Final Exam Study Guide by Peng Sun

 

0.   ALWAYS CHECK YOUR ANSWER, ESPECIALLY AFTER YOU SOLVED AN EQUATION (FOR EXTRANEOUS SOLUTIONS)

1.     Basic Equations

a)       Involving fractional expressions (common denominator)

b)      Using radicals

2.     Modeling with equations (guideline Page 83)

3.     Quadratic Equations

a)       Completing the square

b)      Factoring

c)       Quadratic formula

d)      Discriminant

e)       Complex roots

4.     Complex numbers

a)       Arithmetic operations

b)      Square roots of negative numbers

c)       Complex conjugate

5.     Other types of equations

a)       Factoring, factoring by grouping

b)      Involving radicals (Isolate then square)

c)       Quadratic type (substitution)

6.     Inequalities

a)       Rules (Page 125, reverse direction when multiplied by negative)

b)      Nonlinear inequalities (Guideline Page 128)

c)       Modeling

7.     Absolute Value

a)       Equations

b)      Inequalities

8.     Coordinate plane

a)       Distance formula

b)      Midpoint formula

c)       Finding intercepts

d)      Circles

                        i.              Finding the equation

                      ii.              Finding the center and radius from equation

e)       Symmetry (Page 165)

9.     Lines

a)       Slope

b)      Point-slope form

c)       Slope-intercept form

d)      Vertical and horizontal lines

e)       General equation

f)        Rewrite equation in different forms (b, c, e)

g)       Parallel and perpendicular lines

h)       Slope as rate of change

10. Variation

a)       Direct variation

b)      Inverse variation

c)       Joint variation

11. Function

a)       Evaluating a function

b)      Domain and range

c)       Piecewise defined functions

d)      Graph

e)       Vertical line test

                        i.              Function or not

                      ii.              Domain

f)        Horizontal line test

                        i.              One-to-one or not

                      ii.              Range

g)       Equations that define functions

h)       Increasing and decreasing functions; Average rate of change

12. Transformation of functions

a)       Shifting

b)      Reflecting graphs

c)       Stretching and shrinking

d)      Even and odd functions

13. Quadratic functions

a)       Standard form (completing the square)

b)      Maximum or minimum value

14. Combining functions

a)       Domain of the combination

b)      Composition and the domain

c)       Recognizing a composition

15. Inverse functions

a)       One-to-one or not

b)      Property of inverse functions

c)       Finding the inverse (Page 283)

d)      Domain and range of the inverse

e)       Graphs of inverse functions

16. Polynomials

a)       Graphs; transformations

b)      End behavior (Page 313)

c)       Graphing (Guideline Page 316)

d)      Shape of the graph near a zero

e)       Dividing polynomials

                        i.              Long division

                      ii.              Synthetic division

                    iii.              Remainder and factor theorems

17. Zeros of polynomials

a)       Rational zeros

                        i.              Rational zero theorem

                      ii.              Finding rational zeros (Guideline Page 335)

b)      Number of real zeros: Descartes’ rule

c)       Complex zeros: Fundamental Theorem of Algebra; Complete factorization

d)      Zeros and their multiplicities

e)       Complex zeros come in conjugate pairs

f)        Finding polynomials with given zeros (and their multiplicities)

g)       Factoring into linear and quadratic factors with real coefficients

18. Rational functions

a)       Vertical and horizontal asymptotes

b)      Transformations of y=x-1

c)       Graphing (Guideline Page 362); behavior near vertical asymptotes

d)      Slant asymptotes and end behavior

19. Exponential functions

a)       Graphs (domain, range, intercepts, asymptotes, etc.)

b)      The natural exponential function

c)       Compound interest

                        i.              n times per year

                      ii.              Continuously compounded interest

20. Logarithmic functions

a)       Definition; Inverse of exponential function

b)      Change between exponential form and logarithmic form

c)       Graphs (domain, range, intercepts, asymptotes, etc.)

d)      Common log and nature log

e)       Laws of logs

                        i.              Expanding and combining logs

                      ii.              Change of base

21. Exponential and logarithmic equations

a)       Exponential equations (Isolate then take log)

b)      Logarithmic equations (Isolate then rewrite in exponential form)

c)       Other types (quadratic type, factoring, etc.)

d)      Compound interest; Annual Percentage Yield

22. Modeling with exponential and logarithmic functions (Section 5.5)

You are required to know the formula for exponential growth model. Other formulas from this section will be given if needed.

23. Sequences

a)       Finding the terms

b)      Recursively defined sequences

c)       Finding the partial sums

d)      Sigma notation (count the number of terms)

e)       Arithmetic sequences

                        i.              Definition; common difference

                      ii.              Formula for the nth term

                    iii.              Formula for the nth partial sum

f)        Geometric sequences

                        i.              Definition; common ratio

                      ii.              Formula for the nth term

                    iii.              Formula for the nth partial sum

                   iv.              Sum of infinite geometric series; writing a repeated decimal as a fraction