Web viewDerive the equation of a parabola given the focus and ... ellipse, parabola, or hyperbola and graph the ... What is the equation for an ellipse with one focus at

2015 - 2016

Honors Precalculus & TrigonometryHonors Precalculus & Trigonometry is a more rigorous and extensive course (especially in regards to the study of Trigonometry). This course includes advanced techniques using a graphing calculator with special emphasis in the areas of polynomial and rational functions, exponential and logarithmic functions, trigonometry, systems of equations, matrices, vectors, and conics. Students will cover all topics covered in Precalculus & Trigonometry, but will also be expected to apply problem-solving techniques to more challenging real-world problems (mixture, half-life, uniform motion, etc). Additional topics unique to Honors Precalculus & Trigonometry will include, but not be limited to:

Graph cot x , sec x , csc x transformations. Graph all six inverse trigonometric functions and state their restricted

domains. Use double angle and half angle formulas for sine, cosine, and tangent. Use the AAS area formula to calculate the area of any triangle. Calculate linear and angular velocity. Convert between D°M'S' ' and decimal degree. Solve inverse trigonometric equations. Prove trigonometric identities involving double angles. Show that composition of functions is closed under addition, subtraction,

multiplication, and division. Write the domain of a resultant composition function. Given a rational function, determine when a slant asymptote exists, and find

the equation of the asymptote using an appropriate algebraic method. Find the inverse and determinant of a 3 x 3 matrix algebraically (without

technology). Use Cramer’s Rule to solve a system of equations. Probability Chapter 13

Course Information:

Frequency & Duration: Daily for 42 minutesText: Sullivan, M. Precalculus: Graphing and Data Analysis, Second Edition. Prentice Hall: Upper Saddle River, NJ, 2001.

Honors Precalculus & Trigonometry v. 2015 - 2016

Content: Trigonometric Functions Duration: Aug./ Sept. (4 weeks)

Essential Question:

What is trigonometry?How is trigonometry used to find unknown values?Why are certain values undefined for certain functions?

Skill:

Find values of trigonometric functions for acute angles of right triangles. Solve right triangles including special right triangles (30-60-90, 45-45-90). Use special right triangles to determine geometrically the values of sine,

cosine, and tangent for π3, π4,∧π6

, and use the unit circle to express the values of sine, cosine, and tangent for π−x ,π+x ,∧2 π−x in terms of their values for x, where x is any real number.

Find missing angles of a right triangle using the inverse trigonometric functions.

Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle (s=rθ).

Convert degree measures of angles to radian measures, and vice versa. Convert between D°M'S' ' and decimal degree. Identify all angles that are coterminal with a given angle. Find the length of an intercepted arc in a circle with a given central angle

measure and radius using the formula s=rθ. Find area of sectors of circles using the formula A=1

2r2θ.

Calculate linear and angular velocity. Explain how the unit circle in the coordinate plane enables the extension of

trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle. (Find the values of trigonometric functions of any angle including angles greater than 360 ° (2π ).)

Apply radian measure of an angle and the unit circle to analyze trigonometric functions. (Find the values of trigonometric functions using the unit circle.)

Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

Assessment:

If cosθ=25

, find the exact value of the other 5 trigonometric functions of θ. Given a 30-60-90 triangle and one side length, find the other two side

lengths. Given a 45-45-90 triangle and one side length, find the other two side

lengths. Given a right triangle, a side length and an acute angle, find all other

missing sides and angles. A competitor in a triathlon is running along the course shown. Determine

the length in feet that the runner must cover to reach the finish line.


Write 120° in radians leaving your final answer as a multiple of π. Write 5π

6 in degrees.

Find the length of the intercepted arc in the circle with the central angle and given radius. Round to the nearest tenth.

Find the area of the sector of a circle that has a central angle of 7π12

and a

radius of 3.4cm . Let (8, -6) be a point on the terminal side of an angle θ in standard position.

Find the exact values of the six trigonometric functions of θ . Sketch the angle, and then find its reference angle if θ=300°. Find the exact value of each of the six trigonometric functions if θ=120° by

first finding its reference angle. Let tanθ= 5

12, where sin θ<0, find the exact value of the five remaining

trigonometric functions of θ. Rationalize the denominator if necessary. Find the exact value of sin π

3 without using a calculator.

Find the exact value of cos 11 π4

without using a calculator.

Find the exact value of tan(−5π4

) without using a calculator.

Resources: Precalculus: Graphing and Data Analysis, pages 367 – 407.


Standards: This content is beyond the scope of the PA Core Mathematics Standards.

Vocabulary:

Angle– two noncollinear rays that share a common endpoint known as a vertex. Formed by rotating a ray about its endpoint; Coterminal Angles– two angles that have the same initial side and terminal side but different measures; Initial Side– the starting position of a ray; Period- the smallest number c for which f is periodic is called the period of f; Periodic Functions– functions that repeat their values at regular intervals. A function y=f (t ) is periodic if there exists a positive real number c such that f ( t+c )= f (t ) for all values of t in the domain of f; Quadrant Angle– an angle whose terminal side lies on one of the coordinate axes; Radians– a unit of an angle, angle at the center of a circle whose arc is equal in length to its radius; Reference Angle– if θ is in standard position, its reference angle θ', is the acute angle formed by the terminal side of θ and the x-axis; Standard Position– in the coordinate plane, an angle with its vertex at the origin and its initial side along the positive x-axis; Terminal Side– the ray’s position after rotation; Unit Circle– a circle with a radius of 1 centered at the origin;

Comments:


Content: Graphing Trigonometric Functions

Duration: October (2 weeks)

Essential Question:

Why are graphs useful?How can graphs of trigonometric functions be valuable?

Skill: Graph transformations of sine, cosine, tangent, secant, cosecant, and

cotangent functions identifying the domain, range, intercepts, symmetry, amplitude, period, frequency, phase shift, vertical shift, midline, and end-behavior of the graph.

Assessment:

Describe how the graphs of f ( x )=sin xand g ( x )=12sin xare related.

Sketch two periods of both functions on the same coordinate plane. Describe how the graphs of f ( x )=cos x and g ( x )=cos 1

5x are related.

Find the period of each and sketch one period of each on the same coordinate plane.

State the amplitude, period, frequency, phase shift, and vertical shift of: y=3sin (x−π4 ), and then graph two periods of the function.

Locate the vertical asymptotes, and sketch the graph of y=tan 2x .

Recommended Activity: Precalculus: Graphing and Data Analysis, pages 408 – 419.


Vocabulary:

Amplitude– half the distance between the maximum and minimum values of the function or half of the height of the wave; Frequency– the number of cycles the function completes in one interval. The reciprocal of the period; Midline– a new horizontal axis created after a vertical shift; Period– the distance between any two sets of repeating points on the graph of the function; Phase Shift– the difference between the horizontal position of the


function and that of an otherwise similar sinusoidal function; Sinusoid– any transformation of a sine function; Vertical Shift– an translation either up or down

Comments:

Content: Applying Trigonometric Functions (The Law of Sines and Cosines)

Duration: October (2 weeks)

Essential Question:

How do you solve for sides or angles in oblique triangles?How do you find the area of oblique triangles?

Skill:

Solve oblique triangles by using the Law of Sines and the Law of Cosines. Apply the Law of Sines and Cosines to find unknown measurements in right

and non-right triangles (e.g. surveying problems, resultant forces). Find the area of oblique triangles using Heron’s Formula. Find the area of a triangle given SAS by deriving and using the formula A=1

2ab sinC

Find the area of a triangle given AAS.

Assessment:

Solve the following triangle:

Two ships are 250 feet apart and traveling to the same port as shown. Find the distance from each port to each ship.

Find all solutions for the given triangle, if possible. If no solution exists, write no solution. Round side lengths to the nearest tenth and angle measures to the nearest degree. a=15 , c=12 , A=94 °

When a hockey player attempts a shot, he is 20 feet from the left post of the


goal and 24 feet from the right post, as shown. If a regulation hockey goal is 6 ft wide, what is the player’s shot angle to the nearest degree?

Find the area of the triangle:

Find the area of the triangle to the nearest tenth:

Resources: Precalculus: Graphing and Data Analysis, pages 507 - 544.


Vocabulary:

Ambiguous Case– given the measures of two sides and a nonincluded angle, one of the following will be true (1) no triangle exists, (2) exactly one triangle exists, (3) two triangles exist. In other words, there may be no solution, one solution, or two solutions; Heron’s Formula– used to find the area of an oblique triangle A=√s (s−a)(s−b)(s−c) where s is the semiperimeter and

found by s=12(a+b+c ); Law of Cosines- a2=b2+c2−2bc cosA ;

b2=a2+c2−2ac cosB; c2=a2+b2−2abcosC; Law of Sines- sin Aa

= sinBb

= sinCc ; Oblique Triangles– triangles that are not right triangles;

Comments:



Content: Trigonometric Identities and Equations

Duration: November (3 weeks)

Essential Question: What approaches can be used to verify a trigonometric identity?

Skill:

Identify and use basic trigonometric identities to find trigonometric values. These should include: reciprocal and quotient identities, Pythagorean identities, cofunction identities, double and half angle, and odd-even identities.

Use basic trigonometric identities to simplify and rewrite trigonometric expressions. Utilize skills such as factoring, combining fractions, and eliminating fractions.

Prove the Pythagorean Identity (sin2θ+cos2θ=1¿¿ and use it to find sin θ, cosθ, or tanθ given sin θ, cosθ, and tanθ and the quadrant of the angle.

Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

Solve trigonometric equations using algebraic techniques. Solve trigonometric equations using basic identities. Use sum and difference identities to evaluate trigonometric functions. Use sum and difference identities to solve trigonometric equations. Use double and half angle identities to evaluate and solve trigonometric

equations. Solve inverse trigonometric equations.

Assessment:

If csc θ=74,find sin θ.

If cot x= 25√5

and sin x=√53

, find cos x.

If tanθ=−8∧sin θ>0 , find sin θ∧cosθ.

If tanθ=1.28 , find cot(θ−π2 ) Simplify: csc θsec θ−cot θ

Simplify by factoring: sin2θ cosθ−sin( π2−θ) Simplify by combining fractions: sin xcos x

1−sin x−1+sin xcos x

Rewrite 11+cos x

as an expression that does not involve a fraction.

Verify that csc2x−1csc2x

=cos2 x

Verify that 2csc x= 1csc x+cot x

+ 1csc x−cot x

Verify that sin x1−cos x

=csc x+cot x

Verify that cot x sec xcsc2 x−cot3 x sec x=csc x Solve: 2 tan x−√3=tan x


Solve: 4 sin2 x+1=4 Solve: cos x sin x=3cos x Solve: cos4 x+cos2 x−2=0 Find all solutions of 2cos2 x−sin x−1=0 on the interval ¿ Find all solutions of csc x−cot x=1 on the interval ¿ Find the exact value of sin 15 ° Find the exact value of tan 7 π

12

Find the exact value of tan 32°+ tan 13°1−tan32 ° tan 13°

.

Resources: Precalculus: Graphing and Data Analysis, pages 446 – 506.


Vocabulary:

Identity– an equation is an identity if the left side is equal to the right side for all values of the variable for which both sides are defined; Pythagorean Identities- sin2θ+cos2θ=1, tan2θ+1=sec2θ, cot2θ+1=sec2θ; Sum and Difference Formulas- sin (α± β )=sin α cos β± cos α sin β,

cos (α ±β )=cosα cos β∓sinα sin β, tan (α ±β )= tan α ± tan β1∓ tan α tan β ; Trigonometric

Identities– identities that involve trigonometric functions;Comments:


Content: Word Problems Duration: December (2 weeks)

Essential Question: How are mathematical applications all around us?

Skill:

Solve mixture problems. Solve uniform motion problems involving upstream and downstream

movements. Solve dilution problems. Solve concentration problems. Solve “working-together” problems.

Assessment:

It takes Billy 15 minutes to cut the grass and Sally 20 minutes to cut the grass. How long will it take working together?

How many ounces of a 10% acid solution should be added to 20 ounces of an 8% acid solution to create a solution that is 9% acid.



Vocabulary: There are no applicable vocabulary terms.

Comments:

Content: Characteristics of Functions Duration: Dec. /Jan. (3 weeks)


Essential Question: How can mathematical ideas be represented?

Skill:

Interpret key features of graphs (x-intercepts, y-intercepts, increasing, decreasing, local and absolute maxima and minima, symmetry, even, odd, continuity, domain, range, and end behavior (at ±∞ or a specific value).

Analyze the graph of a function. Sketch graphs given key features. Use the key features of the graph of a function to solve real-world problems

involving optimization. Calculate the average rate of change of a function, and be able to estimate

average rate of change from the graph of a function.

Assessment:

Given the graph of a function, give its y-intercept and x-intercept(s) as ordered pairs.

If f ( x )=−2 x3+43

, find the y-intercept.

If f ( x )=2x2+ x−15, find the zeros algebraically and estimate graphically, and then find the coordinates of the relative maximum or minimum (vertex).

Given the graph or equation of a function, determine if it is even or odd. Given the graph of a function, find all points of discontinuity. Given the equation of a function, use the three-part continuity test to

determine if the function is continuous at a given point. Then state the type of discontinuity present.

If f ( x )= x+3x2−9

, determine if the function is continuous at x=3. If it is not, state the type of discontinuity as infinite, jump, or removable.

Given the graph of a function, write its end-behavior in limit notation. Given the graph of a function, state the intervals on which it is increasing,

decreasing, and/or constant. Given the graph of a function find the coordinates of all critical points. Sketch the graph of a cubic function that has x-intercepts at (3, 0), (5, 0),

and (-6, 0) and has an end-behavior of down-up. Suppose each of the 75 orange trees in a Florida grove produces 400

oranges per season. Also suppose that for each additional tree planted in the orchard, the yield per tree decreases by 2 oranges. How many additional trees should be planted to achieve the greatest total yield?

Find the average rate of change of f ( x )=−x3+3 x on the interval [-2, -1]. The height of an object that is thrown straight up from a height 4 feet above

the ground is given by h (t )=−16 t 2+30t+4. Find and interpret the average speed of the object from 1.25 seconds to 1.75 seconds.

Resources: Carter, John, et. al. Glencoe Precalculus. 2014. McGraw Hill: Columbus, OH.



Vocabulary:

Average Rate of Change– the slope of a line between any two points; Continuous Function– has no breaks, holes, or gaps. You can trace the graph of a continuous function without lifting your pencil; Critical Points– points at which a line drawn tangent to the curve is horizontal or vertical; Discontinuous Function– a function that is not continuous and will contain an infinite discontinuity, jump discontinuity, or a removable discontinuity; End-behavior– describes how a functions behaves at either end of the graph as x increases without bound towards negative or positive infinity; Extrema– critical points at which a function changes its increasing or decreasing behavior. At these points, the function has a maximum or minimum value; Line Symmetry– can be folded along a line so that the two halves match exactly; Nonremovable Discontinuity– infinite and jump discontinuities. Cannot be eliminated by redefining the function at that point, since the function approaches different values from the left and right sides at that point or does not approach a single value at all. Instead it is increasing or decreasing indefinitely; Point of Inflection– a critical point at which the graph of a function changes its shape, but not its increasing or decreasing behavior. Instead it changes its concavity; Point Symmetry– can be rotated 180° with respect to a point and appear unchanged; Secant Lin – a line through two points; Turning Points– indicate where the graph of a function changes from increasing to decreasing, and vice versa; Zeros (Roots)– x-intercepts/solutions of equations

Comments:


Content: Composite Functions Duration: January (2 weeks)

Essential Question:

How can you extend algebraic properties and processes to composite functions?

Skill:

Perform algebraic operations (including compositions) on functions. Write an expression for the composition of one given function with another

and find the domain, and range of the composite function. Determine, using the horizontal line test, if a function has an inverse Verify by composition that one function is an inverse of another. Show that composition of functions is closed under addition, subtraction,

multiplication, and division. Write the domain of a resultant composition function.

Assessment:

If f ( x )=x2+4 x∧g ( x )=√x+2, find ( f ∙ h ) ( x ). If f ( x )=x2+1 and g ( x )=x−4, find [ f ∘g] (x) and [ f ∘g](6).

If f ( x )= 1x+1

and g ( x )=x2−9, find [ f ∘g] (x) and state the domain of the new function.

Show that f ( x )= 6x−4

and g ( x )=6x+4 are inverse functions using

composition.



Vocabulary:

Composite Function – a function such that [ f ∘g ] ( x )=f [g (x )].; Inverse Function – a function that reverses another function. If one function contains the point (a, b) then its inverse would contain (b, a)

Comments:


Content: Graphing Functions Duration: February (4 weeks)

Essential Question: How can mathematical ideas be represented as graphs?

Skill:

Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. The library of functions will include: linear, quadratic, cubic, quartic, radical, rational, exponential, logarithmic, and absolute value.

Graph rational functions, identifying zeros, intercepts, holes, and asymptotes (including slants) when suitable factorizations are available.

Model data that exhibit linear, quadratic, cubic, and quartic behavior using technology.

Graph exponential and logarithmic functions, showing intercepts, and end-behavior.

Model data that exhibit exponential, logarithmic, and logistic behavior using the regression feature of a graphing calculator.

Assessment:

Given the equation of a polynomial function, find all zeros by factoring (including grouping, GCF, sum/differences of cubes). Use the zeros, the concepts of repeated-zeros, and end-behavior to sketch a graph of the function.

Given the equation of a higher-degree polynomial function that is not easily factored, find at least one zero using technology, and then find the rest of the zeros using synthetic division.

Given a rational function, find any vertical, horizontal, or slant asymptotes, intercepts, holes, and end-behavior. Then, graph the function.

Given a data set, enter the data into a graphing calculator. Run an appropriate regression, and use that information to make inferences. Sketch the graph of f ( x )=3x. Identify all intercepts.

Sketch the graph of f ( x )=2−x. Identify all intercepts. Sketch the graph of f ( x )=e4x . Identify all intercepts. Sketch the graph of f ( x )=log3 x. Identify all intercepts. Sketch the graph of f ( x )=log(x+4) using transformation rules. Given a set of data, use the calculator to run the appropriate regression.

Resources: Precalculus: Graphing and Data Analysis, pages 137 – 148, 172 – 252, 292 - 315.



Vocabulary:

Algebraic Functions– functions with values that are obtained by adding, subtracting, multiplying, and dividing constants and the independent variable or raising the independent variable to a rational power.; Asymptotes– imaginary lines that the graph of a function approaches; Common Logarithm– log base 10; Exponential Functions– a function in which the base is a constant and the exponent is a variable; Holes– a removable discontinuity in the graph of a rational function; Horizontal Asymptote– the line y=c is a horizontal asymptote of the graph of f (x) if lim ¿x→−∞

❑f ( x )=c or lim ¿x→∞

❑f (x )=c;

Logarithmic Functions- ; Multiplicity– the number of times a zero is

repeated (even versus odd); Natural Base- e=lim ¿ x→∞❑ (1+ 1x )

x

; Natural

Logarithm– log base e; Polynomial Function- ; Rational Function- f ( x ) is quotient of two polynomial functions a (x) and b (x), where b is nonzero; Repeated Zero– if a factor (x−c) occurs more than once in the completely factored form of f (x), then its related zero c is called a repeated zero; Transcendental Functions– functions that cannot be expressed in terms in terms of algebraic expressions; Vertical Asymptote– a nonremovable discontinuity such that the line x=c is a vertical asymptote of the graph f (x) if lim ¿x→c

❑

−¿ f ( x )=±∞¿

or lim ¿x→c❑

+¿ f ( x )=±∞¿

Comments:


Content: Matrices Duration: March (4 weeks)

Essential Question:

How can you use a matrix to organize data?How can a matrix represent a transformation of a geometric figure in the plane?How can you use a matrix equation to model a real-world situation?

Skill:

Add and subtract matrices. Multiply a matrix by a scalar. Use matrices in applications. Multiply matrices of appropriate dimensions. Understand that the zero and identity matrices play a role in matrix

addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.

Find the inverse of a matrix and use it to solve systems of linear equations (using technology for matrices of dimensions 3 x 3 or greater).

Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.

Solve matrix equations. Use matrices to represent and manipulate data. Determine if a 2 x 2 matrix is invertible by finding the determinant, and if it

is, find the inverse algebraically. Find the determinant of a 3 x 3 matrix and then find the inverse if it exists. Work with 2 × 2 matrices as transformations of the plane, and interpret the

absolute value of the determinant in terms of area. Represent a system of linear equations as a single matrix equation in a

vector variable. Find the inverse of a matrix if it exists and use it to solve systems of linear

equations (including 3 × 3). Use Cramer’s Rule.

Assessment:

If A=[ 8 3−5 14] and B=[12 −7

6 −23], find A + B.

Find 3[−6 −3 710 2 −15].

If A=[3 −14 0 ] and B=[−2 0 6

3 5 1], find AB.

Find 4 (P+Q )if P=[ 3 8 −2−5 5 −4 ]∧Q=[−4 5 7

3 −10 −6 ] Given A=[−9 15 4

2 −10 −5]∧B=[ 5 −7 814 10 −3] , solve 4 X−B=A for X .

Allison took a survey of her high school to see which class sent the most text messages, pictures, and talked for the most minutes on their cell phones each week. The averages for freshmen, sophomores, juniors, and seniors are shown.


a) If each text message costs $0.10, each picture costs $0.75, and each minute on the phone costs $0.05, find the average weekly cell phone cost for each class. Express your answer as a matrix.

b) If there are 100 freshmen, 180 sophomores, 250 juniors, and 300 seniors that use cell phones at Allison’s school, use her survey results to estimate the total number of text messages sent, pictures sent, and minutes used on the cell phone each week by these students. Express your answer as a matrix.

Determine whether A=[−3 2−2 1]∧B=[1 −2

2 −3 ]areinverse matrices . Find A−1, if it exists, if A=[ 8 −5

−3 2 ] Find the determinant of the matrix, then find the inverse if it exists. A=[ 2 −3

4 4 ] Using technology, find the determinant of the matrix, then find the inverse if

it exists. C=[−3 2 41 −1 2

−1 4 0] . Use matrices to find the area of a polygon with vertices: (1, 1), (1, -4), (3, -

5), (3, -2). Use an inverse matrix to solve the system of equations, if possible. 2 x−3 y=−1 −3 x+5 y=3 Solve a system of equations using Cramer’s Rule.

Resources: http://www.onlinemathlearning.com/determinant-area-hsn.vm12.htmlPrecalculus: Graphing and Data Analysis, pages 709 - 769.


Vocabulary:

Column matrix– a matrix with one column; Determinant of a Matrix– Let

A=[a bc d ] . A is invertible if∧only if ad−cb≠0. If A is invertible, then

A−1= 1ad−cb [ d −b

−c a ]. The number ad−cb is called the determinant of the 2 x

http://www.onlinemathlearning.com/determinant-area-hsn.vm12.html


2 matrix and is denoted det (A )=|A|=|a bc d|=ad−cb. The determinant provides

a test for determining if the matrix is invertible; Dimensions– rows x columns of a matrix; Element– each value in a matrix; Equal matrices– each element in one matrix is equal to the corresponding element in the other matrix; Identity matrix– an n x n matrix consisting of all 1’s on the main diagonal, from upper left to lower right, and 0’s for all other entries; Inverse matrix– the multiplicative inverse of a square matrix. Let A be an n x n matrix. If there exists a matrix B such that AB = BA = Identity Matrix then B is called the inverse of A; Invertible matrix– a matrix that has an inverse;Matrix– a rectangular array of variables or constants in horizontal rows and vertical columns, usually enclosed in brackets; Row matrix– a matrix with one row; Singular matrix– a matrix that does not have an inverse; Square matrix– a matrix that has the same number of rows and columns Square System– If a system of linear equations has the same number of equations as variables, then its coefficient matrix is square and the system is said to be a square system. If this square coefficient matrix is invertible, then the system has a unique solution; Zero matrix– a matrix in which every element is zero

Comments:


Content: Conics Duration: April (4 weeks)

Essential Question:

How does mathematics help us describe the physical world?

What do graphs of relationships that are not functions look like?

Skill:

Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

Derive the equation of a parabola given the focus and directrix. Derive the equations of ellipses and hyperbolas given the foci, using the fact

that the sum or difference of distances from the foci is constant. Given a quadratic equation of the form ax2 + by2 + cx + dy + e = 0, use the

method for completing the square to put the equation into standard form; identify whether the graph of the equation is a circle, ellipse, parabola, or hyperbola and graph the equation.

Assessment:

What is the equation of the circle with center (4, -3) and radius 2? What are the center and radius of the circle described by the equation

( x+7 )2+( y−8 )2=9? What are the center and radius of the circle described by the equation x2+ y2−18 x+12 y+68=0?

What are the center and radius of the circle described by the equation 2 x2+2 y2+12x+20 y+36=0?

What is the equation of the parabola with focus (5, 7) and directrix y = 2? What is the equation for an ellipse with foci (±2 ,0) and constant distance 5? What is the equation for an ellipse with one focus at (5, 8), center at (9, 8),

and constant distance 16? What is the equation for a hyperbola with foci (±13 ,0) and transverse axis

length of 10?



Vocabulary:

Conic Sections- or conics, are the figures formed when a plane intersects a double-napped right cone (a double-napped right cone is two cones opposite each other and extending infinitely upward and downward); Degenerate Conics- when the plane intersects the vertex of the cone, it can create a point, a line, or intersecting lines; Ellipse- is the locus of points in a plane such that the sum of the distances from two fixed points, called foci, is constant; Latus rectum- the


line segment that passes through the focus, is perpendicular to the axis of the parabola, and has endpoints on the parabola; Locus- a set of all points that fulfill a geometric property; Hyperbola- is the locus of all points in a plane such that the absolute value of the differences of the distances from two foci is constant; Parabola- represents the locus of points in a plane that are equidistant from a fixed point, called the focus, and a specified line, called the directrix

Comments:


Content: Probability Duration: May (4 weeks)

Essential Question: How does probability influence decisions?

Skill:

Calculate a theoretical probability. Calculate an empirical probability. Understand mutually exclusive events, and calculate their probabilities. Use the binomial theorem and Pascal’s triangle to calculate the probability

of a Bernoulli event. Understand independent and dependent events, and calculate their

probabilities. Use the Law of Large Numbers to describe the relationship between

theoretical and empirical probability. Create a probability model. Calculate permutations and combinations.

Assessment:

Evaluate 10P7 Evaluate 10C7 Suppose a bag contains 7 red balls, 4 white balls, and 3 green balls. Two

balls are drawn without/with replacement. Find P(red and green), Find P(red or green), Find P(red|green).

What is the probability that you randomly guess on a 10-question true/false test, and get at least 7 right?



Vocabulary:

Empirical Probability- probability based on data; Independent Events- two events where the outcome of the first does not affect the probability of the second; Mutually Exclusive Events: two events that have no outcomes in common.

Comments:


Documents

Web viewDerive the equation of a parabola given the focus and ... ellipse, parabola, or hyperbola and graph the ... What is the equation for an ellipse with one focus at