Download Pre-Calculus Aims and Objectives

Townsend Harris High School @ QC
Kenneth Bonamo, Principal
Susan Brustein, AP Math, Science and Technology
Pre-Calculus Aims and Objectives
Lesson
Aim and Objectives
1.
Aim: What is a sequence?
Students Will Be Able To:
 Define a sequence
 Use summation notation
 Define an arithmetic sequence
 Find the first n terms of a sequence given a formula
2.
Aim: How do we find the nth term of an arithmetic sequence?
SWBAT:
 Derive the formula for the nth term of an arithmetic sequence
 Solve problems involving the nth term
 Define arithmetic mean
 Find the arithmetic mean(s) given two numbers
3.
Aim: How do we find the sum of the terms of an arithmetic sequence?
SWBAT:
 Derive the formula for the sum of n terms of an arithmetic sequence
 Solve problems involving the sum of n terms of an arith. sequence
4.
Aim: How do we apply formulas in word problems?
SWBAT:
 Translate word problems into symbols and variables
 Solve problems leading to arithmetic sequences
5.
Aim: What is a finite geometric sequence?
SWBAT:
 Define a geometric sequence
 Derive the formula for the nth term of a geometric sequence
 Solve problems involving the nth term of a geometric sequence
 Define a geometric means in a geometric sequence
 Find geometric mean(s) given two numbers
6.
Aim: How do we find the sum of n terms of a finite geometric sequence?
SWBAT

Derive the formula for the sum of n terms of a geometric sequence

Solve problems that involve finding the sum of n terms
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7.
Aim: How do we find the sum of an infinite geometric sequence?
SWBAT:
 Derive the formula for the sum of an infinite geometric sequence
 Apply the formula to find the sum of an infinite geometric sequence
8.
Aim: What is the Binomial Theorem? (optl. two days for negative and rational powers)
SWBAT:
 State the binomial theorem
 Expand binomials raised to powers
 Find a given term of an expansion
9.
Aim: How can we identify polynomials and define related terms?
SWBAT:
 Define a polynomial as anxn + an-1xn-1 + an-2xn-2 + …+ a1x + a0
 Determine if expressions are polynomials in one variable
 Determine the degree of a polynomial
 Write polynomials in standard form
10.
Aim: How can we use the information we already know about quadratic functions?
 Define a quadratic function and describe its graph
 Solve a quadratic by factoring or using the quadratic formula
 Find non-real, complex zeros of a quadratic function
 Write a quadratic equation and find its roots
 Solve quartic equations that can be reduced to quadratic equations.
11.
Aim: How do we find the zeros of higher degree polynomials?
SWBAT:
 Distinguish between zeros of polynomials and roots of polynomial equations
 Determine the zeros of third-, fourth- or fifth-degree polynomials using
methods such as grouping, and factoring a quartic in the form of a quadratic
(ax 4  bx 2  c   ) .
 Sketch the basic shape of a polynomial (cubic, quartic, even-, odd-degree)
12.
Aim: How can we perform operations on polynomials? (or How can we find the quotient
of two polynomials?)
SWBAT:
 Perform operations on polynomials including division
 Express the dividend in terms of the quotient, the divisor and the remainder
 State that when the remainder is 0, the divisor is a factor of the dividend
13.
Aim: How can we use synthetic division to find quotients?
 Use synthetic division to divide a polynomial by a binomial in the form of x –
a
 Apply the process for synthetic division for more complex problems
 Use synthetic division to factor the sum and difference of two cubes
 Determine factors of a polynomial
2

(optl.)
Include other special factoring
14.
Aim: How can we determine the remainder of the division of a polynomial by a binomial
without actually finding the quotient?
SWBAT:
 Find the value of a polynomial
 Write the remainder theorem
 Find the remainder when P(x) / (x - k) by finding P(k)
 Find P(k) by finding the remainder when P(x) / (x - k)
15.
Aim: How can we determine if a binomial is a factor of the polynomial?
SWBAT:
 State the Factor Theorem
 Apply the theorem to polynomials to determine if a given binomial is a factor
16.
Aim: How can we determine the possible rational roots of a polynomial equation?
SWBAT:
 State the Fundamental Theorem of Algebra
 State the Rational Root and Integral Root Theorems
 Determine possible rational roots for a given polynomial equation
17.(optl.) Aim: How can we determine sum and product of roots of a polynomial equation of any
degree?
SWBAT:
 Find the sum and product of the roots of a polynomial of any degree
18.
Aim: How do we find imaginary or non-real roots? (or How does knowing one root of
P(x) = 0 in the form of a  b or a  bi help us to find remaining roots?)
SWBAT:
 State the conjugate pairs theorem
 Apply the theorem to find the remaining roots of a polynomial equation
 Use synthetic division to form a depressed equation and solving it
19.
Aim: How can we locate the real roots of a polynomial equation using the graphing
calculator?
SWBAT:
 State the location principle to determine the existence of roots
 Illustrate the location principle using the graphing calculator
 Use the graphing calculator to locate real roots using “trace”, “calculate” and
“table”
20.
Aim: How do we resolve a polynomial into partial fractions?
SWBAT:
 Resolve polynomials into partial fractions
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21.
Aim: How do we resolve polynomials with repeated factors into partial fractions?
SWBAT:
 Resolve more difficult polynomials into partial fractions
22.
Aim: How can we perform operations on complex numbers? How can we represent
complex numbers graphically?
SWBAT:
 Perform operations on imaginary numbers
 Express powers of i in simplest form
 Distinguish real numbers, imaginary and complex numbers
 Perform operations on complex numbers
 Represent a complex number as a vector in the complex plane
23.
Aim: How do we represent points in the polar coordinate plane? (2 days?)
SWBAT:
 Define pole, polar axis, polar angle () and polar distance (r)
 Plot points in the polar plane (include negative values of r and )
 Write the general form of a complex number in polar form
 Write the formulas to change from rectangular to polar and vice versa
24.
Aim: How can we write a complex number in polar form?
SWBAT:
 Find the corresponding values of r and  for a complex number in
a + bi form
 Express complex numbers in polar form
25.
Aim: How do we change complex numbers from polar to rectangular form?
SWBAT:
 Substitute values in x  r cos and y  r sin  to find values of x and y
 Express complex number in a + bi form
26.
Aim: How do we find the product of complex numbers in polar form?
SWBAT:
 Derive the formula for the product of two complex numbers in polar form
 Apply the formula to find the product
27.
Aim: How do we find the quotient of complex numbers in polar form?
SWBAT:
 Derive the formula for the quotient of two complex numbers in polar form
 Apply the formula to find the quotient
28.
Aim: How do we raise a complex number to a power?
SWBAT:
 Derive the formula for raising a complex number to a power
 Find a power of a complex number by applying formula
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29.
Aim: How do we find all the nth roots of a complex number in polar form?
SWBAT:
 State DeMoivre’s Theorem for raising a complex number to a power
 Use the theorem to deduce a formula to find the principal root
 Use the formula to find principal roots of complex numbers
 Write the formula to find all the nth roots
 Find all the roots of a complex number in polar form
 Find all the roots of a polynomial
30.(optl) Aim: How can we graph equations in polar form on the graphing calculator?
SWBAT:
 Transform an equation from rectangular form to polar form
 Graph equations such as r  2a cos , r  1 cos , r  a , r  2  3cos ,
r  sin2
31.
Aim: What is the equation of a circle?
SWBAT:
 State the definition of a circle
 State the equation of a circle whose center is at the origin having a given
radius.
 Sketch a circle of the form x 2  y 2  r 2
 Write the standard form of the equation of a circle whose center is not the

 
2
origin x  h  y  k

 r
2
2
Write the equation of a circle given the center and radius
32.
Aim: How do we complete the square in order to write the equation of a circle in standard
form?
SWBAT:
 Solve a quadratic equation by completing square
 Complete the square in order to determine the center and radius of a circle in
general form
33.
Aim: What is a parabola?
SWBAT:
 State the definition of a parabola
 Write the standard for m of the equation of a parabola y2 =4px, x 2 =4py
 Write definitions of directrix, focus, vertex, axis of symmetry
(optl. Latus rectum and its endpoints)


Determine the directrix and focus for parabolas whose vertex is the origin and
sketch
Write the equation of a parabola whose vertex is the origin given its focus and
directrix.
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34.
Aim: How do we determine the focus and directrix of a parabola whose vertex is not the
origin?
SWBAT:
 Write the standard form of the equation of a parabola whose vertex is not the



2


origin x  h  4 p y  k and y  k





  4 p x  h
2
Determine the directrix and focus for parabolas whose vertex is not the origin
(h, k) and sketch
Write the equation of a parabola whose vertex is (h, k) given its focus and
directrix and sketch
Write the coordinates of the vertex, equation of the directrix and coordinates
of the focus for any parabola.
Show how a parabola with vertex not at the origin is a translation
Complete the square in order to determine the coordinates of the vertex of a
parabola in general form
35.
Aim: What is an ellipse?
SWBAT:
 Write the definition of an ellipse.
 Write the standard form of the equation of an ellipse (center at the origin).
x2 y2

1
a 2 b2
 Define vertex, foci, major and minor axis.
 State that the sum of the distances from any point on the ellipse to the foci is
equal to the distance between the vertices.
 Determine the foci and major and minor axis given its equation and sketch.
 Write the equation of an ellipse given its foci and length of major/minor axis.
36.
Aim: How do we determine the equation of an ellipse whose center is not at the origin?
SWBAT:
 Write the standard form of an equation of an ellipse whose center is not the
x  h  y  k   1
origin.
2



2
a2
b2
Determine the vertices, foci and eccentricity of ellipses whose center is not the
origin.
Complete the square where necessary (i.e. general form) in order to determine
the center.
Sketch ellipses whose center is not the origin by finding the center and length
of the axes.
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37.
Aim: What is a hyperbola?
SWBAT:
 Write the definition of a hyperbola.
 Write the standard form of the equation of a hyperbola whose center is at (0,
x2 y2
0). 2  2  1
a
b
 Write the definition of conjugate axis, transverse axis, asymptote.
 Write the relationship c 2  a 2  b2 for a hyperbola.
 Find the values of a, b, c, given an equation
 Determine the coordinates of the foci, the length of the transverse axis, the
length of the conjugate axis.
 Define eccentricity for the hyperbola and describe its meaning.
 Write equations of the asymptote.
 Sketch equilateral hyperbolas.
38.
Aim: How do we determine the equation of hyperbola whose center is not at the origin?
SWBAT:
 Write the standard form of the equation of a hyperbola whose center is (h, k).
x  h  y  k   1
2


2
a2
b2
Use completing the square to put an equation into standard form.
Write the equations of asymptotes to hyperbolas whose center is
(h, k).
39.
Aim: How do we evaluate and simplify expressions with rational exponents?
SWBAT:
 State the laws of exponents for rational numbers.
 Write the meaning of zero and negative exponents.
 Evaluate or simplify expressions with rational exponents.
40.
Aim: How do we solve equations involving exponents?
SWBAT:
 Define an exponential equation.
 Write a procedure to solve equations of the form:
m
a) a x  a y , b) x a  y a , c)

41.
xn  k
d) x a  y b where x and y can be expressed in the same base
Find the solution to equations of the forms listed above.
Aim: How can we describe the characteristics of the exponential function?
SWBAT:
 Define the exponential function.
 Graph the exponential function.
 State the characteristics of the exponential function.
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42. Aim: How can we solve word problems involving e ?
SWBAT:
 Define e by introducing compound interest
 Solve compound interest problems using e
 Solve other problems such as population growth
43. Aim: What is the inverse of the exponential function?
SWBAT:
 Write the inverse of the exponential function in log form.
 Graph the logarithmic function.
 State the characteristics of the logarithmic function graph.
 Convert from exponential form to logarithmic and vise versa.
 State the laws of logarithms.
 Simplify log expressions using these laws.
 Find the value of log expressions.
44. Aim: How can we solve log equations?
SWBAT:
 Describe how to change from log form to omit exponential form.
 Solve equations of the type: log b a  x , log x a  b , log b x  a , y  aloga b
 State the laws of logs.
 Apply these laws to simplify log equations and solve.
45. Aim: How can we express the log of a number in a different base?
SWBAT:
log a
 Show that log b a 
log b
 Express the log of any number in terms of a logarithmic expression in any
other base.
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