Final Exam Review Spring 2012-2013 (Rennell) Name _____________________________________
For numbers 1-4, find the roots, vertex and axis of symmetry for the following:
1. g(x) = (2x – 3)(x + 4) 2. f(x) = -2(5x + 3)(6 x –5)
3. 4. y y
5
4
3
5
4
3
2 2
(–2, 0)
–7 –6 –5 –4 –3 –2
1
(6, 0)
1 2 3 4 5 6 7 x
(–5, 0)
–5
1
(–1, 0)
–4 –3 –2 –1
–1
1 2 3 4 5 x
–2
–3
–2
–3
–4
(0, –4)
–4
–5 –5 (0, –5)
For numbers 5 – 10, the quadratic functions are given in standard form, find the following: a.
Write in factored form. b.
State the zeros of the function. c.
Find the y-intercept.
5. f ( x )

81 x
2 
64 6.
g ( x )

3 x
2 
3 x

6 7. h ( x )
  x
2 
8 x

15
8. f ( x )
 
6 x
2 
27 x

15 9. 20

2 x
2 
11 x

20 10. 18

2 x
2  x

10
11. Given the function f ( x )

2 ( x

3 )
2 
2 , a. state the vertex
b. write the function in standard form
c. find the y-intercept.
12. Given the function k ( x )

3 x 2 
12 x

7 , a. find the vertex form b. state the vertex

For questions 13-14, use the quadratic formula to find the roots. State the answers in decimal form and round to the nearest hundredth.
13. h ( x )
 
9 x
2 
24 x

16 14. f ( x )

5 x
2 
13 x

7
Rewrite the following expressions as in imaginary number.
15.

99 16.
 
24
Rewrite each quadratic function with imaginary numbers as zeros in standard form.
17. f
 

( x

2 i )( x

3 i ) 18. g ( x )

3 ( x

4 i )( x

4 i )
For numbers 28-30, find the zeros of the following quadratics. Write the imaginary roots as complex numbers in standard form.
19. f ( x )

5 x
2 
3 x

1 20. g ( x )

4 x
2 
53 21. h ( x )
 
2 x
2 
3 x

2
1. Use the Leading Coefficient Test to determine the end behavior of the graph of the given polynomial function. a. f(x) = -x
4
+ x
2 b. f(x) = x
3
– 4x
2 c. f(x) = (x – 3)
2 d. f(x) = -x
3
– x
2
+ 5x – 3
2. Find the zeros for each polynomial function. a. f(x) = 3(x + 5)(x + 2)
2 b. f(x) = 4(x – 3)(x+ 6)
3
3. For each function, i. Use the Leading Coefficient Test to determine the graph’s end behavior. ii. Find the x-intercepts. iii. Find the y-intercept. iv. Find a few additional points and graph, the function. a. f(x) = x
3
+ 2x
2
– x – 2 b. f(x) = x
4
– 2x
3
+ x
2

4. Divide using long division. a. (6x
3
+ 7x
2
+ 12x – 5) ÷ (3x – 1) b.
4𝑥
4
−4𝑥
2
+6 𝑥−4
5. Divide using synthetic division. a. (x
2
– 5x – 5x
3
+ x
4 ) ÷ (5 + x) b.
𝑥
5
+𝑥
3 𝑥−1
−2
6. Use synthetic division and the Remainder Theorem to find the indicated function value. f(x) = 2x 3 – 11x 2 + 7x – 5; f(4)
7. Solve the equation 2x
3
– 5x
2
+ x + 2 = 0 given that 2 is a zero of f(x) = 2x
3
– 5x
2
+ x + 2.
8. Use the Rational Zero Theorem to list all possible rational zeros for each given function. a. f(x) = 3x 4 – 11x 3 – x 2 + 19x + 6 b. f(x) = x 5 – x 4 – 7x 3 + 7x 2 – 12x – 12
9. Given f(x) = 2x
3
– 3x
2
– 11x + 6 a. List all possible rational zeros. b. Use synthetic division to test the possible rational zeros and find an actual zero. c. Use the quotient from part (b) to find the remaining zeros of the polynomial function.
10. Given f(x) = 6x
3
+ 25x
2
– 24x + 5 = 0 a. List all possible rational roots. b. Use synthetic division to test the possible rational roots and find an actual root. c. Use the quotient from part (b) to find the remaining roots of the polynomial function.

11. Find the domain of each rational function. a. 𝑔(𝑥) =
3𝑥
2
(𝑥−5)(𝑥+4) b. ℎ(𝑥) = 𝑥+7 𝑥 2
−49
12. Find the vertical asymptotes, if any, of the graph of each rational function. a. 𝑔(𝑥) = 𝑥+3 𝑥(𝑥+4) b. ℎ(𝑥) = 𝑥 𝑥(𝑥+4)
13. Find the horizontal asymptotes, if any, of the graph of each rational function. a. 𝑓(𝑥) =
12𝑥
3𝑥 2 +1 b. 𝑔(𝑥) =
12𝑥
3𝑥 2
2
+1 c.
ℎ(𝑥) =
12𝑥
3𝑥 2
3
+1
14. Follow the seven steps to graph the rational function. f ( x )
 x
4 x

2
15. Solve each polynomial inequality and graph the solution set on a real number line. Express each solution set in interval notation. a. (x – 4)(x + 2 )> 0 b. 3 x
2
+ 10x – 8

0 c. x
3 
3 x
2 
9 x

27

0
16. Solve each rational inequality and graph the solution set on a real number line. Express each solution set in interval notation. a. x x

4

3

0 b.
 x x


4
2

0
1.
Bigtown has an initial population of 5000 people and grows by 12% every year. Write a function to describe the amount of people in the town after any amount of years.
2.
Given f
 

34

1 .
082
 x find: a.
The growth factor b.
The percent change c.
The initial value
3.
Simplify

6 a
2 b

2 c
4
3
 
5 a

1 b
6 c
2

2
.

4.
Given that the table shows exponential decay, find the following information. x f(x)
1
51.94
2
27.5282 a.
What is the decay factor? b.
What is the percent change? c.
What is the initial value? d.
Write a function to describe this relationship.
3
14.5899
5.
A certain bacteria grows 12 % every 8 minutes. Initially, there was 30 micrograms of this bacteria. a.
What is the 8 minute growth factor? b.
Write a function to determine the amount of bacteria after n 8 minute intervals. c.
What is the 1 minute growth factor? d.
Write a function to determine the amount of bacteria after t minutes. e.
What will the mass of the bacteria be after 35 minutes? f.
When will the bacteria’s mass be 400 micrograms?
6.
Given the equation h
  
943
  t
4 , where t is in days, answer the following questions.
7.
Write a function for a bank account that has an advertised 4.3% APR, compounded quarterly, if $300 is invested initially. Leave your answer in terms of t. a.
4 day growth factor:_____________ b.
1 day growth factor:_____________ c.
1 day percent change:____________
8.
For the above problem, how much money will the account have after 8 years?
9.
Write a function for a CD in each situation, if $500 is invested initially. a.
4.6% APR compounded quarterly. b.
4.5% APR compounded semiannually. c.
4.3% APR compounded monthly.
10.
Determine how much money you would have in each account after 20 years.
11.
How much money would you need to invest into an account offering 3% APR compounded monthly if you wanted to have $10,000 after 10 years?
12.
Sasha invested $450 in an account that offered 3.76% APR compounded quarterly. a.
What is the quarterly growth factor? b.
What is the quarterly percent change? c.
What is the annual growth factor? d.
What is the annual percent change?

13.
Write each in logarithmic form. a.
2 3 = 8 b.
𝑚 𝑘 c.
8
3
1
= 3
= 2
14.
Write each in exponential form. a.
𝑙𝑜𝑔
2
64 = 6 b.
𝑙𝑜𝑔
7 𝑎 = ℎ c.
𝑙𝑜𝑔 𝑓 𝑒 = 𝑚
15.
Condense and simplify. No calculator. a.
𝑙𝑜𝑔
4
12 − 𝑙𝑜𝑔
4
3 b.
log 3 − 𝑙𝑜𝑔300 c.
2𝑙𝑜𝑔
6
12 − 𝑙𝑜𝑔
6
4
16.
Rewrite each of the following as sums and differences of a logarithm of some number. a.
𝑙𝑜𝑔
3
(8𝑥𝑦𝑧) b.
𝑙𝑜𝑔
7
( 𝑎𝑏 𝑑 3
)
17.
Solve the following, and round to three decimal places. a.
𝑙𝑜𝑔
9
54 = 𝑥 b.
𝑙𝑜𝑔
6
(−8) = 𝑥
18.
Solve the equation 2100

700

1
.
035

12 x
12
, rounding your answer to three decimal places.
19.
Adam invests $3000 at an interest rate of 2.4% compounded monthly, while Brenda invests $3500 at an interest rate of 2.5% compounded semiannually. Who will have more money after 15 years? How much money will each person have after 15 years?