Download Jan 26, 2004 - Angelo State University

Name _________________________________ Math 1302 – Short Quiz #1 – Jan 26, 2004
1.
Identify each set
a) { 0, 1, 2,3 , .... } → ____________________________________
b) { m/n | m and n are integers with n ≠ 0 } → _______________________
2. Give me an example of
a ) a real number that is not rational → ______________________
b) an integer that is not a whole number → ____________
c) a whole number that is not a natural number → _____________
3.
Which of these is the best example of the commutative law of addition ?
4. Use the distributive law to complete the right side of the expression
5. Find the GCF of 12 and 20. → GCF ( 12, 20 ) = ___________
6. What is the LCM of 12 and 16 → LCM ( 12, 16) = _______________
7. What is the sum of the smallest three prime numbers ? _____________
2 ( x + y ) = __________________
Name __________________________________________________ Math 1302 – Quiz #2 – Jan. 28, 2004
1. Find each of the following absolute values. Exact values are required.
a) | - 3 – ( - 4 ) | = ___________
2.
b) | 5 -
13 | = __________________
How many terms does the following literal expression have ? 3x + 2xy + y2 + 3 → ________________
How many factors does 3xy have ? ________________
4. Simplify the following
a) 3 – 2 ( x – 2 ) = ____________
b) ( 2x) ( 3x ) = ________
c) ( - 4 )o = ________
d) ( 0 5 ) = ________
e) - 42 = _________
5.
More problems
a) ( - ¾ )2 = _________
6.
12 x 2
= __________
4x
b) ( - ¼)-2 = ____________
Name _______________________________________ Math 1302 – Quiz #3 – Feb. 2, 2004
1.
Complete the exponent rules (formulas)
a) xn • xm = __________
2.
b) ( xn ) m = _________
Use the rules of exponents to simplify. Leave answers with nonnegative exponents.
a)
( 2xy-2 ) -4 = ____________
b)
2 x 2 y 3
= _____________
6 x 3 y 1
c)
6 xy 3
2 x 1
•
= __________
3xy  2
2  2 xy
d)
x2/3 • x1/4 = __________
e) ( 3x1/2y1/3 )6 = ___________
Name _________________________________ Math 1302 – Quiz #4 – Feb. 4, 2004
1. Complete by using the rules of exponents.
x1/ 2
x1/ 3
= ___________
2. What are the two square roots of 64 ? ________________________
3. How many square roots does the number – 9 have in the real number system ? ___________
4. What are the
principal nth roots of
a) 81 , n = 4 ( fourth roots) → ____________________
b) - 64 , n = 3 ( cube roots ) → _______________
c) - 25, n =2 (square roots) → ________________
5.
Simplify each of the following radicals
a)
3
d)
 27 = _________
4
x 12 y 4 = __________
b)
121 = ________
c) x12 = _________
Name _________________________________________ Math 1302 – Short Quiz #5 – Feb. 9, 2004
Simplify each of the following radicals.
a)
45 = __________
b)
3
16x 4 = _____________
c) 8 - 2 3 = _______________
6
= ____________
8x
d)
6
e)
x 2 y 4 = _____________
3
f)
= ____________
6x
g)
3
2
= ___________
3x 2
Name ___________________________________ Math 1302 – Quiz #6, Feb. 11, 2004
1. What is the degree of each of the following polynomials ?
a) 2x – 3 → ________
b) 4 → ______
c) 4x3y5 + 2x7 + 212 → ______
2. Which of these are polynomials ? (circle answer )
1/3
5
– x/y
3x2y-2 + 3
3x
21/2y
all are
none of them are
3. Simplify – combine similar terms
a) 3 – 2x ( x – 2 ) = _______________________
b) ( 5x – 2y) ( 5x + 2y ) = _____________________
c) ( x + 3y)2 = ______________________________
4. Factor each of the following polynomials.
a) x + 2xy = ___________________
b) x2 – 9y2 = ___________________
c) 3 ( x – 2y) + 5x( x – 2y ) = ___________
d)
e)
2x3 - 18xy2 = __________________________
x2 + 16 = _________________
5. Simplify the following radicals(leave answer as a single radical).
a)
c)
3
3
4 = __________
1
= _______
4
2 •
b)
d) 4 + 3
3
3
2 = _____________
8 = _________
e)
 4 = ________
Name ___________________________ Math 1302 – Quiz #7 ( maybe 8) – Feb. 23
1.
3
 8 = ____________
2. – a2 = __________ if a = - 3
3.
Factor.
a) x (x – y) - 2a ( x – y ) = __________________
b) 4x2 - 3x - 10 = _____________________
c)
x2 – xy - x + y = __________________
d) x2 + 10x + 25 - y2 = ___________________
4.
Simplify the following fractions by factoring and reducing.
a)
b)
c)
490 / 700 = _________
x2  2
= _____________
x4  4
x 2  10 x  16
= ___________
x 2  2x
Name __________________________________ Math 1302 – Quiz – Feb. 25, 2004
Simplify – find common denominator and reduce to simplest form.
1.
a)
1/6 +
5/8 = ____________
c)
4
2x
x
= ____________
4
d)
e)
b)
x2
x5
2x  3
= __________________
x5
2 x
x 1
= ____________
4x
4x  1
2x 1 x  5
+
4 x
x4
2. Use the rules of division and simplify.
2x  6
4x
÷ 2
= ____________
2
x  3x
x 9
a)
b)
x 2  4x  5
x 2  10 x  25
÷
x3 1
x 2  x 1
3. Simplify the following complex fraction
a)
1-
1
= __________
1 x
2
x = _________
b)
2
4
x
2
Name _____________________________ Math 1302 – Quiz #9 – March 1, 2004
1. List the four basic values of in: _________
________
___________ ________
Write in simplest form
i23 = ________
i49 = ________
2. Find
 16 = __________
3
 8 = _______
 20 = ___________
3. Find ( a complex number should always be written in the form a + bi; 2 + 3i, -2 – 5i, ½ - ¼ i,...
a) ( 3- 2i) + ( 4 – i ) = _____________________
b) ( -3 + 2i) - ( -2 – 3i) = _____________________
c) ( 2 + i) • ( 1 + 2i) = _______________________
4. Graph the numbers
a) the real number: - 16
b) the complex numbers:
1) 3 + 2i
2) - 16 + 0i
Name _________________________________ Math 1302 – Quiz #10 – March 3, 2004
1. Find
a) ( 3 -2i) •(3 +2i ) = _____________
b)
2i
= __________
2i
2. Find x and y.
a) x + 2y + yi = 4 – 2i → x = ________
y = _________
b) x + 2yi = 4 + ( 1-x)i → x = ________
3.
y = ____________
Find the conjugate of
-2i → _________
4+3i → __________
4 – 0i → ______________
4. What is the real part of - 2 – 7i → ____________
5. What is the imaginary part of 4-5i → __________
6. What is the modulus ( actual value) of the complex number 12 – 5i → ____________
7.
Write the quadratic equation 2x2 – 4 = 5x in standard form → _____________________
8. Find the value of a = ________
b= __________ c = __________ of 4x = 1 – x2
9. Find the solution of 4 (x – 3 ) = 0 → ________________
Name __________________________________ Math 1302 – Quiz #11, March 8, 2004
1. State the quadratic formula used to solve equations of the form ax2 + bx + c = 0
2. Write the following quadratic equation in standard form 2x2 = 1 – 3x
→ _____________________________
c = _______
3. Find the solution of each of the following quadratic equations. The solutions may be imaginary.
a) x(x +4 ) = 5 →
b) x2 = 2x →
x = ____________________________
x = _____________________
c) 4x2 + 9 = 0 →
x = __________________
4. What must be added to make the following polynomial a perfect square ?
x2 + 12 x + __________
x2 - __________ + 25
5. Solve the following equation by completing the square.
2x2 + 9x + 9 = 0
1) divide by 2 → _________________________
2) move the constant to the right side → _______________________
3) add ? to both sides → ____________________________________
more:
6. Find a, b, c in the equation 4x2 – 5x - 2 = 0 → a = _____ , b = _______, c = ______
Quiz #12
Name ________________________________ Quiz #13 Math 1302 – March 31, 2004
1. Find the solution of
| x| =2
→ ______________________
2. Find the solution of
| 3 - 2x | = - 4 → __________________
3. Find the solution of
→ ___________________
|x|< 3
4. Solve.
| 2 -x | < 4
→ __________________________
5. What is the solution of the inequality
| 3 + 2/3 x | > - 4 → ____________________
6. Identify as functions or just relations.
________________________
b) y2 = 1 – x2
a)
1
4
____________________
2
3
c)
1
____________
7. Sketch the graph of
y=| x–2|
x + 2y = 4
Name __________________________________ Math 1302 – Quiz #14 April 7, 2004
1) f(x) = x – 3
g(x) = x2 + 1
a) (f + g) ( 2) = ____________________
b)
( g • f ) ( 1 ) = ___________________
c)
(f / g ) ( 0 ) = _________________
d)
(g o f ) (2) = g ( f (2) ) = ______________
2. Sketch the graph of f (x) = - 2x2 - 8x + 1
Name ____________________________________ Math 1302 – Quiz # 15, April 12, 2004
1.
The line x = 3 has slope m = ? ______________
2. A line that is horizontal will always have slope = ? ______________
3. Find the slope of the line 2x – 4y = 1.
m = _________________
4. What is the y-intercept of y = x2 – 2x + 3 ? → ________________
5. What is(are) the x-intercepts of y = x 2 - 2x - 3 → __________________
6. Find the vertex of y = x2 – 2x – 3 → ___________________
7. Graph the function y = x2 – 2x - 3
9. True or false.
________________ a) all functions are relations
________________ b) all lines are functions
10. What is the domain of
a) y =
x
→ ______________________________________________
x3
b) y = 3 → _________________________________________________
11. What is the range of
a) y = x2 -2x – 3 → ________________________________
b) y = | x + 2 | → ______________________________
Name _________________________________ Math 1302 – Quiz #16, April 19, 2004
1) How many solutions (roots ) does the following equation have ?
3x4 + 2x3 + 2x + 1 = 0 → _____________________
How many are positive ? → _____________
2) How many solutions does P(x) = 0 have if P(x) = x7 – 1 = 0 ? ____________________
How many are positive ? __________
How many are negative ? ________
3) Use synthetic division to find
( x2 + 3 ) ÷ ( x + 2 )
4) Find the remainder of (x20 - 2x + 3 ) ÷ ( x + 1 ) . remainder = _________
5) IF x = 2 is known to be a solution of x3 – 3x - 2 = 0 , then find the other two solutions.
Name _____________________________________ Math 1302 – Quiz #17 – April 21, 2004
1.
How many negative roots does P(x) = 0 have if P(x) = x4 – 2 = 0 ? _____________
What are they ? _____________
How many positive ? _________
What are they ? ___________
Find the remaining roots. ________________
2. If 2, -3, and 1 are the only roots of P(x) = 0, then what is the degree of P(x) ? ____
Find P(x) . ____________________________
3. IF - 2i, 3 +
4.
2 , and 5 are solutions of P(x) = 0, then P(x) must be of degree ___ or more.
Given P(x) = x8 + 2x + 1. How many
a) roots does P(x) = 0 have ? _________
b) rational roots ? ___________
5. Given 2x4 + 3x – 5. List all possible rational numbers that should be tested to find the rational
roots.
___________________________
How many roots are negative ? ____________
6. Find all of the roots of x3 – 2x + 1 = 0
7.
Find (2x – y)4. _____________________________________
Name _________________________________ Math 1302 - Quiz #18 -- April 26, 2004
1. Solve 2 – 3(x+1) ≥ 1
2.
Sketch the graph of x + 2y = 4
4. Sketch the graph of x + 2y ≤ 4
5. Sketch the graph of x > 3
6. Sketch the graph of
x + 2y ≤ 4
x≥3
3.
Sketch the graph of x = 3
Answers Quiz #1
1. a) Whole Numbers,
3,
7, π
b) rational numbers
2.
a)
b) -3, -13, - 2, ...
c) 0
3.
None → a + b = b +a is the commutative law of addition
4. 2(x + y ) = 2x + 2y
5. GCF (12, 20 ) = 4
6. LCM (12, 16) = 48
7. Prime Numbers → 2, 3, 5, 7, → sum of first three = 10
Answers Quiz #2
1. Find each of the following absolute values. Exact values are required.
a) | - 3 – ( - 4 ) | = ___________
ans. | - 3 + 4 | = | 1 | = 1
2.
13 | = __________________
b) | 5 -
13
ans. 5 -
How many terms does the following literal expression have ? 3x + 2xy + y2 + 3 → ________________ ans. 4
How many factors does 3xy have ? ________________ ans. 3
4. Simplify the following
a) 3 – 2 ( x – 2 ) = ____________ ans. 3 – 2x +4 = 7 – 2x
b) ( 2x) ( 3x ) = ________ ans. 6x2
c) ( - 4 )o = ________ ans. – 40 = - 1 but ( -4)o = 1
d) ( 0 5 ) = ________ ans. 05 = 0
e) - 42 = _________ ans. – 42 = - (42) = - 16
5.
More problems
a) ( - ¾ )2 = _________ans. 9/16
6.
12 x 2
= __________ ans. 3x
4x
b) ( - ¼)-2 = ____________ ans. ( -1/4)-2 = ( -4/1)2 = 16
Quiz #3 Answers.
1.
Complete the exponent rules (formulas)
a) xn • xm = __________
b) ( xn ) m = _________
ans. xn + m
2.
ans.
xnm
Use the rules of exponents to simplify. Leave answers with nonnegative exponents.
2xy-2
)
-4
= _______ ans.
y8
16 x 4
a)
(
c)
6 xy 3
16
2 x 1
•
= ____ ans. 2 2
2
2
x y
3xy
2 xy
e) ( 3x1/2y1/3 )6 = ___________
2 x 2 y 3
b)
= __________
6 x 3 y 1
ans.
x2/3 • x1/4 = _____
ans. x11/12
d)
ans. 36x3y2
y2
3x 5
or 729x3y2
Quiz #4
Answers
2) 8 and – 8
1) x 1/6
4)
3) none, there is no real root
a) 3 and -3 are the 2 4th roots of 81 in the real number system ( only ones )
but only one is called the principal 4th root → 3
b) the cube roots of -64, there is only one → - 4
c) the square roots of -25 → there is not one → there is no principal square root of -25 ( not in the real number system)
5.
3
a)
 27 = _________
ans. – 3
d)
4
b)
121 = ________
ans. 11
x 12 y 4 = __________
ans. x3 y ( divide exponents inside radical by the index, 4)
c) x12 = _________
ans. x6
Name _________________________________________ Math 1302 – Short Quiz #5 – Feb. 9, 2004
Simplify each of the following radicals.
a)
45 = __________
b)
3
ans. 3 5
16x 4 = _____________ ans. 2x 3 2x
c) 8 - 2 3 = _______________ ans. 8 – 2
6
= ____________ ans.
8x
d)
6
e)
x 2 y 4 = _____________ ans.
3
f)
3x
2x
6x
2x
= ____________ ans.
6x
g)
3
2
= ___________ ans.
3x 2
3
18 x
3x
3
x y2
3
Quiz # 6 Answers
1. What is the degree of each of the following polynomials ?
a) 2x – 3 → ________
b) 4 → ______
ans.
1,
c) 4x3y5 + 2x7 + 212 → ______
0, 8
2. Which of these are polynomials ? (circle answer )
1/3
5
– x/y
3x2y-2 + 3
3x
21/2y
ans.
1/3
all are
none of them are
21/2y
and
3. Simplify – combine similar terms
a) 3 – 2x ( x – 2 ) = _______________________
b) ( 5x – 2y) ( 5x + 2y ) = _____________________
c) ( x + 3y)2 = ______________________________
answers:
7 – 2x2 ,
25x2 – 4y2 ,
x2 + 6xy + 9y2
4. Factor each of the following polynomials.
a) x + 2xy = ___________________
b) x2 – 9y2 = ___________________
c) 3 ( x – 2y) + 5x( x – 2y ) = ___________
d)
2x3 - 18xy2 = __________________________
e) x2 + 16 = ___________
answers:
(x – 3y)(x + 3y ),
x( 1 + 2y) ,
(x -2y)(3 + 5x)
2x(x-3y)(x + 3y)
Prime
5. Simplify the following radicals(leave answer as a single radical).
a)
c)
3
3
4 = __________
2 •
b)
1
= _______
4
d) 4 + 3
3
3
ans. a)
3
2
b)
6
32
c)
3
2 = _____________
8 = _________
2
2
e)
d) 10
 4 = ________
e) no real solution
Name ___________________________ Math 1302 – Quiz #7 ( maybe 8) – Feb. 23
 8 = ____________ ans. – 2
1.
3
3.
Factor.
2. – a2 = __________ if a = - 3 ans. – (-3)2 = - 9
a) x (x – y) - 2a ( x – y ) = __________________ ans. (x – y) ( x – 2y)
b) 4x2 - 3x - 10 = _____________________
c)
ans. ( 4x – 5) ( x – 2 )
x2 – xy - x + y = __________________ ans. ( x2 –xy) + ( - x + y) = x(x – y) – (x – y)
= (x – y)( x – 1 )
d) x2 + 10x + 25 - y2 = ___________________ ans. (x2 + 10x + 25 ) - y2 = (x + 5)2 – y2
= [(x + 5) – y] [ (x+5) + y ] =
= [ x + 5 - y] [ x + 5 + y ]
4.
Simplify the following fractions by factoring and reducing.
a)
b)
c)
490 / 700 = _________
x2  2
= _____________
x4  4
ans. 490 / 700 = 49 / 70 = 7/ 10
ans.
x2  2
1
= 2
2
2
( x  2)( x  2)
x 2
( x  8)( x  2) ( x  8)
x 2  10 x  16
= ___________ ans.
=
2
x
x( x  2)
x  2x
Name __________________________________ Math 1302 – Quiz – Feb. 25, 2004
Simplify – find common denominator and reduce to simplest form.
1.
a)
1/6 +
x2
2x  3
= __________________
x5
x5
ans.:
( x  2)  (2 x  3)
 ( x  5)
 x 5
=
=
 1
x5
x5
x5
5/8 = ____________
b)
ans.: 4/24 + 15/24 = 19/24
c)
4
2x
x
8 x( x) 8  x 2
= ____________ ans.


4
4x 4x
4x
(4 x)( 2  x) (4 x  1)( x  1) 8 x  4 x 2  (4 x 2  3x  1)
2 x
x 1
d)
= ____________ ans.:


4x
4x  1
4 x(4 x  1)
4 x(4 x  1)
4 x(4 x  1)
=
e)
 8 x 2  11x  1
4 x(4 x  1)
(2 x  1)  x  5 2 x  1  x  5 x  6
2x 1 x  5
+
= ________________ ans.:



4 x
x4
x4
x4
x4
x4
2. Use the rules of division and simplify.
2( x  3)
x( x  3) 2
2x  6
4x
÷ 2
= ____________ ans.

  1/ 2
2
( x  3)( x  3)
4x
4
x  3x
x 9
a)
b)
( x  5)( x  1)
( x 2  x  1)
x 2  4x  5
x 2  10 x  25
1
÷
=
_______
ans.


3
2
2
x 1
x  x 1
( x  1)( x  x  1) ( x  5)( x  5) x  5
3. Simplify the following complex fraction
a)
ans.
1-
1
= __________
1 x
(1  x)  (1)  x
x

or 
1 x
1 x
1 x
2
x = _________
b)
2
4
x
2
ans.
2( x  1)
2x  2
x
x 1



x
4 x  2 2(2 x  1) 2 x  1
Name _____________________________ Math 1302 – Quiz #9 – March 1, 2004
1. List the four basic values of in: _________ ________
ans. i, -1, -i, 1
Write in simplest form
i23 = ________ ans. i3 = -i
___________ ________
i49 = ________ans. i1 = i
2. Find
 16 = __________
3
ans. 4i
 8 = _______
ans.: -2
 20 = ___________
ans. 2i 5
3. Find ( a complex number should always be written in the form a + bi; 2 + 3i, -2 – 5i, ½ - ¼ i,...
a) ( 3- 2i) + ( 4 – i ) = _____________________ ans.: 7 – 3i
b) ( -3 + 2i) - ( -2 – 3i) = _____________________ ans.: - 1 - 5i
c) ( 2 + i) • ( 1 + 2i) = _______________________ ans. 2+5i+2i2 = 5i
4. Graph the numbers
a) the real number: - 16
ans. since it is a real number → use a real line
|
-16
|
0
b) the complex numbers:
1) 3 + 2i
2) - 16 + 0i
3+2i
ans. Use a plane
-16+i
Name _________________________________ Math 1302 – Quiz #10 – March 3, 2004
1. Find
a) ( 3 -2i) •(3 +2i ) = _____________ ans. 32 - (2i)2 = 9 + 4 = 13
b)
(2  i)( 2  i) 4  4i  i 2 3  4i 3 4
2i
= __________ ans.:


  i
2i
(2  i)( 2  i)
5
5 5
4i2
2. Find x and y.
a) x + 2y + yi = 4 – 2i → x = ________
y = _________
ans.: x + 2y = 4 and yi = - 2i → y = -2 so → x + 2(-2) = 4 → x = 8 and y = -2
b) x + 2yi = 4 + ( 1-x)i → x = ________
y = ____________
ans.: x = 4 and 2y = 1-x → 2y = 1 – (4) → x = 4 and y = -3/2
3.
Find the conjugate of
-2i → _________
ans.: 2i
4+3i → __________
4 – 0i → ______________
ans.: 4 – 3i
ans.: 4 + 0i = 4
4. What is the real part of - 2 – 7i → ____________ ans.: real part = -2
5. What is the imaginary part of 4-5i → __________ ans.: imaginary part = - 5
6. What is the modulus ( actual value) of the complex number 12 – 5i → ____________
ans.: | 12 – 5i | = 12 2  (5) 2 = 13
7.
Write the quadratic equation 2x2 – 4 = 5x in standard form → ________ ans.: 2x2 – 5x – 4
8. Find the value of a = ____
b= ____ c = ______ of 4x = 1 – x2
9. Find the solution of 4 (x – 3 ) = 0 → ________________ ans.: x = 3
ans.: a=1, b= 4, c= -1
Name __________________________________ Math 1302 – Quiz #11, March 8, 2004
1. State the quadratic formula used to solve equations of the form ax2 + bx + c = 0
ans.:
 b  b 2  4ac
2a
x=
2. Write the following quadratic equation in standard form 2x2 = 1 – 3x
ans.: 2x2 + 3x – 1 = 0 c = - 1
3. Find the solution of each of the following quadratic equations. The solutions may be imaginary.
ans.: x2 + 4x - 5 = 0 → (x – 5 ) ( x + 1) = 0 → x = 5 or x = -1
a) x(x +4 ) = 5 →
b) x2 = 2x →
ans.: x2 – 2x = 0 → x(x – 2 ) = 0 → x = 0 or x = 2
c) 4x2 + 9 = 0 → ans.: 4x2 = - 9
→ x2 = -9/4 → x = 
3
i
2
4. What must be added to make the following polynomial a perfect square ?
x2 + 12 x + __________
ans.: 36
x2 - __________ + 25
ans.: 10
5. Solve the following equation by completing the square.
1) divide by 2 → x2 + 9x/2 + 9/2 = 0
2x2 + 9x + 9 = 0
2) move the constant to the right side → x2 + 9x/2 = -9/2
3) add ? to both sides → x2 + 9x/2 + (9/4)2 = - 9/2 + 81/16
more:
( x + 9/4)2 = - 9/2 + 81/16 = -72/16 + 81/16 = 9/16
x + 9/4 =  ¾ → x = - 9/4  ¾ → x = -9/4 + ¾ = -6/4 = -3/2 or x = -12/4 = - 3
ans.: -3/2 or -3
6. Find a, b, c in the equation 4x2 – 5x - 2 = 0 → a = _____ , b = _______, c = ______
ans.: a = 4
b=-5
c=-2
Name ________________________________ Quiz #13 Math 1302 – March 31, 2004
1. Find the solution of
| x | = 2 → ______________________
ans.:
| x | = 2 means that x = 2 or x = - 2
2. Find the solution of
| 3 - 2x | = - 4 → __________________
answer: absolute value can not be negative → no solution.
3. Find the solution of
4. Solve.
→ ___________________
ans.: |x | < 3 means that - 3 < x < 3
|x|< 3
→ __________________________
| 2 -x | < 4
answer: - 4 < 2 – x < 4
→ -6<-x<2 →
6>x>-2
| 3 + 2/3 x | > - 4 → ____________________
5. What is the solution of the inequality
ans.: the absolute is never negative ( nonnegative ) → all real numbers will work.
6. Identify as functions or just relations.
________________________
a)
1
b) y2 = 1 – x2
4
____________________
2
3
1
c)
____________
b) relation (y2 )
a) function
c) relation
7. Sketch the graph of
y=| x–2|
x + 2y = 4
-- 2
|4
|
|
Name __________________________________ Math 1302 – Quiz #14 April 7, 2004
1) f(x) = x – 3
g(x) = x2 + 1
a) (f + g) ( 2) = ____________________
ans.: (f +g) (x) = x2 + x – 2 → (f + g)(2) = (2)2 + ( 2) – 2 = 4
b)
( g • f ) ( 1 ) = ___________________
ans.: (g •f) (x) = (x -3 )(x2 +1 ) → (g • f ) (1) = ( 1 -3)(12 +1) = - 4
c)
(f / g ) ( 0 ) = _________________ ans.: (f/g)(x) =
d)
(g o f ) (2) = g ( f (2) ) = ______________
ans.: g ( f(x) ) = g ( x – 3 ) → g (f ( 2) ) = g( 2 -3) = g( -1) = (-1)2 + 1 = 2
x 3
→ (f /g ) (0 ) = - 3/1 = - 3
x 2 1
2. Sketch the graph of f (x) = - 2x2 - 8x + 1
ans.: this is a parabola that opens downward
with vertex at V(x,y) where x = - b/2a
-- 9
x = - b/2a = - ( -8) / (- 4) = - 2
y = f( -2) = - 2( -2)2 – 8 (-2) + 1
| -2
= - 8 + 16 +1 = 9
Vertex at ( - 2, 9 ) opens downward.
Name ____________________________________ Math 1302 – Quiz # 15, April 12, 2004
The line x = 3 has slope m = ? ______________ ans.: undefined – it’s a vertical line
1.
2. A line that is horizontal will always have slope = ? ______________ zero
3. Find the slope of the line 2x – 4y = 1. m = ____
ans.: write in the form y = mx + b → y = ½ x – ¼ → m = 1/2
4. What is the y-intercept of y = x2 – 2x + 3 ? → ________________ ans.: 3
5. What is(are) the x-intercepts of y = x 2 - 2x - 3 → __________________
ans.: x2 – 2x - 3 = 0 → ( x – 3) (x + 1) = 0
6. Find the vertex of y = x2 – 2x – 3 → ___________________
x = - b/2a = - ( -2)/ 2(1) = 1 → y = 1 -2 – 3 = - 5 V(1, -5)
7. Graph the function y = x2 – 2x - 3
vertex at V( 1, -5) opens upward
with x-intercepts: -1, 3
y-intercept: - 3
9. True or false.
________________ a) all functions are relations ans.: true → but not all relations are functions.
________________ b) all lines are functions → ans.: false – vertical lines are not
10. What is the domain of
a) y =
x
→ ___________________________ ans.: all real numbers except x = - 3
x3
b) y = 3 → _________________________________________________ ans.: all real numbers
11. What is the range of
a) y = x2 -2x – 3 → ________________________________ all real numbers y ≥ - 5
b) y = | x + 2 | → ______________ ans.: all real numbers y ≥ 0
Name _________________________________ Math 1302 – Quiz #16, April 19, 2004
1) How many solutions (roots ) does the following equation have ?
3x4 + 2x3 + 2x + 1 = 0 → _____________________ ans.: four
How many are positive ? → _____________ ans.: none
2) How many solutions does P(x) = 0 have if P(x) = x7 – 1 = 0 ? __________ ans.: seven
How many are positive ? __________
ans.: 1 – there is one sign variation
3) Use synthetic division to find
-2 1
______
0
-2
-2
( x2 + 3 ) ÷ ( x + 2 ) =
3
4
---------------- →
1
How many are negative ? ________
ans.: NONE -- P(-x) has no sign variations
x -2 +
7
x2
7
4) Find the remainder of (x20 - 2x + 3 ) ÷ ( x + 1 ) .
ans.: remainder = P( -1) = (-1)20 - 2 (-1) + 3 = 1 + 2 + 3 = 6
5) IF x = 2 is known to be a solution of x3 – 3x - 2 = 0 , then find the other two solutions.
ans. Use synthetic division to get the following depressed equation.
2 | 1
0
-3
-2
2
4
2
--------------------------1
2
1
0
→ x2 + 2x + 1 = 0 contains the other two solutions.
(x + 1) ( x +1 ) = 0 → x = - 1, - 1
So the three solutions are: -1, -1, and 2
Name _____________________________________ Math 1302 – Quiz #17 – April 21, 2004
1.
How many negative roots does P(x) = 0 have if P(x) = x4 – 2 = 0 ? _____________
What are they ? _____________
How many positive ? _________
What are they ? ___________
Find the remaining roots. ________________
answer: P(x) has four roots - one is negative, one is positive →  4 2
2. If 2, -3, and 1 are the only roots of P(x) = 0, then what is the degree of P(x) ? ____
Find P(x) . ____________________________
answer: three roots → degree is three and P(x) = ( x -2) (x + 3 ) ( x -1 )
3. IF - 2i, 3 +
2 , and 5 are solutions of P(x) = 0, then P(x) must be of degree ___ or more.
Since some roots come in pairs- we do not have three roots – we have five roots – so
P(x) must be of degree ≥ 5.
4.
Given P(x) = x8 + 2x + 1. How many
a) roots does P(x) = 0 have ? _________ → 8 roots
b) rational roots ? ___________ →
the only possible rational roots are  1 and only -1 works. Thus, only one rational root
5. Given 2x4 + 3x – 5. List all possible rational numbers that should be tested to find the rational
roots.
___________________________ ans.: p = 5 → 1, 5
q = 2 → 1, 2
p/q =  1,  5,  1/2,  5/2 ---- these are the only rational numbers that should be tested
How many roots are negative ? ______ P( - x ) = 2x4 -3x – 5 → 1 sign variation → 1 neg. root
6. Find all of the roots of x3 – 2x + 1 = 0
ans.: P( 1) = 13 – 2(1) + 1 = 0 -- that’s one root – need the other two – use synthetic division
1 | 1
0 -2 1
1 1 -1
-------------------1
7.
1
-1
0 →
x2
+ x - x =0 →
x=
 (1)  (1) 2  4(1)(1)
Find (2x – y)4. _____________________________________
2(1)
pril 26, 2004
1. Solve 2 – 3(x+1) ≥ 1
ans.: 2 – 3x – 3 ≥ 1 → -3x ≥ 2 → x ≤ -2/3
2.
Sketch the graph of x + 2y = 4
3.
Sketch the graph of x = 3
-2
/4
3
|
4. Sketch the graph of x + 2y ≤ 4
shaded
5. Sketch the graph of x > 3
shade
shade
shade
shade
6. Sketch the graph of
x + 2y ≤ 4
x≥3