ACCUPLACER REVIEW*

PERRY HIGH SCHOOL APRIL 2015

* The information in this packet is based on the practice tests available at: http://www.riosalado.edu/testing/placement/Pages/Math-Practice-Placement.aspx

1 (http://www.learnnc.org/reference/number+sense) 2 http://dl.uncw.edu/digilib/mathematics/algebra/mat111hb/pandr/quadratic/quadratic.html

3 https://partiallyderivative.wordpress.com/2013/12/04/function-transformations-card-matching-activity/ https://secure-media.collegeboard.org/digitalServices/pdf/accuplacer/accuplacer-texas-success-initiative-assessment-sample-questions.pdf

ANOTHER REVIEW CAN BE FOUND AT: http://www.ccm.edu/admissions/placementTesting/cal/mathreviewsheets.aspx

THINGS TO REMEMBER: http://regentsprep.org/regents/math/algtrig/formulasheetalgebra2trig.pdf

NUMBER SENSE: An intuitive understanding of numbers, their magnitude, relationships, and how they are affected by operations. 1 Write < or > between the pair of numbers to make the statement true. -43, -41

> <

Write < or > between the pair of numbers to make the statement true.

-46.8 ___________ -46.08

If x ≠ 0, then =

-4x - 4x

If x ≠ 0, then 10𝑥

320𝑥.

Explain why x cannot equal 0:

Which of the following number sentences is true?

-6 < 0 < 3 < < -3

-6 > -5 > -3 -3 > 0 > -5

Order the following from least to greatest:

48/3 9.1 √75 15 3

7

5

8

Which of the following is the greatest?

3 ÷ (-4) 3 × (-4)

3 - (-4) 3 + (-4)

Which is greatest?

7 + (-12) -7 - (-12)

7 - (-12) -7 + (-12)

=

6 15 14 -5

−12 + (−13)

5

Which of these expressions is negative?

-(2 - 3)

3(2 - 3) 3 - (2 - 1)

PROPERTIES OF EXPONENTS:

Product Quotient Power Negative Exponents Zero

𝑎𝑏4 × 𝑎𝑏−2 4𝑥5

𝑥3 (𝑎𝑏2)4

𝑥−4

𝑥3 6x0

3m 3m2 6m 9m2

(9𝑥

4)3

2x5y4 2x4y3 -2x4y3 2x4y2 x5y4

If (3𝑥7𝑦2𝑧)2 = 12𝑚(𝑥𝑦4𝑧) then m = ______.

=

9a2b 4ab2 4ab a2b2

(3𝑥𝑦2

𝑧) (

2𝑦

𝑎3)

2

4x(3x2 - 2x + 6) =

12x2 - 8x + 24

7x2 - 8x + 10

12x3 - 2x + 6

12x3 - 8x2 + 24x

6xy (2x2 + 5z + 4)

LINEAR EQUATIONS: There are 3 forms of a linear equation that you should have memorized. You

should be able to graph quadratic equations, identify the x- and y-intercepts, and find the slope.

Standard form: Slope-Intercept Form Point-Slope Form

Graph each of the following. Identify the slope and y-intercept.

3x + 6 y = 12 m = ________ (0, ______)

y = -2/3 x + 5 m = ________ (0, ______)

(y+5) = 3 (x-2) m = ________ (0, ______)

SOLVE: http://tutorial.math.lamar.edu/Classes/Alg/SolveLinearEqns.aspx

(a) (b)

Given that a linear function passes through the points (-3, 8) and (8, 6),

determine the formula of the function.

Given that a function has a constant rate of change of -4 and passes

through the point (-3, 4), find the equation of the line. What is the

value of y when x = –5.6?

Graph the following two points. f(2) = 10 f(6) = 15 Find the

average rate of change between those points:

If 8x - 2 = 6x, then x =

-1

- 1

Solve for x:

5x – 7 = 8x

If m + 9 = 36, what is the value of m ?

15 45 23 27

-x + 72 = 31

Which of the following graphs could represent 3x + 2y = 12?

Graph the functions using the x- and y-intercepts. Then, identify the slope of the line.

6x + 4y = 12

The line y = 3x + 1 intersects the line y = kx - 2 where x = 1. What is the value of k?

4 7 6 1 2

The line y = 5x2 + 4 intersects the line y = kx - 3 where x = 1. What is the value of k?

If Tanya could increase her running speed by 1 mph, she could complete a six mile race in 18 minutes less time. What is her current rate in mph?

4 5 3.5 4.5 3

The variables x and y are directly proportional, and y = 2 when x = 3. What is the value of y when x = 9 ?

A. 4

B. 6

C. 8

D. 12

Which of the given linear equations could be represented by this graph?

x + y = 0 y = 2 – x

y = x +2 y = x

Graph 3x + 2y = 12

ABSOLUTE VALUE EQUATIONS

http://grockit.com/blog/gre-absolute-value-refresher/ STEPS: 1) Solve to get the absolute value alone

2) Split into two equations & solve (remember to flip the inequality sign When you divide by a negative)

3|x +4| - 5 > 16 5 |x + 2| > -15

Simplify the absolute value expression. |-23|

23 - -23

Simplify

−2|−54 + 23|

If = 13, then x =

3 only 3 or only 3 or -3 3 or

-

Find x: 4|𝑥 + 5| + 2 = 18

What is the solution set for ?

x < 1 or x > 3 x > 3

-1 < x < 1 1 < x < 3 x < 1

What is the solution set of ?

QUADRATIC FUNCTIONS: A quadratic function is one of the form f(x) = ax2 + bx + c, where a, b, and c are

numbers with a not equal to zero. The graph of a quadratic function is a curve called a parabola. 2

GENERAL _____________________ VERTEX: __________________ FACTORED: _____________________

HOW TO FACTOR: HOW TO GRAPH: y = x2+8x+7

y = 3x2 -5x -2 1. Identify the vertex (find axis of symmetry and corresponding y value.)

2. Find the x and y-intercepts. 3. Connect the points.

What is the quadratic formula?

What is the discriminant? What does it tell us?

Find the roots of the functions (solve by factoring, quadratic formula, or complete the square, you choose)

f(x) = 4x2 + 20 f(x) = 3x2 + 6x - 8 375 2 xxxg

Re-write the function 223 xxf in standard form.

Given that g(x) = x2 + 6x + 10 and f(x) = -3x-4, determine the solutions to g(x) = f(x)

Given that g(x) = 3x2 + 5x + 1 and f(x) = 4x+5, determine the solutions to g(x) = f(x)

Solve x2+8x+7=0 by…

….Factoring …Completing the Square …Quadratic formula

Find the discriminant. State the number and type of solutions. 882 2 xxy

Identify the vertex. State whether it is a maximum or minimum.

y = x2 + 8x +7 y=5(x-4)2 + 6

____ Suppose a parabola has vertex (–8, –7) and also passes through the point (–7, –4). Write the equation of the

parabola in vertex form.

a. c.

b. d.

What is the maximum or minimum value of the function? What is the range?

____

a. minimum value: – 9

range:

c. minimum value: – 9

range:

b. minimum value: –1

range:

d. minimum value: 1

range:

If 6x2 + 5x - 21 = (2x - 3)(ax + b). What is the value of a + b?

-11 -4

7 10

Factor

21x2 – 8x - 5

x2 - 4x - 5 = 0 when x =

4 or -1 1 or 9

-1 or 5 1 or 5

4x2 -7x -2 = 0 when x =

(5x - 3)2 =

25x2 - 30x + 9

25t2 -15t + 9

25x2 + 9

5t2 - 3

(7x + 10)2

x2 - =

Factor:

(49x2 –1

16)

(2t - 3)(t + 4) =

2t2 + 5t – 12 2t2 + t – 12

t2 + t – 12 2t2 - 5t - 12

(5x + 4) (x -7)

r2 - 25 =

(r - 5)(r - 5) (r - 5)(r + 5)

(5 + r)(5 - r) (5 - r)(5 - r)

Factor

(9x2 – 121)

Factor: (x + 2)2 - 4y2

(x + 2 - 2y)(x + 2 + 2y) (x + 2 - 2y)2

(x + 2 - 4y)(x + 2 + 4y) (x + 2 + 2y)2

x2 + 4 - 4y2

Factor: 25a2 – 9b2

If a = 3b , what is the value of 9(b - 1)2 in terms of a?

3a2 - 6a + 9 a2 - 3a + 1

a2 - 6a + 9 a2 - 3a + 9 27a2 - 9

If a = 5x, what is the value of 25(x+2)2 in terms of a?

If f is a quadratic function with no real zeros, which of these is f?

x2 + x + 12 x2 + x – 12

x2 - 3 x2 + x x2 - 3x + 2

If f is a quadratic function with exactly one real zero, which of these is f?

a) x3 – 2x2 –x + 2 b) x2 + 5x + 4 c) 9x2 -6x +4 d) x2 + 2x + 3

POLYNOMIAL FUNCTIONS:

A polynomial function is a function with whole number integer powers. It has one or more

roots (x-intercepts). You should know:

1. The first term determines the end behavior. How?

2. To graph, identify the end behavior, y-intercept, and x-intercepts. Then, connect the

dots.

3. Not all polynomial equations can be solved by factoring in the traditional sense. When

you cannot factor by another method, you may have to use synthetic division to factor.

Consider the leading term of each polynomial function. What is the end behavior of the graph?

Write the polynomial in factored form & identify the roots of the function: 4x3 + 8x2 – 96x x3 + 9x2 + 18x Use synthetic division and factor the equation completely and then graph the function. Note: x = 5 is one root. f(x) = x3 – 5x2 +16x -80

If x3 + ax – 8 is divided by x - 1, the remainder is -4. what is the value of a?

7 -76 3 8 -7

Use synthetic division to evaluate f(2) if f(x) = 3x4 -5x2 + 2.

PROBABILITY

Your friend is having a party and has 15 games to chose from. There is enough time to play 4 games. In how

many combinations of 4 games can games be chosen?

How many different numbers can be made using any three digits of 12,378?

Find the mean, mean, mode, and range of the data: 4, 8, 6, 7, 5, -4, 4, 6, 7, 7, 9, 10

Ryan has grades of 87%, 78%, and 95% on his last 3 tests. What is the minimum grade he can get on his 4th test in order

to get an A average (90% or above) on his tests?

Kyle has grades of 85, 92, 96, and 89% on his first tests. What score does he need on his final test to have an average of

92? (assume all tests are weighted equally).

In a school election, there are 3 candidates for president, 4 for vice-president and 4 for treasurer. How many different outcomes are there for the election?

48 12 24 11 10

Teresa has 5 shirts, 3 pairs of shorts and 2 hats. How many different outfits can she wear consisting of one of each?

15. There are 20 children in the cast of a class play, and 8 of the children are boys. Of the boys, 4 have a speaking part in the play, and of the girls, 8 do not have a speaking part in the play. If a child from the cast of the play is chosen at random, what is the probability that the child has a speaking part? A. 2/5 B. 1/2 C. 3/5 D. 3/4

City High Temp

A t°F

B 87°F

C 81°F

D 62°F

E 93°F The table above shows the high temperature last Thursday for five cities, A through E. If the median of the Thursday high temperatures for these cities was 81 F, ° which of the following could NOT have been the high temperature last Thursday for City A ? A. 85 F° B. 75 F° C. 65 F° D. 55 F

IMAGINARY NUMBERS

What are imaginary numbers?

Simplify the following:

(4+10i)(3-2i) – 𝟒𝒊(𝟐 + 𝟑𝒊) − √−𝟑𝟔

𝟓 + √−𝟏𝟔 √−𝟓𝟐 𝟒

𝟑−𝒊

If (2 + i)(3 + i) = a + bi, what is the value of a + b?

3 2 10 5 9

If (-6+i)(4+7i) = a + bi, what is the value of a + b?

COMPOSITE & INVERSE FUCNTIONS

1. What is a composite function?

2. How do you find an inverse function?

3. How can you prove that two functions are inverse functions using composite functions?

1) Given: f(x) = 3x + 5 g(x) = 4x2 +3 2) Find the inverse of 3) Are f(x) and g(x) inverse functions?

f + g g – f 𝑓(𝑥) =𝑥−3

5 f(x) = 3x + 5 𝑔(𝑥) =

1

3𝑥 − 5

f(g(x)) g(f(x))

Given the following table, find f-1(-2).

If f(x) = 2x + 1 and g(x) = x2 + 1, then f(g(x)) =

2x2 + 2 (2x + 1)2

2x2 + 1 2x2 + 3 2(x + 1)2

f(x) = 4x + 5 g(x) = x2 – 7. Find f(g(x)) and g(f(x))

f contains the points (3, 1) and (1, -3). If f-1 is the inverse of f, what is the value of f-1(f(3))?

- 3 -1 1

Use the table to find the value of g(f(3))

x f(x) g(x)

3 5 12

1 17 -20

5 8 -15

12 15 14

17 20 23

x f (x)

–2 -5

0 -3

1 -2

2 -1

RATIONAL FUNCTIONS

In rational functions, you’ll want to eliminate the denominators by multiplying both sides of the equation

by the Least Common Multiple.

What is special about the domain and range rational equations?

What does it mean to simplify and combine like radicals?

Simplify: 𝒙+𝟗

𝟐𝒙−𝟑×

𝟐𝒙𝟐+𝒙−𝟔

𝒙𝟐−𝟖𝟏

𝟓

𝒙+𝟔÷

𝟓𝒙

𝒙+𝟔

Solve:

45 16

5√12 + √24 − √6

- + =

0

5

𝑥+

6

𝑥𝑦−

10

𝑦2=

Simplify:

What is the domain of g(x) = ?

x > 2 or x < -2 All Real numbers All

Real numbers except 2 and -2 -2 < x < 2All Real numbers except -2

What is the domain of f(x) = ?

Which of the following does not intersect the line y = -1?

y = y =

y = 2 - y = x2 - 4 y = 3 - 2x

What is a horizontal asymptote? How do you find it?

+ - =

p

5𝑝

15+

𝑝

5−

8𝑝

3=

=

1

5𝑥2 − 9𝑥 − 2

𝑥2 + 2𝑥 − 8

SOLVING & SIMPLIFYING OTHER EQUATIONS

Solving a system of 3 variables?

3x + 2y + 4z = 11 -3x +2y +6z = 11 5x +4y – 3z = -12

Solve the system using any method:

y = x + 5 2x + 3y = 12

4x + y = 20 5x – y = 13

4x < x + 6 has the solution

x < 2 x < 3 x > 2 x < 5

5x + 8 > 4(x+1)

The difference of two numbers is seven and their sum is 23. What is the product of the two numbers?

120 56 184 161

The difference of two number is 95. The product of the two numbers is -2226. What are the two numbers?

If 10x - 6 = 7, then x =

1

3x – 8 = 24

If y = 2x - 3 and x = 2y, then what is the value of x?

-2 -3 2 1

If 2y = 3x -14 and x = 3y. Then, x =

Use the given values of the variables to find the value of the expression. 6x + 2y for x = 5, y = 9

72 40 48 64

Evaluate -8y + 7z for x = -3 and y = 2

If = a + b , then a + b =

7 4 11 3 9

If f(x) = x2 + 2(x + 1), then f(2) =

12 16 8 4 20

If g(x) = 3x2 + 7x-1, find g(3)

If , what is the value of a?

5 3 -2 7 4

What is the value of y?

For the given system of 3 equations, find x.

-1 1 2 3

Solve the system:

{

𝑥 + 2𝑦 + 𝑧 = 16−𝑦 + 3𝑥 = 5

𝑧 + 𝑦 = 9

Which of the following functions contains the points (0, 1), and (2, 1)?

Which of the following represent a function that passes through (2,3) and (4, 2)?

a. y = 0.5x +3 c. y =|x-1| b. x - 2y = 4 d. –x -2y = 4

GEOMETRY & TRIGONOMETRY

SOH CAH TOA

1. The angle of depression from a ship to a rock

on the ocean floor is 21 . After sailing 245 m,

the ship is directly above the rock. At this point,

what’s the distance between the rock and the

ship?

2. The angle of elevation from a leaf on the

ground to the top of a tree is 23 . The leaf is

54m from the base of the tree. How tall is the

tree?

Draw the angle and find a coterminal angle that is less than 360°: Find the csc30° 530° 23∏/4

Find the tan135° Graph the function sinϴ

Graph the points (1,2) and (7, 10). Find the distance between the two points and label the midpoint. Find the volume of a sphere with diameter 20. Find the volume of a sphere with radius of 6 and height of 9.

ABC is equilateral with each side having length 6, and DEFG is a square.

If D and E are the midpoints of AB and AC, the the perimeter of DEFG is

24 18 12 20 16

Given AC = 42, CB = 46, AB = 48. D, E, F are midpoints. Find the perimeter of triangle DEF.

http://www.regentsprep.org/regents/math/geometry/gp10/pracmid.htm

In the right triangle shown

tanθ =

In the right triangle shown, find cos θ =

=

2 sin 3θ

2 tan 3θ 6 tan θ

Simplify: 3sinϴ cotϴ

If cos θ = , then tan θ =

In a right triangle, the cscϴ= 9/4, find the cotϴ.

What is the difference between the largest and the

smallest y values of the function y = ?

4 2 0 1

Complete a table of values for the function 𝑦 =𝑐𝑜𝑠𝜃.

What is the difference between the largest and smallest y values of the function?

If cos θ = , then θ could not be

420 300 -60 60 30

If csc θ is undefined, then sin θ =

Which of the following functions contains the

point ?

y = 2sin x y = csc x y = tan x

y = 2sec x y = 2cot x

Evaluate: sec 𝜋

3

Arithmetic Review

Four people are going to equally share a lunch tab of $38.40. How much does each person owe?

1.2 × 0.03 =

22.3 - 8.46 =

Which of these is not equivalent to 45/100?

0.55 0.45 45%

1.025 x 0.04 1580 × 0.044 is closest to

A runner's time for a 100 meter race improved from 18.3 seconds to 16.4 seconds. By how many seconds did he reduce his time?

70% of the animals in the zoo are mammals. If there are 80 mammals. How many animals are there in the class?

A car gets 42 miles to the gallon. How many miles will it be able to go on 6.5 gallons?

In Mr. Smith's math class, 2/3 of the 27 students are girls. How many girls are in the class?

Karl’s average time to finish a book was 3.5 weeks. After completing a speed reading course, his average time to finish was 2 weeks. By how much did he improve his reading time?

Use long division to find:

0.5488 ÷ 5.6

73.5 × 125 is closest to

100 1,000

10,000 100,000

What is 85% of 200? The distance

between and is

An item originally priced at $120 is sold at a 20% discount. What is the new price?

$96 $24 $100 $240

What percent is 14 of 35?

21 40 35 25

20% of a number is 15. What is the number?

75 300 30 3

+ =

÷ =

22

5÷ 3

is equivalent to

3.75% 3.8% 37.5%0.375%

÷ =

-

6

Which of these inequalitites is false?

0.04 < 0.045 < 0.05

0.1 < (0.4)2 < 0.2

0.1 < 0.2 < 0.25

1 < 1.25 < 1.2

The bill for 8 people to eat pizza was $54.70. Since Tommy bought a special glueten-free meal, he paid his own dinner of $8.50. Everyone else split the remaining bill evenly. How much does each person owe?

13.2 × 0.2 =

15.4 2.64 0.264 26.4

=

+ is closest to

4 13 5 3

LOGARITHMIC FUNCTIONS & EXPONENTIAL EQUATIONS

Logarithmic functions are the inverse “undo” exponential functions.

Logbx = y by = x

logb 1 = _____ logbb = _____ logbbx = ______ lnex = _____

3 Rules for Condensing: Rules for solving: Product – 1. Condense into one log function on either side of equal sign. Quotient Rule – 2. Use the Power Rule to move variables out of the exponent Power Rule - 3. Solve for the variable.

Write the equation in logarithmic form. Write the equation in exponential format: log3x+4 = 16

Evaluate the logarithms:

log81 3

Write the expression as a single logarithm.

Expand the logarithmic expression.

Solve the exponential/logarithmic equation.

Does the function represents exponential growth or exponential decay?

In a particular region of a national park, there are currently 330 deer,

and the population is increasing at an annual rate of 11%.

a. Write an exponential function to model the deer population.

b. Explain what each value in the model represents.

c. Predict the number of deer that will be in the region after five years. Show your work

INTEREST EQUATIONS:

Continuously By Period A = Pert A = P(1+ r/n)nt

If Carlos has $8,000 to invest for the next 5 years, should he put it in account A or B?

A: 6.5% annual interest rate, compounded quarterly

B. 4.8% annual interest rate, compounded continuously

If Becky has $12,500 to invest for the next 12 years, should she put it in account A or B?

A: 7.5% annual interest rate, compounded semi-annually

B. 6.8% annual interest rate, compounded continuously

FUNCTIONS: DOMAIN AND RANGE:

A function has one, and only one, input for each output. You can use a vertical line test to

verify if a relation is a function.

List the domain and range of the relation. Is the relation a function?

{(14, 15), (5, 7), (3, 10), (11, 1), (5, 8)}

List the domain and range of each relation. Then, use the vertical-line test to determine which graph represents

a function.

a.

c.

b.

d.

2 4–2–4 x

2

4

–2

–4

y

O 2 4–2–4 x

2

4

–2

–4

y

O 2 4–2–4 x

2

4

–2

–4

y

O 2 4–2–4 x

2

4

–2

–4

y

GEOMETRIC AND ARITHMETIC SEQUENCES & SERIES

What is an arithmetic Sequence? What is a geometric sequence?

What is summation notation?

Generate the first five terms in the sequence using the explicit formula.

a. –30, –25, –20, –15, –10

b. 30, 25, 20, 15, 10

c. –10, –15, –20, –25, –30

d. 10, 15, 20, 25, 30

What is the term in the sequence using the given formula?

a. 14

b. 57

c. 44

d. –44

Write a recursive formula for the sequence 7, 13, 19, 25, 31, ... Then find the next term.

a. ; 37

b. ; 7

c. ; –23

d. ; 37

Write an explicit formula for the sequence 8, 6, 4, 2, 0, ... Then find .

a. ; –16 c. ; –20

b. ; –18 d. ; –18

Suppose you drop a tennis ball from a height of 8 feet. After the ball hits the floor, it rebounds to 80% of its

previous height. How high will the ball rebound after its third bounce? Round to the nearest tenth.

a. 3.3 feet b. 4.1 feet c. 5.1 feet d. 1 feet

Is the sequence arithmetic? If so, identify the common difference.

13, 20, 27, 34, ...

a. yes; 7 b. yes; –7 c. yes; 13 d. no

Find the 50th term of the sequence 5, –2, –9, –16, ...

a. –352 b. –343 c. –338 d. –331

Is the sequence geometric? If so, identify the common ratio.

2, –4, –16, –36, ...

a. yes; –2 b. yes; 2 c. yes; –3 d. no

What is the fifth term of the geometric sequence? 5, 15, 45, ...

a. 1215 c. 405

b. 1875 d. 3645

Use summation notation to write the series 49 + 54 + 59 + ... for 14 terms.

a.

c.

b.

d.

What is for 6 – 24 + 96 – 384 + ... ?

a. 19,662 b. –78,642 c. –4914 d. 1230

=

3 2 12 9 24

∑ 3𝑔 − 5

6

𝑔=1

If an is a sequence whose nth term is 2(n + 1) - 2(n - 1), then what is a4?

32 4 64 8 24

an = 3(n-3) + 5n+1

a3 =

GRAPHING FUNCTIONS & THEIR SHIFTS:

Describe how each piece transforms the graph: What order should transformations occur in? y = -3(-5x+4)-8

1. Label the parent function of each graph:

2. Write the equation of each transformation: f(x) = |x| Left 4 Reflect across x-axis Down 2

q(x) = x3 Reflect across y-axis Vertically stretch by a factor of 3

g(x) = log3x Reflect across y-axis Stretch horizontally by a factor of 2

h(x) = 5x Reflect across y-axis Right 7 Up 12

List the parent function and the transformations in order that they occur: h(x) = 2 (x-5)3 + 2 k(x) = sin(3x -5)+6

g(x) is given. Graph: g(x) = f(1/2 x-1) f(x) is given. Graph: f(x) = 2 f(x) -3

Match each equation with its graph:3

1. Which graph would represent 𝒇(𝒙) = 𝟐−𝒙 − 𝟐 A.

B.

C.

D.