit Results Students will... Use The Laws of Sines to find missing sides & angles of oblique triangles. Use The Laws of Cosines to find missing sides & angles of oblique triangles. What if we're given all three sides of an oblique triangle...can we use The Law of Sines? Suggested Activities The following activities can be incorporated into the daily lessons: Find the remaining sides and angles in the oblique triangle. Determine that no triangle or multiple triangles exist given certain combinations of sides and angles in a triangle Find the remaining sides and angles in the oblique triangle (including a reallife application), given SSS and SAS values Content Area: Math Unit Title: Topics in Analytic Geometry Target Course/Grade Level Advanced Math Duration: 4 weeks Unit Overview Description This chapter, students will learn how to find the inclination of a line, the angle between two lines, and the distance between appoint and a line. They will learn how to write the standard form of the equation of a parabola, and an ellipse. And also, they will learn how to rewrite a set of parametric equations as a rectangular equation and find a set of parametric equations for a graph. The student will get evaluation by quizzes and by tests after completion of sections and the chapter. Concepts Lines Introduction to conics: Parabolas Ellipses Concepts & Understandings Understandings Find the inclination of a line Find the angle between two lines. Find the distance between a point and a line.
14 Parametric equation. Recognize a conic as the intersection of a plane and a doublenapped cone. Write the standard form of the equation of a parabola. Use the reflective property of parabolas to solve reallife problems. Write the standard form of the equation of an ellipse. Use properties of ellipses to model and solve reallife problems. Find the eccentricity of an ellipse. Evaluate a set of parametric equations for a given value of the parameter. Sketch the curve that is represented by a set of parametric equations. Rewrite a set of parametric equations as a single rectangular equation. Find a set of parametric equations for a graph. CPI Codes APR.HS.05 CED.HS.01 CED.HS.03 SSE.HS.01 SSE.HS.03 FBF.HS.01 See Addendum Learning Targets 21 st Century Themes and Skills Guiding Questions How do you find the inclination of a line? How do you find the angle between two lines? How do you find the distance between a point and a line? How do you recognize a conic as the intersection of a plane and a doublenapped cone? How do you write the standard form of the equation of a parabola? How do you use the reflective property of parabolas to solve reallife problems? How do you write the standard form of the equation of an ellipse? How do you use properties of ellipses to model and solve reallife problems.? How do you find the eccentricity of an ellipse? How do you evaluate a set of parametric equations for a given value of the parameter? How do you sketch the curve that is represented by a set of parametric equations? How do you rewrite a set of parametric equations as a single rectangular equation? How do you find a set of parametric equations for a graph?
15 Unit Results Students will... Find the inclination of a line Find the angle between two lines. Find the distance between a point and a line. Recognize a conic as the intersection of a plane and a doublenapped cone. Write the standard form of the equation of a parabola. Use the reflective property of parabolas to solve reallife problems. Write the standard form of the equation of an ellipse. Use properties of ellipses to model and solve reallife problems. Find the eccentricity of an ellipse. Evaluate a set of parametric equations for a given value of the parameter. Sketch the curve that is represented by a set of parametric equations. Rewrite a set of parametric equations as a single rectangular equation. Find a set of parametric equations for a graph. Suggested Activities The following activities can be incorporated into the daily lessons: Finding the inclination of a line. Finding the angle between two lines. Finding the distance between a point and a line. Finding the standard equation of a parabola Finding the focus of a parabola. Finding the tangent line at a point on a parabola. Finding the standard equation of an ellipse. Writing an equation in standard form. Analyzing an ellipse. Graphing a polar equation by point plotting. Using symmetry to sketch a polar graph.
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