1)    Definitions of the real numbers

a)     Natural numbers

b)    Whole numbers

c)     Integers

d)    Rational numbers

e)     Irrational numbers

f)     Real number system

2)    Order of operations

3)    Scientific notation

a)     Decimal notation to Scientific Notation

b)    Scientific notation to Decimal Notation

Problems to try from 1.1: {9,10,59,60,66,73,75,78,82,85,90}

1)    Mean, median and mode, range

2)    The initial concept of Domain and Range

3)    Distance formula:

4)    Midpoint formula

5)    Circles: where (h, k) is the center and r is the radius.

a)     Standard form and Graphing

6)    One and Two variable data

Problems to try from 1.2: {6,13,23,29,33,41,63,70,71,73,77}

1)    Definition of a function.

2)    More precise definition of Domain

3)    What conditions are required for a relation to be a function

a)     Vertical line test

b)    Every element in the domain is paired with one and only one element in the range

4)    The “four” ways to describe a function

a)     Verbal

b)    Symbolic

c)     Numerical

d)    Graphical

Problems to try from 1.3: {1,3,10,12,21,24,26,31,33,35,39,41,45,61,66,67,79,80,83,85}

1)    Slope of a line

a)     Positive slope

b)    Negative slope

c)     Zero slope

2)    Functions

a)     Constant function

b)    Linear function

c)     Nonlinear function

3)    Interval Notation

4)    Increasing and decreasing

5)    Average rate of change from a graph

6)    Average rate of change from a function

7)    Difference quotient, example 1

8)    Difference quotient, example 2

Problems to try from 1.4: {4,10,15,147,19,21,23,31,33,48,49,53,57,73,77,81}

1)    Models

2)    Linear models

3)    Graphs of linear functions

4)    Least-squares regression

5)    Correlation coefficient r

Problems to try from 2.1: {3,13,15,25,30,33,37,45-48,69,73,75,77,79,87,89}

1)    Point-slope form of the equation of a line: or

2)    Slope-intercept form of the equation of a line:

3)    Direct variation:

4)    Horizontal line: , where b is a constant

5)    Vertical line: , where a is a constant

6)    Parallel or perpendicular lines

Problems to try from 2.2: {5,11,13,21,27,33,37,39,45,47,49,51,57,69,82,84,89,97,101,109}

1)    Linear equations

2)    Properties

a)     Addition property

b)    Multiplication property

c)     Distributive property

d)    Identity, contradiction and conditional

e)     Percentages

3)    Solving Equations

a)     Symbolic (or Algebraic) Method

b)    Graphical Method

c)     Numerical Method

Problems to try from 2.3: {9,11,23,31,33,369,41,44,55,57,65,77,84,86,89,96,100,105,108,113,120,125}

1)    Interval notation

2)    Compound inequality

3) Graphs of Piecewise-defined functions

4) Evaluating Piecewise-defined functions

5)    Solving inequalities

a)     Symbolic (or algebraic) method

b)    Intersection-of-graphs method

c)     The x-intercept method

d)    Numerical method

Problems to try from 2.4: {5,6,11,15,23,3147,49,86,93}

1)    Absolute value function

2)    Absolute value equations

a)     Symbolic (or algebraic) method

b)    Intersection-of-graphs method

c)     The x-intercept method

d)    Numerical method

 

3)    Absolute value inequalities

a)     Symbolic (or algebraic) method

b)    Intersection-of-graphs method

c)     The x-intercept method

d)    Numerical method

Problems to try from 2.5: {11,15,19,21,31,33,39,43,47,61,67}

1)    Quadratic functions

2)    Parabola, general form

3)    Completing the square to find the vertex (or standard) form

4)    Vertex formula

5)    Vertex form or standard form

Problems to try from 3.1: {7,9,15,17,21,25,29,35,49,59,67,83,87,91,107}

1)    Quadratic Equations

a)     Solve by factoring

b)    Solve using the square root property

c)     Solve by completing the square

d)    Solve using the quadratic formula

e)     Solve by graphing

f)     Solve numerically

g)     The discriminant

Problems to try from 3.2: {5,11,13,17,21,27,31,37,53,55,61,63,65,69,89,91,97,101,105,118}

1)    The imaginary unit ,

2)    The expression with a > 0, simplifies to

3)    Standard form of a complex number a + bi, where a and b are real numbers

4)    Complex conjugates

5)    Arithmetic operations on complex numbers

6)    Complex solutions to equations

Problems to try from 3.3: {9,19,25,31,39,41,47,55,63,67,73,75,77}

1)    Quadratic inequality

a)     Solve graphically

b)    Solve symbolically or algebraically

Problems to try from 3.4: {1,7,11,21,37,39,47,53}

1)    Vertical translations

If c > 0

a)     y  =  f (x) + c shifts f (x) up  c units

b)    y  =  f (x) - c shifts f (x) down  c units

2)    Horizontal translations

If c > 0

a)     y  =  f (x + c) shifts f (x) left  c units

b)    y  =  f (x – c) shifts f (x) right  c units

3)    Vertical stretch or shrink

If c > 0

a)     y = cf (x) is stretched vertically if c > 1

b)    y = cf (x) is shrunk vertically if c < 1

4)    Horizontal stretch or shrink

If c > 0

a)     y = f (cx) is stretched horizontally if c < 1

b)    y = f (cx) is shrunk horizontally if c > 1

5)    Reflections

a)     y = - f (x) is a reflection of  y =  f (x) across the x – axis

b)    y =  f (-x) is a reflection of  y =  f (x) across the y – axis

6)    Finding the equation from a transformed graph.

Problems to try from 3.5: {1-8,11,15,17,19,27,35,41,47,51,55,75}

1)    Absolute (or global) maximum or minimum

2)    Relative (or local) maximum or minimum

3)    Even function f ( - x ) =  f ( x )

4)    Odd function f ( - x ) =  - f ( x )

Examples:

a)     Determining if a function is even odd or neither.

b)    Determining if a function is even odd or neither.

Problems to try from 4.1: {11,20,27,31,33,41,47,49,51,57,65,69,73,77,93,97}

1)    Polynomials or varying degree

Degree = n , at most  n  x – intercepts, and at most n – 1 turning points.

(For the case of degree 1, there are no turning points, and for degree 2, there is exactly one turning point)

Problems to try from 4.2: {3,5,7,9,15,17,19,25,33,37,39,45,49,53,59,69,71,75,77,81,87}

1)    Division by a monomial. Each term in the numerator is divided by the monomial denominator

2)    Division by a polynomial. This process is similar to that of long division of natural numbers or integers

3)    Division algorithm: (Dividend) = (Divisor)(Quotient) + (Remainder)

4)    Synthetic division. This method works only when the denominator is of the form x – k

5)    Remainder Theorem: If a polynomial is divided by x – k, the remainder is f ( k )

Problems to try from 4.3: {1,7,9,19,21,25,29,31,39,45,47,49,51}

1)    Factor theorem

2)    x- intercepts, zeros and factors

The following are equivalent:

a)     The graph of f has x – intercept k

b)    A real zero of f  is k. That is,  f ( k ) = 0

c)     A factor of f is (x – k)

3)    Complete factored form: , where the are zeros of f, with a distinct zero listed as many times as its multiplicity. This form is unique.

4)    Real zeros:

a)     The graph of y = f ( x ) crosses the x – axis at each real zero of odd multiplicity.

b)    The graph of y = f ( x ) intersects, but does not cross, (touches) the x – axis at each real zero of even multiplicity.

5)    Factoring a polynomial graphically with only real zeros. Graph y =  f (x) and locate all of the zeros (or x – intercepts). If the leading coefficient is a and the zeros are , then

6)    Solving polynomial equations

a)     Symbolically

b)    Graphically

c)     Numerically

7)    Rational zero test

Let , where , represent a polynomial function f with integer coefficients. If is a rational number in lowest terms and if is a zero of f, then p is a factor of the constant term and q is a factor of the leading coefficient .

8)    Descartes’ Rule of Signs

Let P (x) define a polynomial function with real coefficients and a nonzero constant term, with terms in descending power of x.

a)     The number of positive real zeros either equals the number of variations is sign occurring in the coefficients of P (x) or is less than the number of variations by a positive even integer.

b)    The number of negative real zeros either equals the number of variations is sign occurring in the coefficients of P (- x) or is less than the number of variations by a positive even integer.

9)    Intermediate Value Theorem

Let and , with  and , be two points on the graph of a continuous function f. Then on the interval , f assumes every value between  at least once.

Problems to try from 4.4: {1,3,5,9,13,17,19,23,25,31,35,37,43,45,47,49,51,57,65,69,73,81,95,97,103,105,110,115}

1)    Number of zeros theorem: A polynomial of degree n has at most n distinct zeros. These zeros can be real or imaginary.

2)    Fundamental theorem of algebra: A polynomial of degree n, with n > 1, has at least one complex zero.

3)    Conjugate zeros: If a polynomial has real coefficients and a + bi is a zero, then a – bi is also a zero.

Problems to try from 4.5: {1,5,7,9,17,21,25,31,33,39,43,45,47}

1)    Rational functions: , where p (x) and q (x) are polynomials

2)    Asymptotes:

a)     Vertical

b)    Horizontal

c)     Oblique

3)    Graphs of Rational functions

4)    Rational equations

Problems to try from 4.6: {13,15,17,23,27,29,33,37,39,41,45,47,55,73,77,91}

1)    Rational equations

2)    Variation

a)     Direct: We say y varies directly as the n^th power of x and we write:

b)    Inverse: We say y varies inversely as the n^th power of x and we write:

3)    Polynomial Inequality

a)     Solving graphically

b)    Solving symbolically

c)     Solving numerically

4)    Rational Inequality

a)     Solving graphically

b)    Solving symbolically

c)     Solving numerically

Problems to try from 4.7: {1,7,11,13,16,17,19,25,29,33,35,41,43,53,57,61,65,67,71,75,83,95,97}

1)    Rational exponents

2)    Radical notation

3)    Solving radical equations and those with rational exponents:

a)     Solving graphically

b)    Solving symbolically

c)     Solving numerically

d)    In quadratic form

4)    Power function

5)    Root function

Problems to try from 4.8: {1,13,17,19,23,27,33,39,42,47,51,61,65,67,71,75,79,81}

1)    Combining functions

a)     Sum

b)    Difference

c)     Product

d)    Quotient

e)     Composition

Problems to try from 5.1: {7,15,19,31,33,41,45,53,63,67,73,75,81,89,91}

1)    One-to-one functions

2)    Inverse functions

a)     Verbal representation

b)    Symbolic representation

c)     Numerical representation

d)    Graphical representation

3)    Domains and Ranges of functions and their inverses: Example of restricting the domain of a function.

Problems to try from 5.2: {13,15,17,23,25,31,33,35,41,50,57,59,63,67,91,93,95,103,107,109}

1)    Exponential functions , where a > 0, and C > 0

2)    Graphs of exponential functions

3)    Linear growth vs. exponential growth

4)    Compound interest

a)     Compounded n times per year:

b)    Compounded continuously:

5)    Natural number e and the natural exponential function

6)    Radioactive decay: , where k is the number of years in the half-life of an initial quantity of C units. A will be the amount after x years.

Problems to try from 5.3: {3,7,9,13,27,31,53,55,59,61,63,65,67,79,84,94}

1)    Logarithmic functions , where a > 0,

2)    Graphs of logarithmic functions

3)    Inverse properties of logarithms and their relationship to exponential functions

4)    Exponential equations

5)    Logarithmic equations

6)    Converting from logarithmic to exponential and vice versa

Problems to try from 5.4: {1-12,21-49,53,63,67,71,75,77,81,93,95,101,103,113,115,121,129}

1)    Properties of logarithms

2)    Change of base formula

3)    Graphing logarithmic functions

Problems to try from 5.5: {11,12,27,29,33,45,47,48,55,57,61,67,73,75,77,79,89,91}

1)    Exponential equations

2)    Logarithmic equations

a)     Solve symbolically

b)    Solve graphically

Problems to try from 5.6: {5,9,17,23,25,27,35,47,55,58,61,65,69,72,91,96}

1)    Exponential regression model

2)    Logarithmic regression model

3)    Logistic regression model

Problems to try from 5.7: {5,6,9,12,18,24}

1)    Functions of two variables

2)    Linear system of two variables and two equations

a)     Solve by substitution

b)    Solve by elimination

c)     Solve graphically

3)    Nonlinear systems

a)     Solve by substitution

b)    Solve by elimination

c)     Solve graphically

4)    Consistent system

5)    Dependent system

6)    Inconsistent system

Problems to try from 6.1: {5,11,12,13,17,21,25,31,33,35,39,43,51,55,57,69,71,75,79,93,101,107,109,119,126,129,139}

1)    Systems of linear inequalities in two variables

2)    Linear programming

Problems to try from 6.2: {11,15,19,21,25,31,33,41,45,47,51,55}

1)    Linear systems of 3 or more variables and equations

2)    Dependent system

Problems to try from 6.3: {5,9,15,17,29,31,35,36,37}

1)    Using matrices to solve linear systems of equations

2)    Augmented matrix

3)    Row echelon form vs. Reduced row echelon form

Problems to try from 6.4: {7,9,17,21,23,27,31,35,49,53,61,65,67}

1)    Ellipse with center at the origin

a)     Foci

b)    Vertices

c)     Major axis

d)    Minor axis

2)    Ellipse with center at the point (h, k)

a) Complete the square to write the equation of an ellipse in standard form.

b)     Foci

c)    Center

d)     Vertices

e)    Major axis

f)     Minor axis

 

Problems to try from 7.2: {1,21,25,33,38,47,49,53,57}

1)    Hyperbola with center at the origin

a) Complete the square to write the equation of a hyperbola in standard form.

b)     Foci

c)    Vertices

d)     Transverse axis (vertical vs. horizontal)

e)    Conjugate axis

f)     Asymptotes

 

2)    Hyperbola with center at the point (h, k)

a)     Foci

b)    Center

c)     Vertices

d)    Transverse axis (vertical vs. horizontal)

e)     Conjugate axis

f)     Asymptotes

 

 

Problems to try from 7.3: {1,7,17,21,23,33,39,43,47,53}

1)    Infinite sequence

2)    Recursive sequence

3)    Arithmetic sequence

a)     General term of an arithmetic sequence: or

4)    Geometric sequence

a)     General term of a geometric sequence:  or

Problems to try from 8.1: {1,9,15,23,27,29,35,41,47,51,57}

1)    Series

2)    Arithmetic Series

a)     Sum of a finite arithmetic series: or

3)    Geometric Series

a)     Sum of a finite geometric series: or

b)    Sum of an infinite geometric series: ,

4)    Summation notation

Problems to try from 8.2: {7,13,15,20,25,33,35,41,51,61,67,69,85}