9 Pre calculus questions

TEST #2: CHAPTERS 4-5

(POLYNOMIALS, POWER FUNCTIONSEXPONENTIAL AND LOG FUNCTIONS)

ORGANIZATION OF TEST

Part I: Multiple choice [10 @ 3 ] 30 points

Part II: Open ended answers: (6 questions) 64 points

Part III: Explain questions: 6 points 6 points

Please review all of your online homework assignments for chapters 4 & 5, and the quizzes for

those chapters. Here is a list of topics by name to review:

1. Recognize and create Graphs of Polynomial functions: [Q4 #1, 16

a. Complex roots (How many Real roots, how many imaginary?)

b. Odd/even functions (note that symmetry also feeds into odd/even functions)

c. Positive and negative exponents

d. Fundamental theorem of Algebra (min number of roots) [Q4 #4, 5, 6,7]

e. Finding roots of polynomial functions [Q4 #9, 10. 11, 12, ]

2. Recognize and create Graphs of Polynomial functions graphs of rational functions

a. Asymptotes (Horiz & Vertical)

b. Graphing behavior at end points and intercepts [Q4 #13, 14, 15]

c. Dividing polynomials and using synthetic division [Q4 #17]

3. Recognize and create Graphs of Polynomial functions graphs of Power functions and

exponential functions

a. Know and recognize graphs of ( ) ( ) and transformations of

these

4. Inverse functions and compositions of functions

a. Why do the bases “cancel out”?

b. Given ( ) find ( ) graphically and algebraically (Q5, 13, 14,16,18

c. Understand the notation of compositions of functions in algebraic and graph

form(Q5 #12,

5. Simplifying

a. Powers

b. Logs [Q5, #7,10, HW 5.6 #13,

6. Solving of equations

a. Power functions (Lab 7, Q 5 #17, Q4 #20

b. Log functions (Quiz 5 #11,15, HW 5.6 #19]

c. Exponential functions [HW 5.6 #16, 17, 18, 20]

d. Applications of exponentials and their inverses [HW 5.6 #14]

CONCEPTUAL QUESTIONS FOR CHAPTERS 4 AND 5

Chapter 4 Questions

1. If an odd function f has one local maximum of 5 at x = 3, then what else can be said about f?

Explain.

2. Explain how to determine graphically whether a zero of a polynomial is a multiple zero. Sketch

examples.

3. Show that the following equation has no rational roots. Explain how you did it. 5 4 3 2

2 3 0x x x x x     

4. Could a cubic function with real coefficients have only imaginary zeros? Explain

5. Give an example of a polynomial function that has only imaginary zeros and a polynomial

function that has only real zeros. Explain how to determine graphically if a function has only

imaginary zeros.

6. Suppose that  f x is a polynomial of degree 3 whose coefficients are real numbers; further

suppose that its zeros are 2, ,i and 3 .i Is this possible?? Why or why not??

7. Name all the “tools” at your disposal that play a role in the graphing of rational functions.

Which tools are indispensable and always used? Which are used only as the situation merits?

8. Consider the definition of a power function. Can a function be both a polynomial function and a

power function? Explain

Chapter 5 Questions

9. Describe differences between   fg x and   .f g x Give an example. [Q5, #1]

10. For   2 7f x x  and   2

1 g x

x 

 , discuss/explain how the domain of     h x f g x

is determined. In particular, why is  1h not defined even though  1 3f  ?

11. Can a one-to-one function have more than one x-intercept or more than one y-intercept?

Explain.

12. Let  f x compute the height in feet of a rocket after x seconds of upward flight. Explain what

 1f x computes.

13. Discuss/explain the statement, “For 0k  , the y-intercept of x

y ab k  is  0, a k .

14. Explain how linear and exponential functions differ. Give examples.

15. Describe the relationship among exponential and logarithmic functions. Explain why

logarithms are needed to solve exponential equations.

16. Explain how the graph of  log 3bY x  can be obtained from logby x . Where is the

“new” x-intercept? Where is the new asymptote?

17. A student insists that log x

y

     

and log

log

x

y are equal. How could you convince the student

otherwise? [begin with HW 5.6 #13, then think of a counter example]