HONORS REVIEW PACKET SEMESTER 1 : Advanced Algebra w/ Trig

CHAPTER 1 – Linear and Quadratic Equations

Solve for x.

1.) x

2

x

6

5

4=− 2.) ( ) 20x25x2x8 −=+−−

3.) 23x

2

12x

5

−

=

+

4.) 2x

3

4x

4

4x

x2

22+

+

−

=

−

5.) 2x2 + 5x + 3 = 0 6.) 49x42=

7.) 4x2 + 9 = 12x 8.) (2x + 3)2 = 9

9.) 2x2 – 2x + 5 = 0 10.) 8x4x2

=+

11.) 010xx32

=−+ 12.) 207x67x1022−=−

Perform the indicated operation.

13.) ( ) ( )i28i54 +−−+ 14.) (5 + 3i)(2 – i) 15.) ( )2i23 +

Solve each equation or inequality. For the inequalities, give answer in interval notation.

16.) 1053x =− 17.) 33x23

−=+

18.) 8xx42x23+=+ 19.) 2x – 5 < 3(x – 2)

20.) 123x5 <−≤− 21.) 4

3

3

1x

2

1<

+≤

22.) 534 <++x 23.) 243x ≥+

24.) 9785x =+−

25.) A bank loaned out $12,000, part of it at the rate of 8% per year and the rest at the rate of 18% per year. If the interest received totaled $1,000, how much of it was loaned at 8%? CALC. 26.) A coffee manufacturer wants to market a new blend of coffee that sells for $3.90 per pound by mixing two coffees that sell for $2.75 and $5 per pound respectively. What amounts of each coffee should be blended to obtain 100 lbs of the desired mixture? CALC. (round to whole number) 27.) A townhouse that was bought for $275,000. The value has increased in value by 8%. What is the value of the townhouse? CALC. 28.) An electrician charges $50 for a house visit and $75 an hour for the work that he completes. What is the total charge if the electrician works for 6.5 hours? CALC.

CHAPTER 3 – Functions

1.) Find the domain of each function.

a.) f(x) = x2 + 14x + 40 b.) f(x) = 16

23

2−

−

x

x

c.) f(x) = 3x +

2. Find the following for the functions f(x) = 2 – x and g(x) = 3x2 + 5

a.) g(-3) b.) f(-x) c.) (f + g) (5)

d.) (g – f)(-3) e.) (f ⋅g)(x) f.) -f(x)

g.) Find the average rate of change of g from -1 to 3.

3.) Determine (algebraically) whether the function f(x) = x3 – x – 2 is even, odd, or neither.

4.) The graph of f(x) = x is shifted to the right 4 units, stretched vertically by a factor of 3, reflected

over the x-axis and shifted up 5 units. Write the equation of the new function g(x).

5.) Graph using transformations for g(x) = (x + 1)3 – 3.

6. Use the graph of y = f(x) to the right to answer the following questions.

a.) Find the domain of f.

b.) Find the range of f.

c.) Determine f(-2).

d.) Solve for f(x) = 4.

e.) List the x-intercepts, if any exist.

f.) List the y-intercepts, if any exist.

g.) Find the zeros of f.

h.) Solve f(x) > 0.

i.) List the intervals where f is decreasing.

j.) List the intervals where f is increasing.

k.) List any local maximum value(s) and the x at which it occurs.

l.) List any local minimum value(s) and the x at which it occurs.

m.) List any absolute maximum value(s) and the x at which it occurs.

n.) List any absolute minimum value(s) and the x at which it occurs.

7.) Graph the function f(x) =

≥−

<+−

3x,5x3

2

3x,4x

8) The graph of a piecewise defined function is given. State the piecewise function.

CHAPTER 4 – Linear and Quadratic Functions

1.) Find the vertex and axis of symmetry of the quadratic function.

f(x) = x2 – 6x + 1

2.) Find the vertex and intervals where the function is increasing and decreasing.

86xxf(x)2

−+−= 3 .

3.) Determine whether the given function has maximum or minimum value. Also state the value.

48xxf(x)2

−+−=

4.) Find the quadratic function in vertex form with a vertex of (-1,2) and containing the point (3,6).

5.) Write the equation of this quadratic function in vertex form.

6.) Graph the quadratic function ( ) 9++−=2

2xf(x) using transformations. Plot 5 points.

Vertex: __________ Axis of Symmetry: __________

Domain: __________ Range: __________

7.) Graph the quadratic function f(x) = x2 – 2x – 3 using the vertex and intercepts. Plot 5 points.

vertex: ___________ y-intercept: _____________

x-intercept(s): _____________

8.) Solve each quadratic inequality. Put answer in interval notation.

a.) 10x2x2

+< b.) 0166xx2

≥−+

9.) CALC. A projectile is thrown upward so the height H in feet after t seconds is represented by the

function t100t16)t(H2+−= . In how many seconds will the object be at the maximum height?

10.) CALC. The height h in feet of an object after t seconds is represented by 10t24t16)t(h2

++−= .

Find the maximum height of the object.

11.) An engineer collects data showing the speed x of a given car model & its average miles per gallon M.

CALC. Determine whether a linear or quadratic model best fits the data.

a.) Make a scatter diagram on your calculator.

b.) Find this model. (Round to 4 decimal places)

Speed, x Miles per gallon

M

20 18

30 20

40 23

50 25

60 26

70 24

80 22

CHAPTER 5 – Polynomial and Rational Functions

Write a polynomial function of degree 3 in standard form with the given zeros.

1. Zeros: 1 (mult. 2), 4 2. Zeros: 5, 2i

Graph the polynomial function.

3. 2

3)1)(x(xf(x) −+= 4 4. 2

2)3)(x(xf(x) −+−= x

Real Zeros: Real Zeros:

Cross / Touch x-axis: Cross/Touch x-axis:

Y-Int: Y-Int:

End Behaviors: End Behaviors:

5. Graph the rational function 10x3x

4)x(R

2−−

=

VA: HA:

X-Int: Y-Int:

Domain: Range:

Find the domain and the horizontal asymptote of each rational function.

6. 7x3

5x2)x(R

2

+

−= 7.

145xx

2xG(x)

2

2

−+

=

Domain: Domain:

HA: HA:

8. Find the x-intercepts and y-intercept of the graph of the function 187xx

109xxf(x)

2

2

−+

−−= .

9. Find the vertical and horizontal asymptotes of the functions

a.) 2410xx

5xR(x)

2

2

−+

−= b.)

9x

2xR(x)

2−

+= .

10. Find all the zeros of the function 1816x7xxf(x)23

+−+= given that one zero is 9− .

11. Find all the zeros of the function 1223x3x2xf(x)23

+−−= given that one zero is 3− .

12. CALC. Find all the zeros of the function 6x10x7x)x(f23

−++= .

13. CALC. Find all zeros and linear factors of the function 53xx3x2xf(x)234

−+−+= 3 .

CHAPTER 6 – Exponential and Logarithmic Functions

1. For the functions f(x) = 3x – 5 and g(x) = 1 – 2x2, find the following:

a.) g(f(-2)) b.) f(g(2)) c.) f(g(x)) d.) g(f(x))

2. Is the function {(-1,2), (-3,4), (4,2), (5, -3)} one-to-one? Explain your answer.

3. Find the inverse of each one-to-one function.

a.) f(x) = 3x – 5 b.) g(x) = x3

x2

+

−

4. Change into a log expression. a) 1512x

= b) 19eM =

5. Change into an exponential expression. a) x375log4

= b) 34log9n

=

6. Write as a single log. ylogylog3xlog2 +−

7. Write as the sum and/or difference of logs.

34

y

xlog

8. Solve. a) x2

3x

82

1=

+

b) 4xlog3

= c) ( ) 1x3xlog 2

10=− d) 16025

3x=⋅

−