Review for Final Exam, Math 103A (B. Olson)

The first page of the final exam is to be done without a graphing calculator.

Chapter 3: Polynomial functions.

A.  Be able to find zeros of polynomial functions using Synthetic division, the Remainder Theorem, the Rational Zero Theorem, Conjugate Theorem, the Fundamental Theorem of Algebra, and the Intermediate Value Theorem. 

B.  Know the relationship between zeros, roots, x-intercepts, and factors of a polynomial.

C.  Be able to graph a function with given information.

D.  Be able to write a polynomial function as a product of linear factors with given information.

E.  Be able to construct a function and solve a word problem with the function. (2.7 & 3.3)

F.  Be able to solve inequalities algebraically and graphically.  (Section 3.6)

Sections 3.4, 3.5, Rational functions.

A.  Be able to find the domain of a rational function.

B.  Be able to find the vertical asymptote(s) of a rational function.

C.  Be able to find the horizontal asymptote of a rational function.

D. Know when there will be an oblique asymptote of a rational function and find it.

E.  Be able to find the x- and y-intercepts of a rational function.

F.  Be able to graph a rational function from information in A, B, C, D, and E.

1. 

            a)  What is the domain of f(x)?

            b)  What are the vertical asymptote(s) of f(x) if any?

            c)  Is there a horizontal asymptote or slant asymptote?

            d)  Find the asymptote in c).

            e)  What are the x-intercept(s)?  What is the y-intercept?

            f)  What is the end behavior of f(x)?

            g)  What is the behavior of f(x) near the vertical asymptote(s)?

h)  Graph f(x).

2.  Find the oblique asymptote of 

H.  Be able to derive a rational function from a graph or information given.

PREVIOUS SECTIONS:

A.   Linear Equations. (Section 2.2)

Be able to write the equation for a linear word problem and solve it.  (Exam 1)

B.  Functions. Chapter 2

Know how to find the difference quotient. (2.3)

Understand the relationship between the difference quotient and the slope of a secant line. 

Be able to graph a step function or determine the function from its graph.  (Section 2.4)

Know how to find domain and range of a function.  (Section 2.3)

Operations of functions, Composite functions and their domain.  (Section 2.6)

Know Odd and Even function characteristics and use it to answer questions. (Section 2.4)

D.  Quadratic Functions (Section 3.1)

Know how to find max and min for quadratic equations.

3.  Put into standard form by completing the square.   F(x) = 3x² - 9x - 5

Be able to graph an equation given information.

4.  f(x) is an EVEN function with a positive leading coefficient of degree 4.  Two of its zeros are 4 and -2.  The y-intercept is 64/7 and the function minimum is f(3) = -5.  Graph the function.  Find the function f(x).

Textbook Review:

1.  Read Chapter 3 Review on pages 264 – 266.

2.  Do Chapter 3 Review Exercise on pages 267 – 269, # 29 – 106.  Do SOME problems from each section.

3.  Go over Review Guide for Exam 1 and 2 that cover sections above.  Go over Quizzes that cover the problems in the above sections.

Solutions:

1. a)  D = (-     b)  x = -3         c)  Horizontal asymptote

d)  y = 1           e)  x-int. (-2, 0), y-int. (0, 2/3) 

f)  as x  f(x) . as x  f(x) .

g) as x  f(x) . as x  f(x) .

h)

2.  y = 2x + 2

3. 

4.  f(x) = (x – 4)(x + 2)(x + 4)(x – 2)