Algebra 1
February 6, 2017 | Author: Katherine Aquino | Category: N/A
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Part 1/2
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10 Reasons Why Online Review continues to Grow in Popularity. 1. Pacing Allow pupils to work at their own pace. 2. Open Scheduling Allow professionals to continue their review while working. 3. Savings Charge less than traditional schools. 4. Lack of Commute Save on gas and commuting time. 5. Connections Connect with top-notch instructors and guest lecturers from around the world. 6. Teaching & Testing Options Students are able to choose a learning and evaluation format that works for them. 7. Effectiveness Online education is effective. 8. Choice Online education allows students to choose from a wide variety of programs 9. Flexibility Online education offers flexibility for students who have other commitments. 10. Networking Opportunities Students enrolled in online education programs network with peers from all over the nation.
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Algebra Part 1/2 Chapter 1: Tools of Algebra Chapter 2: Functions, Equations, and Graphs Chapter 3: Linear Systems Chapter 4: Matrices Chapter 5: Quadratic Equations and Functions Chapter 6: Polynomials and Polynomial Functions Chapter 7: Radical Functions and Rational Exponents
Part 2/2 Chapter 8: Exponential and Logarithmic Functions Chapter 9: Rational Functions Chapter 10: Quadratic Relations Chapter 11: Sequences and Series Chapter 12: Probability and Statistics Chapter 13: Periodic Functions and Trigonometry Chapter 14: Trigonometric Identities and Equations
4
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Tools of Algebra 1. Evaluate 2a − 5b − 3a for a = −2 and b = 5.
•
−23
•
27
•
−35
•
−15 2. Solve 4|3w + 1| − 7 = 17.
•
− ,
•
− ,
•
− ,
•
− , 3. One side of a triangle is 4 cm longer than the shortest side and 2 cm shorter than the longest side. The perimeter is 38 cm. Find the dimensions of the triangle.
•
8, 12, 18
•
9 , 13 , 15
•
9.3, 13.3, 15.3
•
11, 13, 14 4. Solve
= b for x. Find any restrictions.
•
x = nb + 3; n ≠ 0
•
x = n(b + 3) − 3; n ≠ 0
•
x=
•
+ 3; n ≠ 0
x = nb − 6; n ≠ 0 5. The expression 200 − 16t2 models the height in feet of an object that falls from an initial height of 200 feet t seconds after being dropped. Find the height of the object after 2.5 seconds.
•
60 feet
•
80 feet
•
120 feet
•
100 feet
2012 UPCAT, ACET, DLSUET, USTET @ www.fb.com/Mathalino
6
• 6. Simplify the expression |−3| − |−5|.
•
8
•
−8
•
−2
•
2 7. Graph the solution of 7x + 3 ≤ 3 or 6x − 8 > 16.
• • • • 8. Suppose you take a true/false quiz that contains five true/false questions. Let an even digit represent a correct answer and an odd digit represent an incorrect answer. In the five trials shown, how often did you get at least 3 questions correct? (Note: Assume there is a 0 before the number 4908 in the first trial.)
•
2
•
3
•
4
•
• • • •
5 9. Graph the solution of −7x + 4 ≥ 11 and 10 − 3x > 1.
2012 UPCAT, ACET, DLSUET, USTET @ www.fb.com/Mathalino
10. Replace
•
<
•
>
•
= 11. Solve 3(x − 5) − 15 = 48.
•
20
•
26
•
−4
•
with , or = to make the sentence −12
14 12. Solve bx − cx = 7 for x.
•
x = 7(b − c)
•
x=
•
x=
•
−c
x= 13. Graph the solution of 2x − 5 > −3 and 3x + 7 ≤ 19.
• • • • 14. Solve 11x − 7 = 7x + 13.
•
x=
•
x=5
•
x = −2
•
x=7
−18 true.
7
2012 UPCAT, ACET, DLSUET, USTET @ www.fb.com/Mathalino
8
15. Solve |8 − m| + 4 = 21.
•
−9, 25
•
−25, 9
•
−25, −9
•
9, 25 16. Brown is the dominant eye color for human beings. If a father and mother each carry a gene for brown eyes and a gene for blue eyes, what is the probability of their having a child with brown eyes?
• • • •
1 17. A jar contains 12 blue marbles, 8 red marbles, and 20 green marbles. You pick one marble from the jar. Find the theoretical probability, P(blue or green).
• • • • 18. Simplify 9a − (6a − 8b) by combining like terms.
•
−3a − 8b
•
−3a + 8b
•
3a + 8b
•
3a − 8b 19. Jean has $20 and is going shopping to buy 3 scarves and as many pairs of earrings as possible. How many pairs of earrings can she buy if scarves cost $2.55 each and earrings cost $2.99 a pair?
•
Jean can buy less than 4 pairs of earrings.
•
Jean can buy at most 4 pairs of earrings.
•
Jean can buy more than 4 pairs of earrings.
•
Jean can buy at least 4 pairs of earrings.
2012 UPCAT, ACET, DLSUET, USTET @ www.fb.com/Mathalino
20. Solve V =
9
πr2h for h.
•
h=
•
h=
•
h=
•
h=
Tools of Algebra 1. Evaluate 2a − 5b − 3a for a = −2 and b = 5. •
CORRECT: −23 2. Solve 4|3w + 1| − 7 = 17.
•
CORRECT: − , 3. One side of a triangle is 4 cm longer than the shortest side and 2 cm shorter than the longest side. The perimeter is 38 cm. Find the dimensions of the triangle.
•
CORRECT: 9 , 13 , 15 4. Solve
= b for x. Find any restrictions.
•
CORRECT: x = nb + 3; n ≠ 0 5. The expression 200 − 16t2 models the height in feet of an object that falls from an initial height of 200 feet t seconds after being dropped. Find the height of the object after 2.5 seconds.
•
CORRECT: 100 feet 6. Simplify the expression |−3| − |−5|.
•
CORRECT: −2 7. Graph the solution of 7x + 3 ≤ 3 or 6x − 8 > 16.
•
CORRECT: 8. Suppose you take a true/false quiz that contains five true/false questions. Let an even digit represent a correct answer and an odd digit represent an incorrect answer. In the five trials shown, how often did you get at least 3 questions correct? (Note: Assume there is a 0 before the number
2012 UPCAT, ACET, DLSUET, USTET @ www.fb.com/Mathalino
10
4908 in the first trial.)
•
CORRECT: 3 9. Graph the solution of −7x + 4 ≥ 11 and 10 − 3x > 1.
•
CORRECT: 10. Replace with , or = to make the sentence −12
−18 true.
•
CORRECT: > 11. Solve 3(x − 5) − 15 = 48.
•
CORRECT: 26 12. Solve bx − cx = 7 for x.
•
CORRECT: x = 13. Graph the solution of 2x − 5 > −3 and 3x + 7 ≤ 19.
•
CORRECT: 14. Solve 11x − 7 = 7x + 13.
•
CORRECT: x = 5 15. Solve |8 − m| + 4 = 21.
•
CORRECT: −9, 25 16. Brown is the dominant eye color for human beings. If a father and mother each carry a gene for brown eyes and a gene for blue eyes, what is the probability of their having a child with brown eyes?
•
CORRECT: 17. A jar contains 12 blue marbles, 8 red marbles, and 20 green marbles. You pick one marble from the jar. Find the theoretical probability, P(blue or green).
•
CORRECT: 18. Simplify 9a − (6a − 8b) by combining like terms.
•
CORRECT: 3a + 8b 19. Jean has $20 and is going shopping to buy 3 scarves and as many pairs of earrings as possible. How many pairs of earrings can she buy if scarves cost $2.55 each and earrings cost $2.99 a pair?
•
CORRECT: Jean can buy at most 4 pairs of earrings.
2012 UPCAT, ACET, DLSUET, USTET @ www.fb.com/Mathalino
20. Solve V =
•
πr2h for h.
CORRECT: h =
Functions, Equations, and Graphs 1. What are the domain and range of the function below?
•
Domain: {−2, 1}; Range: {−3, −1, 0, 1, 2}
•
Domain: {−3, −1, 0, 1, 2}; Range: {−2, 1}
•
Domain: {all real numbers}; Range: {−2, 1}
•
Domain: {−2, 1}; Range: {all real numbers} 2. What is the vertex of the graph of y = |x + 3|?
•
(3, 0)
•
(0, 3)
•
(0, −3)
•
(−3, 0)
11
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3. What is the graph of y > |x − 3|?
•
•
•
• 4. What point has coordinates of (−5, 4)?
12
2012 UPCAT, ACET, DLSUET, USTET @ www.fb.com/Mathalino
•
N
•
M
•
L
•
O
5. What is the graph of 0.4x + 0.1y ≥ 1?
•
•
•
•
13
2012 UPCAT, ACET, DLSUET, USTET @ www.fb.com/Mathalino
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6. In chemistry Charles Law states that if the pressure remains constant, the volume of an ideal gas varies directly with the temperature in degrees Kelvin. Suppose a gas has a volume of 100 mL at 250°K. Which proportion could you use to find the volume v of the gas if the temperature decreases to 150°K?
•
=
•
=
•
=
• •
=
7. What is the graph of y ≥ − x?
•
•
•
•
2012 UPCAT, ACET, DLSUET, USTET @ www.fb.com/Mathalino
8. What is the graph of y = |x| − |2x|?
•
•
• •
not here 9. The value of y varies directly with x. If y = 64 when x is 16, what is x when yis 288?
•
72
•
864
•
1,152
•
96
15
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10. What is the slope of this line?
• •
−
• • •
− 11. Which line, having the given slope and passing through the given point, represents a direct variation?
•
m=
; (9, 6)
•
m=
; (6, 9)
•
m=
; (9, −6)
•
m = − ; (9, 6) 12. What is an equation of the line whose slope is
•
3x − 6y = −13
•
3x − 5y = −7
•
3x − 6y = 5
•
6x − 3y = −13
and which contains the point (−3,
)?
2012 UPCAT, ACET, DLSUET, USTET @ www.fb.com/Mathalino
13. What is an equation of the inequality?
•
y ≤ |x − 1| − 2
•
y > |x + 1| − 2
•
y ≥ |x + 1| − 2
•
y ≥ |x − 1| − 2
14. Identify the set of linear equations and graph for y = | x + 6| + 3.
•
y = − x − 3, if x ≤ −4; y =
x + 9, if x > −4
•
y = − x − 9, if x ≥ −4; y =
x + 3, if x < −4
17
2012 UPCAT, ACET, DLSUET, USTET @ www.fb.com/Mathalino
•
•
y = − x + 9, if x ≤ −4; y =
18
x − 3, if x > −4
y = − x + 3, if x ≥ −4; y = x − 9, if x < −4 15. The length of an object's shadow varies directly with its height. A boy who is 4 ft 6 in. tall casts a shadow that is 6 ft 3 in. long. What is the height of a tree that casts a shadow that is 25 ft long?
•
18 ft
•
20 ft
•
16 ft
• •
216 ft
16. What is an equation of a line that is parallel to the line whose equation is y = − x + 1?
•
y=− x+5
•
y=− x
•
y=
x−1
•
y=
x+3
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17. What is the graph of the equation 5x − 2y = 6?
•
•
•
•
19
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20
• 18. The capacity of the gas tank in a car is 16.5 gallons. The car uses 10.5 gallons to travel 178.5 miles. After driving 85 more miles, only 1 gallon of gasoline remains in the tank. Which linear equation and graph model the amount of gas gleft in the tank after driving m miles?
•
g=−
•
g=−
m + 16.5
•
g=−
m + 16.5
m + 16.5
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•
21
g = − m + 16.5 19. This graph represents a translation of the graph of y = |x|. What is the equation of this graph?
•
y = |x − 3|
•
y = −|x − 3|
•
y = |−x + 3|
•
y = −|x + 3| 20. What is an inequality for this graph?
•
y − 3x > −5
•
y − 3x ≥ −5
•
y − 3x ≤ −5
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•
22
y − 3x < −5
Functions, Equations, and Graphs 1. In chemistry Charles Law states that if the pressure remains constant, the volume of an ideal gas varies directly with the temperature in degrees Kelvin. Suppose a gas has a volume of 100 mL at 250°K. Which proportion could you use to find the volume v of the gas if the temperature decreases to 150°K?
•
•
CORRECT: = 2. What is an equation for a direct variation whose graph passes through the point (−5, 3)?
CORRECT: y = − x 3. Which statement describes the relationship between the graph of the equationy = | | and the graph of the equation y = | | + 4?
• •
CORRECT: The graph of y = | | was shifted 4 units up. 4. What is the slope of the line through the points P(−5, −6) and Q(−12, 5)?
•
CORRECT: − 5. What is the graph of the equation 4x − y = 5?
•
CORRECT: 6. Which graph represents the equation y = − |x|?
•
CORRECT:
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7. What is a point-slope equation of the line through the points (−1, −5) and (−2, 4)? •
CORRECT: y + 5 = −9(x + 1) 8. What is the equation of y = |x| translated 1 unit up and 6 units to the left?
•
CORRECT: y = |x + 6| + 1 9. What is the graph of y = −|3x|?
•
CORRECT: 10. What is the graph of y > |x − 3|?
•
CORRECT: 11. What are the domain and range of this function?
•
CORRECT: Domain: x > 0; Range: y > 0 12. Which statement about 5y = 2x is true?
•
CORRECT: The equation represents a direct variation whose constant of variation is
.
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13. What is the graph of y >
•
24
+ 1?
CORRECT: 14. What is the constant of variation for the function given in the table? x y 20 50 42 105 64 160 81 202.5
•
CORRECT: 2.5 15. Find f(−5) if f(x) = −8x − 1.
•
CORRECT: 39 16. You can use a linear model to represent the relationship between the hours worked h and the money earned m. When h = 0, then m = 0. If a person earned $39 after working 6 hours, how much will the person earn after working 25 hours?
•
CORRECT: $162.50 17. What are the ordered pairs of the relation? Find the domain and range.
•
CORRECT: {(−2, 5), (−1, 2), (0, 1), (1, 2), (2, 5)}; domain {−2, −1, 0, 1, 2}; range {5, 2, 1} 18. Find an equation for the linear model of the situation below and use it to make a prediction. A train is traveling north at a constant rate. At 3:00 P.M. it is 55 miles north of a city. At 4:15 P.M. it is 80 miles north of the city. If d represents the distance in miles, and t represents the time in hours, how many miles north of the city will the train be at 5:45 P.M.?
2012 UPCAT, ACET, DLSUET, USTET @ www.fb.com/Mathalino
•
•
CORRECT: d = 64t + 55; d = 231 miles 19. What is an inequality for this graph?
CORRECT: y − 3x ≥ −5 20. What is an equation of the line whose slope is
•
and which contains the point (−3,
CORRECT: 3x − 6y = −13
Linear Systems 1. Identify the graph that shows the solution to the system of inequalities.
•
CORRECT:
2. Solve the system •
25
CORRECT: (1, 5, −2)
.
)?
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3. Use elimination to find the solution of the system •
.
CORRECT: (−1, 3, 2) 4. A small fish market sells only tuna and salmon. A tuna costs the fish market $0.75 per pound to buy and $2.53 per pound to clean and package. A salmon costs the fish market $3.00 per pound to buy and $2.75 per pound to clean and package. The fish market makes $2.50 per pound profit for each tuna it sells and $2.80 per pound profit for each salmon it sells. The fish market owner can spend only $159.00 per day to buy fish and $197.34 per day to clean and package the fish. The graph below represents the feasible region for this linear programming model. What are the coordinates of the vertices of the feasible region, and what are the vales of t and s that maximize the objective function?
; P = 2.50t + 2.80s
•
CORRECT: (0, 0), (0, 53), (71.76, 0), (28, 46); t = 28 and s = 46. 5. What equivalent system of two equations do you get when you use substitution to eliminate y and solve this system of three equations?
•
CORRECT: 6. How could you solve this system using substitution?
•
26
CORRECT: Substitute 2r + 3 for t in the equation 5r − 4t = 6. 7. Identify the graph of the system and its solution.
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infinite number of solutions
•
CORRECT: 8. Graph the system of inequalities.
•
CORRECT: 9. Which system has no solution because exactly two of the planes are parallel?
•
CORRECT: 10. This spreadsheet shows the monthly revenue R and monthly expenses E for a new business. In what month will the revenue equal the expenses if the linear models for R and E are R = 4100x + 300 and E = 1900x + 19900? Assume July is month 1.
•
CORRECT: March
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11. Plot the point (−3, −1, 2) in a three-dimensional coordinate system.
•
CORRECT: 12. A factory can produce two products, x and y, with a profit P approximated byP = 12x + 15y. The production of y can exceed x by no more than 200 units. Moreover, production levels are limited by the formula x + 2y ≤ 1000. What production levels yield maximum profit?
•
CORRECT: x = 1000; y = 0 13. What is the solution of this system?
•
CORRECT: (−2, −3) 14. The objective function for this graph is P = 5x + y. What values of x and ymaximize this function?
•
CORRECT: (1000, 0): The maximum is 5000. 15. Tasty Bakery sells three kinds of muffins: chocolate chip muffins at 40 cents each, oatmeal muffins at 45 cents each, and cranberry muffins at 50 cents each. Charles buys some of each kind and chooses three times as many cranberry muffins as chocolate chip muffins. If he spends $8.85 on 19 muffins, how many chocolate chip muffins does he buy?
•
CORRECT: 3 16. One rental car agency charges $35 per day plus 11 cents per mile to rent a mid-size car. Another agency charges $42 per day plus 6 cents per mile to rent an equivalent type of car. Let c represent the cost of renting a car from either agency, and m represent the number of miles traveled daily. Which system of equations represents the cost of renting from each agency? For what value of mare the costs of rental equal?
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•
CORRECT: ; 140 miles 17. Which graph represents this system and its solution?
•
CORRECT: (3, −4) 18. The equation P = 5x + 3y represents the objective function for this linear programming model. What is the vertex of the feasible region shown below that results in the maximum value of P?
•
CORRECT: (12, 14)
19. What steps can you use to solve the system •
CORRECT: Multiply 4x + 5y = 53 by 2 and multiply 7x + 2y = 59 by −5. Then add.
20. Classify the system •
using the method of elimination?
CORRECT: independent
without graphing.
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Matrices 1. Find
.
• • • • ?
2. What is the determinant of
•
10
•
20
•
−10
•
38
3. Identify a23 in
•
23
•
16
•
−21
•
3
.
, and B =
4. Find BA, if A =
• • • •
undefined 5. Which product is undefined?
A=
,B=
,C=
,D=
, if it is defined. If it is not defined, choose undefined.
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•
AB
•
BA
•
DB
•
CD
6. Find the inverse of
, if it exists.
•
•
• does not exist
•
7. Solve
•
•
•
•
X−
=
.
31
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8. Write the coordinates of the image after a translation 3 units right and 2 units down.
•
•
•
•
9. Find 4
.
•
•
•
•
10. What are the dimensions of
•
3 × 2, −4
•
2 × 3, 3
? Identify a12.
32
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• •
2 × 3, −4 3 × 2, 3 11. Write the coordinates of the image after a dilation three times the original size.
•
•
•
• 12. Use matrices to represent the vertices of graph f and graph g.
• • • •
33
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13. What are the coordinates of the image after a rotation of 90°?
• • • •
14. Use an augmented matrix to solve
•
(2, 5, −1)
•
(4, 6, 1)
•
(4, 5, −6)
•
none of these
15. What are the dimensions of
•
4×5
•
7×3
•
5×4
•
6×4
.
?
34
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16. Find
+
35
.
• • • • 17. Assign a number to each letter of the alphabet and a blank space as shown by the table below:
Matrix A was used to encode the cryptogram below.
A= 14 _ 5 18 18 _ 16 _ 11 4 _ 3 2 1 3 _ 17 9 3 17 20 _ 12 16 Use A−1 to decode the cryptogram.
•
LEAPS AND BOUNDS
•
GRIND TO A HALT
•
RUN LIKE THE WIND
•
IN BROAD DAYLIGHT 18. Use matrices A, B, and C. Find B − C if you can. A=
,B=
,C=
• • • •
not possible 19. Determine whether
•
has an inverse. If an inverse matrix exists, find it.
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36
• • does not exist
•
20. Determine whether J =
•
yes
•
no
and K =
are multiplicative inverses.
Matrices 1. Find
. CORRECT:
•
2. What is the determinant of
?
CORRECT: −10
•
3. Identify a23 in
.
CORRECT: −21
•
4. Find BA, if A = •
CORRECT: undefined 5. Which product is undefined?
A= •
, and B =
,B= CORRECT: BA
,C=
,D=
, if it is defined. If it is not defined, choose undefined.
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6. Find the inverse of
•
CORRECT:
7. Solve
•
•
X−
=
CORRECT:
.
CORRECT:
10. What are the dimensions of •
.
CORRECT: 8. Write the coordinates of the image after a translation 3 units right and 2 units down.
9. Find 4
•
, if it exists.
CORRECT: 3 × 2, −4
? Identify a12.
37
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11. Write the coordinates of the image after a dilation three times the original size.
•
CORRECT: 12. Use matrices to represent the vertices of graph f and graph g.
•
CORRECT: 13. What are the coordinates of the image after a rotation of 90°?
•
CORRECT:
38
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14. Use an augmented matrix to solve
.
CORRECT: (2, 5, −1)
•
15. What are the dimensions of
?
CORRECT: 5 × 4
•
16. Find
•
+
.
CORRECT: 17. Assign a number to each letter of the alphabet and a blank space as shown by the table below:
Matrix A was used to encode the cryptogram below.
A= 14 _ 5 18 18 _ 16 _ 11 4 _ 3 2 1 3 _ 17 9 3 17 20 _ 12 16 Use A−1 to decode the cryptogram. •
CORRECT: IN BROAD DAYLIGHT 18. Use matrices A, B, and C. Find B − C if you can. A=
•
,B=
,C=
CORRECT:
20. Determine whether J = •
39
CORRECT: yes
and K =
are multiplicative inverses.
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40
Quadratic Equations and Functions 1. Solve x2 + 6x − 2 = 0 by completing the square.
±3
• •
−3 ±
•
±3
•
11 ± 2. A rocket is launched from atop a 63-foot cliff with an initial velocity of 95 feet per second. The height of the rocket t seconds after launch is given by the equation h = −16t2 + 95t + 63. Graph the equation to find out how long after the rocket is launched it will hit the ground. Estimate your answer to the nearest hundredth of a second.
•
6.54 sec
•
2.97 sec
•
2.97 sec
•
5.94 sec
2012 UPCAT, ACET, DLSUET, USTET @ www.fb.com/Mathalino
41
3. Factor x2 − 15x + 54.
•
(x − 3)(x − 18)
•
(x + 1)(x + 54)
•
(x + 9)(x + 6)
•
(x − 9)(x − 6) 4. Is y = (x + 3)(x − 6) − 2x2 linear or quadratic? Identify the quadratic, linear, and constant terms.
•
linear function linear term: −3x constant term: −18
•
quadratic function quadratic term: −2x2 linear term: −3x constant term: −18
•
quadratic function quadratic term: −x2 linear term: −3x constant term:−18
•
linear function linear term: −9x constant term:−18 5. Solve −11x + 9 + 5x2 = 0 using the Quadratic Formula.
• • • •
,− 6. Identify the vertex and y-intercept of the graph of f(x) = −4(x + 3)2 + 7.
•
(3, −7), y-intercept −5
•
(−3, 7), y-intercept −29
•
(−3, −7), y-intercept −29
•
(3, 7), y-intercept −5
2012 UPCAT, ACET, DLSUET, USTET @ www.fb.com/Mathalino
42
7. A rocket is launched from atop a 54-foot cliff with an initial velocity of 119 feet per second. The height of the rocket t seconds after launch is given by the equation h = −16t2 + 119t + 54. Graph the equation to find out how long after the rocket is launched it will hit the ground. Estimate your answer to the nearest hundredth of a second.
•
3.72 sec
•
3.72 sec
•
7.87 sec
•
7.44 sec 8. Find the absolute value of |−1 + 3i|.
•
2
• •
10
•
8
2012 UPCAT, ACET, DLSUET, USTET @ www.fb.com/Mathalino
43
9. Solve −9x + 6 + 5x2 = 0 using the Quadratic Formula.
•
,−
• • • 10. Solve x2 − 5x − 1 = 0 using the Quadratic Formula. Find the exact solutions. Then approximate any radical solutions. Round to the nearest hundredth.
• • • •
; −0.19, 5.19 ; −4.79, −0.21 ; 0.21, 4.79 ; −5.19, 0.19 11. Find a quadratic model for the values in the table.
•
f(x) = 7x2 − 9x + 3
•
f(x) = 3x2 − 9x + 7
•
f(x) = −9x2 + 3x + 7
•
f(x) = 3x2 + 7x − 9 12. Write the equation of the parabola with vertex (9, −2) and point (12, 16) in vertex form.
•
y = (x + 9)2 + 2
•
y = (x − 12)2 − 16
•
y = (x − 9)2 − 2
•
y = 3(x + 12)2 + 16 13. Factor x2 − 6x + 8.
•
(x + 4)(x + 2)
•
(x − 4)(x − 2)
•
(x − 2)(x − 6)
•
(x + 2)(x + 6)
2012 UPCAT, ACET, DLSUET, USTET @ www.fb.com/Mathalino
44
14. Identify the vertex and the axis of symmetry of the parabola.
•
vertex: ( ,
); axis of symmetry: x =
•
vertex: ( ,
); axis of symmetry: y =
•
vertex: (
); axis of symmetry: x = −
•
,
vertex: ( , ); axis of symmetry: y = 15. Solve 6x2 = 12. If necessary, round your answer to the nearest hundredth.
•
±6
≈ ±8.49
•
±
•
±2
•
none of these
≈ ±1.41 ≈ ±3.46
16. Find the quadratic function y = ax2 + c with a graph that includes (0,
•
y = 4x2 −
•
y = 4x2 +
•
y = −4x2 −
•
none of these 17. Solve 5x2 − 11x + 2 = 0 using the Quadratic Formula.
•
−2, −
•
2,
•
−
•
, ,−
) and (−3,
).
2012 UPCAT, ACET, DLSUET, USTET @ www.fb.com/Mathalino
18. Factor t2 − 10t + 25.
•
(t − 25)(t + 1)
•
(t − 5)(t + 5)
•
(t + 5)2
•
(t − 5)2 19. Rewrite y = −x2 + 5x − 1 in vertex form.
•
y = −(x −
)2 +
•
y = −(x −
)2 −
•
y = −(x +
)2 −
•
•
•
•
y = (x − )2 + 20. Graph the quadratic function with vertex (−3, −1) and y-intercept 2.
45
2012 UPCAT, ACET, DLSUET, USTET @ www.fb.com/Mathalino
46
•
Quadratic Equations and Functions 1. Solve x2 + 6x − 2 = 0 by completing the square. •
CORRECT: −3 ± 2. A rocket is launched from atop a 63-foot cliff with an initial velocity of 95 feet per second. The height of the rocket t seconds after launch is given by the equation h = −16t2 + 95t + 63. Graph the equation to find out how long after the rocket is launched it will hit the ground. Estimate your answer to the nearest hundredth of a second.
•
CORRECT: 3. Factor x2 − 15x + 54.
•
CORRECT: (x − 9)(x − 6) 4. Is y = (x + 3)(x − 6) − 2x2 linear or quadratic? Identify the quadratic, linear, and constant terms.
•
CORRECT: quadratic function quadratic term: −x2 linear term: −3x constant term:−18 5. Solve −11x + 9 + 5x2 = 0 using the Quadratic Formula.
•
CORRECT: 6. Identify the vertex and y-intercept of the graph of f(x) = −4(x + 3)2 + 7.
•
CORRECT: (−3, 7), y-intercept −29 7. A rocket is launched from atop a 54-foot cliff with an initial velocity of 119 feet per second. The height of the rocket t seconds after launch is given by the equation h = −16t2 + 119t + 54. Graph the equation to find out how long after the rocket is launched it will hit the ground. Estimate your answer to the nearest hundredth of a second.
6.54 sec
2012 UPCAT, ACET, DLSUET, USTET @ www.fb.com/Mathalino
47
•
CORRECT: 7.87 sec 8. Find the absolute value of |−1 + 3i|.
•
CORRECT: 9. Solve −9x + 6 + 5x2 = 0 using the Quadratic Formula.
•
CORRECT: 10. Solve x2 − 5x − 1 = 0 using the Quadratic Formula. Find the exact solutions. Then approximate any radical solutions. Round to the nearest hundredth.
•
CORRECT: ; −0.19, 5.19 11. Find a quadratic model for the values in the table.
•
CORRECT: f(x) = 3x2 + 7x − 9 12. Write the equation of the parabola with vertex (9, −2) and point (12, 16) in vertex form.
•
CORRECT: y = (x − 9)2 − 2 13. Factor x2 − 6x + 8.
•
CORRECT: (x − 4)(x − 2) 14. Identify the vertex and the axis of symmetry of the parabola.
•
CORRECT: vertex: ( , ); axis of symmetry: x = 15. Solve 6x2 = 12. If necessary, round your answer to the nearest hundredth.
•
CORRECT: ±
≈ ±1.41
2012 UPCAT, ACET, DLSUET, USTET @ www.fb.com/Mathalino
16. Find the quadratic function y = ax2 + c with a graph that includes (0,
•
CORRECT: y = 4x2 + 17. Solve 5x2 − 11x + 2 = 0 using the Quadratic Formula.
•
CORRECT: 2, 18. Factor t2 − 10t + 25.
•
CORRECT: (t − 5)2 19. Rewrite y = −x2 + 5x − 1 in vertex form.
•
CORRECT: y = −(x − )2 + 20. Graph the quadratic function with vertex (−3, −1) and y-intercept 2.
•
) and (−3,
48
).
CORRECT:
Polynomials and Polynomial Functions 1. Which polynomial function does NOT have 0 as a root?
•
f(x) = x4 − 1
•
h(x) = x4 + 3x2 − x
•
k(x) = x2 − x3
•
g(x) = x2 − 3x 2. Morris Mouse can find his way through the maze easily from entrance A to exit B. However, he receives food only if he finds the exit without going west or south. (North is towards the top of the diagram.) How many different paths can he take through the maze to receive food? (Note: Different
2012 UPCAT, ACET, DLSUET, USTET @ www.fb.com/Mathalino
49
paths have at least one distinct section. See the diagram for an example.)
is different from
•
70 paths
•
140 paths
•
10 paths
•
16 paths 3. Use synthetic division and the Remainder Theorem to find P(−3) if P(x) = x4 + 19x3 + 108x2 + 236x + 176.
•
9
•
8
•
7
•
6 4. The volume of a certain rectangular prism is given by 15x3 + 76x2 − 92x − 48. Give the dimensions of the prism.
•
(5x + 2)(x + 6)(3x + 4)
•
(5x + 2)(x + 6)(3x − 4)
•
(5x − 2)(x + 6)(3x − 4)
•
(5x + 2)(x − 6)(3x − 4) 5. Write a polynomial in standard form with zeros 1, 2, and −5.
•
x3 − 2x2 + 13x − 10 = 0
•
x3 − 4x2 + 17x + 10 = 0
•
x3 + 2x2 − 13x + 10 = 0
•
x3 + 6x2 − 3x + 10 = 0
2012 UPCAT, ACET, DLSUET, USTET @ www.fb.com/Mathalino
6. A polynomial equation with rational coefficients has roots 1 + roots.
•
1−
•
−1 −
,3+
•
−1 +
, −3 +
•
,3+
−1 − , −3 + 7. Divide (2x3 + 3x2 − 4x − 4) ÷ (x + 2) using long division.
•
2x2 − x − 1
•
2x2 + x + 2
•
2x2 − x + 2
•
2x2 + x – 2
8. Solve 8x3 + 27 = 0.
•
x=
,
•
x=− ,
•
x=
•
x=− ,
,
9. Evaluate
•
99
•
110
•
55
•
77 10. Evaluate
.
.
• •
60,480
•
504
• •
84
and 3 −
50
. Find two additional
2012 UPCAT, ACET, DLSUET, USTET @ www.fb.com/Mathalino
51
11. Evaluate 9!.
•
9
•
362,880
•
3,628,800
•
99 12. Divide (3x3 + 4x2 − 5x − 2) ÷ (x + 2) using long division.
•
3x2 + 2x − 1
•
3x2 + 2x + 1
•
3x2 − 2x − 1
•
3x2 − 2x + 1 13. Use the Binomial Theorem to expand (v − u)6.
•
v6 − 8v5u + 17v4u2 − 22v3u3 + 17v2u4 − 8vu5 + u6
•
v6 − 6v5u + 15v4u2 − 20v3u3 + 15v2u4 − 6vu5 + u6
•
v6 − u 6
•
v6 − 6v5u − 30v4u2 + 120v3u3 + 360v2u4 − 720vu5 − 720u6 14. Find the factored form of 4x3 − 8x2 − 12x.
•
4x(x2 − 2x − 3)
•
4(x2 + 1)(x − 3)
•
4x(x + 1)(x − 3)
•
x(4x + 1)(x − 3) 15. Use Pascal's Triangle to expand (3a − b)4.
•
81a4 − 12a3b + 18a2b2 − 12ab3 + b4
•
81a4 − 108a3b + 54a2b2 − 12ab3 + b4
•
81a4 + 108a3b + 54a2b2 + 12ab3 + b4
•
81a4 + 12a3b + 18a2b2 + 12ab3 + b4 16. Assume red and green are equally likely occurrences. What is probability that you will get one green light in a row of five lights?
• •
2012 UPCAT, ACET, DLSUET, USTET @ www.fb.com/Mathalino
52
• • 17. In how many ways can 4 singers be selected from 8 who came to an audition?
•
70 ways
•
4 ways
•
32 ways
•
1344 ways 18. Use Pascal's Triangle to expand (2x + y)4.
•
16x4 − 32x3y + 24x2y2 − 8xy3 + y4
•
16x4 + 32x3y + 24x2y2 + 8xy3 + y4
•
16x4 + 64x3y + 48x2y2 + 16xy3 + y4
•
16x4 − 64x3y + 48x2y2 − 16xy3 + y4 19. Use the Rational Root Theorem to determine all possible rational roots of 5x3 + 4x2 + 4x + 10 = 0. Do NOT find the actual roots.
•
±2, ±5, ±10, ± , ± , ±
•
±1, ±2, ±5, ±10, ± , ±
•
0, ±1, ±2, ±5, ± , ±
•
±2, ±5, ±10, ±50, ± , ± 20. Use synthetic division and the factor x + 3 to completely factor y = 4x3 + 5x2− 23x − 6.
•
y = (x + 3)(x + 2)(4x + 1)
•
y = (x + 3)(x − 2)(4x − 1)
•
y = (x + 3)(x + 2)(4x − 1)
•
y = (x + 3)(x − 2)(4x + 1)
Polynomials and Polynomial Functions 1. Which polynomial function does NOT have 0 as a root? •
CORRECT: f(x) = x4 − 1 2. Morris Mouse can find his way through the maze easily from entrance A to exit B. However, he receives food only if he finds the exit without going west or south. (North is towards the top of the diagram.) How many different paths can he take through the maze to receive food? (Note: Different
2012 UPCAT, ACET, DLSUET, USTET @ www.fb.com/Mathalino
53
paths have at least one distinct section. See the diagram for an example.)
is different from •
CORRECT: 70 paths 3. Use synthetic division and the Remainder Theorem to find P(−3) if P(x) = x4 + 19x3 + 108x2 + 236x + 176.
•
CORRECT: 8 4. The volume of a certain rectangular prism is given by 15x3 + 76x2 − 92x − 48. Give the dimensions of the prism.
•
CORRECT: (5x + 2)(x + 6)(3x − 4) 5. Write a polynomial in standard form with zeros 1, 2, and −5.
•
CORRECT: x3 + 2x2 − 13x + 10 = 0 6. A polynomial equation with rational coefficients has roots 1 + roots.
•
CORRECT: 1 − ,3+ 7. Divide (2x3 + 3x2 − 4x − 4) ÷ (x + 2) using long division.
•
CORRECT: 2x2 − x − 1 8. Solve 8x3 + 27 = 0.
•
CORRECT: x = − , 9. Evaluate
•
.
CORRECT: 55 10. Evaluate
.
•
CORRECT: 504 11. Evaluate 9!.
•
CORRECT: 362,880 12. Divide (3x3 + 4x2 − 5x − 2) ÷ (x + 2) using long division.
•
CORRECT: 3x2 − 2x − 1 13. Use the Binomial Theorem to expand (v − u)6.
•
CORRECT: v6 − 6v5u + 15v4u2 − 20v3u3 + 15v2u4 − 6vu5 + u6
and 3 −
. Find two additional
2012 UPCAT, ACET, DLSUET, USTET @ www.fb.com/Mathalino
54
14. Find the factored form of 4x3 − 8x2 − 12x. •
CORRECT: 4x(x + 1)(x − 3) 15. Use Pascal's Triangle to expand (3a − b)4.
•
CORRECT: 81a4 − 108a3b + 54a2b2 − 12ab3 + b4 16. Assume red and green are equally likely occurrences. What is probability that you will get one green light in a row of five lights?
•
CORRECT: 17. In how many ways can 4 singers be selected from 8 who came to an audition?
•
CORRECT: 70 ways 18. Use Pascal's Triangle to expand (2x + y)4.
•
CORRECT: 16x4 + 32x3y + 24x2y2 + 8xy3 + y4 19. Use the Rational Root Theorem to determine all possible rational roots of 5x3 + 4x2 + 4x + 10 = 0. Do NOT find the actual roots.
•
CORRECT: ±1, ±2, ±5, ±10, ± , ± 20. Use synthetic division and the factor x + 3 to completely factor y = 4x3 + 5x2− 23x − 6.
•
CORRECT: y = (x + 3)(x − 2)(4x + 1)
Radical Functions and Rational Exponents 1. Let f(x) = x2 + 3 and g(x) =
•
. Find (f
g) (2).
9
• • •
•
7 14 2. Graph y = −5
+ 5. Find the domain and range of the function.
domain: x ≥ 0, range: y ≥ −5
2012 UPCAT, ACET, DLSUET, USTET @ www.fb.com/Mathalino
•
domain: x ≥ 5, range: y ≥ 0
•
domain: x ≥ 0, range: y ≤ 5
•
domain: x ≤ −5, range: y ≥ 0 3. Rationalize the denominator of
. Assume that all variables are positive.
• • •
2
• 4. Multiply (3
− 10
)(3
+ 10
•
9x − 100yz
•
9x2 − 100y2z2
•
9x − 60
•
9x2 + 60 xyz − 100y2z2
− 100yz
5. Rationalize the denominator of
•
−
). Assume that all the variables are positive.
. Simplify the answer.
55
2012 UPCAT, ACET, DLSUET, USTET @ www.fb.com/Mathalino
56
• • • + 2 to make it easy to graph using a translation. Describe the graph. 6. Rewrite y = − Find the domain and the range of the function.
•
y = −5
+ 2; it is the graph of y = −5
translated
unit left and 2 units up.
domain: x ≥ − , range: y ≤ 2 •
y = −5 (
+
); it is the graph of y = −5
translated
unit left and
units up.
domain: x ≥ − , range: y ≤ −2 •
y = −5
+ 2; it is the graph of y = −5
translated
unit right and 2 units up.
domain: x ≥ − , range: y ≤ −2 •
y = −5(
−
); it is the graph of y = −5
domain: x ≥ − , range: y ≤ −2 7. Let f(x) = x2 + 7 and g(x) =
. Find (f
g) (−4).
• • • •
8. Find the inverse of y = 3x2 + 4. Is the inverse a function?
•
y=±
, no
•
y=±
, no
•
y=±
, yes
•
y=±
, yes
translated
unit left and
units down.
2012 UPCAT, ACET, DLSUET, USTET @ www.fb.com/Mathalino
9. Graph y =
. Find the domain and range of the function.
•
domain: x ≥ 2, range: y ≥ 0
•
domain: x ≥ −2, range: y ≥ 0
•
domain: x ≥ 0, range: y ≥ 2
•
domain: x ≥ 0, range: y ≥ −2 10. Solve (x + 4) = 16.
•
4
•
64
•
60
•
68
−2
57
2012 UPCAT, ACET, DLSUET, USTET @ www.fb.com/Mathalino
11. Rationalize the denominator of
. Assume that all variables are positive.
• • • • 12. Multiply and simplify
•
6x2 •
•
6x2 •
•
6x2
•
6x2 •
•
•
. Assume that all variables are positive.
13. Simplify x−1(x− )(x ). Assume that all variables are positive.
• • • • 14. Simplify
•
3x5y4
•
3|x5|y4
•
3|x5y4|
•
|3x5|y4
. Use absolute value symbols when needed.
15. Rationalize the denominator of
• • • •
3
. Assume that all variables are positive.
58
2012 UPCAT, ACET, DLSUET, USTET @ www.fb.com/Mathalino
16. Multiply 5
•6
•
3
•
180
•
9
•
360 17. Simplify 7
•
−3
•
3
. Simplify if possible.
−2
.
−4−2
• •
−2
59
−4 18. Let f(x) = 3x + 7 and g(x) = 4x − 2. Find (f
•
−46
•
−72
•
−71
g) (−6).
•
−47 19. For the function f(x) = (4 − 2x)2, find f−1 and the domain and range of f andf−1. Determine whether f−1 is a function.
•
f−1(x) = 2 ± , domain of f: all real numbers, range of f: f(x) ≥ 16, domain of f−1: x ≥ 16, range of f−1: all real numbers; function
•
f−1(x) = 2 ± , domain of f: all real numbers, range of f: f(x) ≥ 0, domain off−1: x ≥ 0, range of f : all real numbers; not a function −1
•
f−1(x) = ± , domain of f: all real numbers, range of f: f(x) ≥ 4, domain off−1: x ≥ 4, range of f : f(x) ≤ 0; function −1
•
f−1(x) = ± , domain of f: all real numbers, range of f: f(x) ≥ 16, domain of f−1: x ≥ −1 16, range of f : all real numbers; not a function
20. Rationalize the denominator of
• • •
. Assume that all variables are positive.
2012 UPCAT, ACET, DLSUET, USTET @ www.fb.com/Mathalino
60
•
Radical Functions and Rational Exponents 1. Multiply (7 + •
CORRECT: 51 + 14
2. Simplify •
−2
CORRECT: −14 4. Simplify
•
. Assume that all variables are positive.
CORRECT: x− y 3. Simplify −2
•
) 2.
−4
.
− 12
. Assume that all variables are positive.
CORRECT: 4a2b 5. Solve (x + 4) = 16.
•
CORRECT: 60 6. The formula for the volume of a sphere isV = a sphere with a volume of 18 in.3.
•
πr3. Find the radius, to the nearest hundredth, of
CORRECT: 1.63 in. 7. Simplify x−1(x− )(x ). Assume that all variables are positive.
•
•
CORRECT: 8. Solve = x − 4. Check for extraneous solutions. CORRECT: 6 9. Rationalize the denominator of
•
•
. Simplify the answer.
CORRECT: 10. Find all the real cube roots of 0.000064. CORRECT: 0.04
2012 UPCAT, ACET, DLSUET, USTET @ www.fb.com/Mathalino
11. Which function matches the graph?
•
CORRECT: y =
−5
12. Multiply and simplify •
•
•
CORRECT: m2 13. Identify the graph of y = 2x2 − 5 and its inverse.
CORRECT: 14. Find the real-number root of
•
.
CORRECT: 0.5 15. Simplify 6
•
. Assume that all variables are positive.
−7
.
CORRECT: −4 16. Let f(x) = 3x2 + 10x − 8 and g(x) = x + 4. Find
•
•
.
CORRECT: −9x + 6, all real numbers x ≠ 4 17. Graph y = 5 + 1. Find the domain and range of the function.
CORRECT:
domain: x ≥ 0, range: y ≥ 1
61
2012 UPCAT, ACET, DLSUET, USTET @ www.fb.com/Mathalino
18. Simplify •
• •
. Use absolute value symbols when needed.
CORRECT: 4|k|5 19. Which function matches the graph?
CORRECT: y =
+4
20. Find the real-number root of
•
CORRECT:
.
62
2012 UPCAT, ACET, DLSUET, USTET @ www.fb.com/Mathalino
Part 2/2
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