Final Review

Unit 1&2

100

Find the average rate of change of the function f(x)=x²−5x+4 over the interval [1,2].

Average rate of change =-2  

200

Given the function f(x)=5x²−6x+3, find and simplify the difference quotient.

10x+5h−6

300

Describe a function g(x) in terms of f(x) if the graph of g(x) is obtained by performing the following transformations:

1. Vertically stretching f(x) by factor of 6
2. Shifting the graph of f(x) to the left 7 units.
3. Shifting upward 4 units.

g(x)= 6f(x+7)+4  

400

Solve the equation. If there is no solution, type DNE.

x^2/5−x^1/5−2=0

x= −1,32

500

Simplify completely. Write your answer using positive exponents only:

(−6x^-15y^-6z²)(−x^-1z¹²)

6z¹⁴/x¹⁶y⁶

Unit 3

100

Find the quotient and remainder.

r⁴−2r³−37r+15/r−4

r³+2r²+8r−5  R= −5

200

Find all solutions to the equation.

x³−6x²+3x+20=0

x = 4,1+√6,1−√6

300

The polynomial of degree 5, P(x) has leading coefficient −4, has roots of multiplicity 2 at x=3 and x=0, and a root at x=−5.
Write a function for P(x) in factored form, and also write P(x) expanded in general form.

Factored form: P(x)= −4x²(x−3)²(x+5)

General (expanded) form: P(x)= −4x⁵+4x⁴+84x³−180x²

400

Write a polynomial with degree 4 that has a zero at x=7i, and a zero at x=−1 with a multiplicity of two, and the x2 coefficient is given to be −50. Write the function using only real values.

P(x)= −x⁴−2x³−50x²−98x−49

500

Given f(x)=(2/3)x+8, find the inverse.

f^-1(x)=3/2(x−8)  

Unit 4

100

Simplify completely:

√48+√27

7√3

200

Perform the indicated operation and leave the answer in factored form:
(−6q²+q+15/15q²+q−2)⋅(25q²−4/−9q²+18q−5)

(2q+3)(5q−2)/(3q−1)²

300

Simplify the expression:

(x/(x+2)(x+3))−(2/(x+2)(x+1))

x−3/(x+3)(x+1)

400

Find the domain of the function: 8x+7/x²+4x−60.

(−∞,−10)∪(−10,6)∪(6,∞)

500

Find the equation of the vertical asymptote and the equation of the slant asymptote of the rational function f(x)=24x²−6x+3/6x−3.

The equation of the vertical asymptote: x=1/2
The equation of the slant asymptote: y=4x+1

Unit 5

100

Write the logarithmic equation in exponential form. Do not simplify. 

1. log₁₆2=1/4  
   
   
2. log₂(1/8)=−3
   
   
3. log(c)=n  

1. 16^1/4=2
2. 2^-3=1/8
3. 10ⁿ=c

200

Simplify the following into a single logarithm with a coefficient of 1:
2log(9)+4log(x)

log(9²x⁴)

300

Write as a single logarithm.

4log₈(x)−5log₈(y)

log₈(x⁴/y⁵)

400

Given ln a=−2, ln b=3, and ln c=5, evaluate the following:

ln(a/b⁴c²)

500

Solve the equation. Determine if the solution(s) are extraneous.

log₂(−2x+6)=log₂(9x+1)

x= 5/11 not extraneous 

Unit 6

100

Solve the equation algebraically. If there are no solutions, enter DNE.

|4x+4|−5=5

x= 3/2

x= −7/2

200

Solve the equation algebraically. If there are no solutions, enter DNE.
∣2/5x+9∣−5=8

x= 10,−55 

300

Solve the system using any method. If there is exactly one solution, write it as an ordered pair. If not, choose one of the other options. 

{2x+4/3y=5  5/4x+2y=4

(2,3/4) 

400

Country Day's scholarship fund receives a gift of $120,000. The money is invested in stocks, bonds, and CDs. CDs pay 4.8% interest, bonds pay 3.2% interest, and stocks pay 7.2% interest. Country day invests $50,000 more in bonds than in CDs. If the annual income from the investments is $6,000 , how much was invested in each vehicle?

• Country Day invested $50,000   in stocks.
• Country Day invested $60,000   in bonds.
• Country Day invested $10,000   in CDs.  

500

Solve for x:

1/5x+1/2=4(5/6x−2)

x=255/94