STUDENT NAME CLASS DAYS/TIME - ProfessorTrimble.com · RADICALS x 3x 4x 5x (8 x3 64 64 12 2....

MATH 102, COLLEGE ALGEBRA UNIT 1 LECTURE NOTES

JILL TRIMBLE, BLACK HILLS STATE UNIVERSITY

STUDENT NAME_________________________________

CLASS DAYS/TIME________________________________

of 59 Math102 Unit 1 Initials__________________

ALL ABOUT EXPONENTS!

2x

Basic Rules of Exponents

0x

1x

2x

a bx x

a

b

x

x

b

ax

a

xy

Math102 College Algebra Unit 1 Outcome/Homework 1

Students will be able to use exponent rules, simplify radical expressions

on examinations, quizzes, and homework problems. (E-text Section P.2, P.3)


a

x

y

2 55 5

2

5

3

3

2x y 2

x y

4

3x

3 5 34 2x y x y

CAUTION!


2 5

5 3

28

21

x y

x y

RADICALS

x 3 x 4 x 5 x ( 8 3x

64 64

2

12


2

38

1

15 25 5243x y

83 1

4 4

1

4

x y

x

3 16


LINEAR EQUATIONS AND INEQUALITIES

Slope =

y

x x

y


Students will be able to find the slope of a line, graph linear equations,

find the intercepts, as well as solve linear inequalities on examinations,

quizzes, and homework problems. (E-text Section 2.3, 1.7)


Find the slope between the coordinates 6,2 & 7,4

Find the slope between the coordinates 9, 2 & 7, 2


Find the slope between the coordinates 6,8 & 6, 7

Negative slope

y

x x

y


Slope ,0a & 0, b

y

x x

y

EQUATIONS OF LINES

Point-Slope Form ⇒

Slope-Intercept Form ⇒


Find the equation of the line: slope = -3 passing through 5, 2

Point-slope form:

Slope-intercept form:

y

x x

y

SOLVING LINEAR EQUATIONS

𝒙 + 𝟑 = 𝟕


𝟐𝟖 − 𝒙

𝟑=

𝒙

𝟒

*HINT: Clear the Fraction

INTERVAL AND SET-BUILDER NOTATION

Interval : (𝟐 , 𝟕)

[𝟐 , ∞)

Set-Builder Notation:

(𝟐 , 𝟕)

[𝟐 , ∞)

SOLVING LINEAR INEQUALITIES

𝟓𝒙 − 𝟖 ≥ 𝟐𝟐

| | | | | | |

1 2 3 4 5 6 7

| | | | | | |2 3 4 5 6 7 8


𝒙 + 𝟐 ≥ 𝒙 + 𝟑

Graph: −𝟐 ( 𝒙 + 𝟑) > 𝟔𝒙 + 𝟐

SPECIAL CASES

𝟑 (𝒚 + 𝟐) − 𝟏𝟎𝒚 > 𝒚 − 𝟒 (𝟐𝒚 + 𝟏) − 𝟏

*__________ Statement: The solution is ________________________________.

y

x x

y


𝟒𝒙−𝟒

𝟕 ≤

𝟖 (𝒙−𝟐)

𝟏𝟒

*______________ Statement: ________ solution.


PARALLEL AND PERPENDICULAR LINES

Parallel Lines =

Perpendicular Lines =

𝑳𝟏 = 𝟐

𝟑𝒙 + 𝟕


Students will be able to find an equation of a parallel or perpendicular

line as well as interpret the meaning of the slope of a line on

examinations, quizzes, and homework problems.

(E-text Section 2.4)

y

x x

y


Write an equation for line L in point-slope form and slope-intercept form.

L is parallel to 𝒚 = 𝟐𝒙.

Point Slope Form:

Slope-Intercept Form:

(𝟏, −𝟑)

L

y

x x

y

𝒚 = 𝟐𝒙

y

x x

y


Write an equation for Line L in point-slope form and slope-intercept form.

L is perpendicular to 𝒚 = 𝟑𝒙.

Point Slope Form:

Slope-Intercept Form:

y

x x

y

(𝟏, 𝟑)

𝒚 = 𝟑𝒙

L


Write the slope-intercept equation of the function f whose graph satisfies the given

conditions. The graph of f passes through (−𝟑, 𝟓) and is perpendicular to the line

whose equation is 𝒙 = −𝟗.

The equation of the function is 𝒇(𝒙) = _______.


Write the slope-intercept equation of the function f whose graph satisfies the given

conditions. The graph of f passes through (−𝟔, 𝟓) and is perpendicular to the line that

has an x-intercept of 1 and a y-intercept of -3.

The equation of the function 𝒇(𝒙) = _______________________.


COMPLEX NUMBERS

𝒊 = √−𝟏𝟔 ⇒

𝒊𝟐 =

𝒊𝟑 =

𝒊𝟒 =

𝒊𝟓 =

. . .

𝒊𝟑𝟏 =

STANDARD FORM

𝒂 + 𝒃𝒊 𝟐 + 𝟑𝒊


Students will be able to add, subtract, multiply and divide complex

numbers on examinations, quizzes, and homework problems.



ADD/SUBTRACT COMPLEX NUMBERS

(𝟕 − 𝟕𝒊) + (𝟏 + 𝟒𝒊) 𝟖𝒊 − (𝟗 − 𝟔𝒊)

MULTIPLYING COMPLEX NUMBERS

𝟖𝒊 (𝟕 − 𝟔𝒊)

(−𝟐 + 𝒊)(−𝟐 − 𝒊) *FOIL

(𝟒 + 𝟗𝒊)𝟐 *FOIL


DIVIDE COMPLEX NUMBERS

*Multiply by the ______________ of the ______________________.

40𝑖

3 + 𝑖

6 + 3𝑖

7 + 3𝑖

4√−9 + 3√−64


Evaluate 𝒙𝟐+𝟏𝟐

𝟏−𝟐𝒙 for 𝒙 = 𝟐𝒊.

3

3+3

𝑖

*common denominator


(−7 − √−6)2

√−12 (√−5 − √7 )


SOLVING QUADRATIC EQUATIONS

Quadratic Equation:

Quadratus is Latin for “___________________”

𝑥2 + 2𝑥 + 3 = 2

2𝑥2 = 0

***QUADRATIC FORMULA***

𝒙𝟐 − 𝟐𝒙 − 𝟐𝟒 = 𝟎

SOLVE BY GRAPHING: SOLVE BY FACTORING:


Students will be able to solve quadratic equations on examinations,

quizzes, and homework problems.



𝒙𝟐 = 𝟒𝒙 + 𝟏𝟐 Solve by 1st: Set __________ to ______.

2nd: Zero _________________________.

*NOTE: x-intercpet = “________________” = “__________”

Solve 𝟒𝟓 − 𝟒𝟓𝒙 = (𝟕𝒙 + 𝟒)(𝒙 − 𝟏)


Solve Using the Square Root Property:

∗ 𝟓𝒙𝟐 = 𝟖𝟎

x = _______

MyMathLab: ____________________

∗ 𝟐 (𝒙 + 𝟐)𝟐 = 𝟗𝟎

x = __________________

MyMathLab: _______________________________


Solve (𝒙 + 𝟓)𝟐 = −𝟒𝟗

x = _______________

MyMathLab: _____________________________

Solve (𝟔𝒙 − 𝟒)𝟐 = 𝟑𝟓

x = ____________________

MyMathLab:


COMPLETING THE SQUARE


RADICAL EQUATIONS

√𝒙 = √𝒙𝟑

= 𝒙𝟑

𝟒 =

**EXTRANEOUS ROOTS MUST ______ _____ SOLUTIONS!!!

√𝟏𝟓 − 𝟐𝒙 = 𝒙

Graph: √𝟏𝟓 − 𝟐𝒙 − 𝒙 = 𝟎

Steps for Solving this Type of Problem: 1. _________ both sides √𝟏𝟓 − 𝟐𝒙 = 𝒙

2. Set equal to ___

3. Solve by ____________

4. __________________ Property

5. Have _________ Solutions

6. Check for ________________________________!


Students will be able to solve radical equations with one or two radical

terms on examinations, quizzes, and homework problems.



√𝒙 + 𝟖 = 𝒙 − 𝟒

Steps for Solving this type of Problem:

1. _________ both sides

2. ________/FOIL the ________

3. Set ________ to 0

4. __________ like terms

5. ________ or ____________ Formula

6. This Case: Zero-__________ Property

7. Have ___________ solutions

8. ________ for ___________ roots!

𝒙 = _______ is the solution!

Graph √𝒙 + 𝟖 = 𝒙 − 𝟒


√𝒙 + 𝟐𝟕 − √𝒙 − 𝟔 = 𝟑

Graph

Steps for Solving this Type of Problem:

√𝒙 + 𝟐𝟕 − √𝒙 − 𝟔 = 𝟑 1. __________ the radical

2. ________ both sides

3. Expand/_______

4. __________ the radical

5. __________ both sides

6. Have __________ solution

7. CHECK for ________________________!


𝒙 = _____ is the solution!

√𝟐𝒙 + 𝟑 + √𝒙 − 𝟐 = 𝟐

Graph

Graph indicates _____________________.

But we MUST _____________!


CHECK for ________________ roots:

There is ______________________!

𝒙𝟓/𝟐 = 𝟑𝟐

Graph Solving: 𝒙𝟓

𝟐 = 𝟑𝟐


𝒙 = _______

(𝒙 − 𝟐)𝟑/𝟐 = 𝟏𝟐𝟓

Graph

Solving: (𝒙 − 𝟐)𝟑/𝟐 = 𝟏𝟐𝟓


Graph the vertex minimum and maximum of each equation

𝒚 = 𝒙𝟐 + 𝟐𝒙 − 𝟖 𝒚 = −𝒙𝟐 − 𝟐𝒙 + 𝟓

Domain for any parabola opening is all real numbers.

D = _________________

Range will change depending on whether the parabola has a minimum or maximum point.

R = ______________ and _________________

Axis of Symmetry, x= x-value of the vertex

X= ________

The x-intercept crosses the x-axis,

y=________

The y-intercept crosses the y-axis,

x=________


Students will be able to find the vertex, the minimum or maximum value,

the intercepts, and then graph a given quadratic function and to solve

quadratic inequalities on examinations, quizzes, and homework

problems.



Formulas

Vertex of a parabola: 𝒙 = −𝒃

𝟐𝒂

Standard Form: 𝒇(𝒙) = 𝒂(𝒙 − 𝒉)𝟐

+ 𝒌

Vertex: (h,k)

Opens up when a>0

Opens down when a<0

Graphing Quadratics

Steps for solving:

1. Vertex = (𝒉, 𝒌) = ______________

2. Standard form = 𝒇(𝒙) = 𝒂(𝒙 − 𝒉)𝟐 + 𝒌 = _________________

3. Simplified standard form = _________________


Steps for solving

1. Find the vertex = 𝒙 = −𝒃

𝟐𝒂 = ______________

2. Find the y-intercept = ___________

3. Find the x-intercept = ___________

Vertex = ________

Standard Form = ____________________


𝒂 = ____

𝒃 = ____

𝒄 = ____

𝒙 = −𝒃

𝟐𝒂= ____________


Vertex = (h,k) = ____________

y-intercept = ______________

x-intercept = ______________

Domain = _______________

Range = ________________


⇔ 𝑫𝒐𝒎𝒂𝒊𝒏: _______________

⇕ 𝑹𝒂𝒏𝒈𝒆: _______________

𝑺𝒕𝒆𝒑𝒔 𝒇𝒐𝒓 𝑺𝒐𝒍𝒗𝒊𝒏𝒈

1. Vertex = (h,k) = __________

2. Axis of symmetry (x-value of the vertex) = x = ____________


__________ Between Two Points (𝒙𝟏, 𝒚𝟏) 𝒂𝒏𝒅 (𝒙𝟐, 𝒚𝟐)

D = √(𝒙𝟐 − 𝒙𝟏)𝟐 + (𝒚𝟐 − 𝒚𝟏)𝟐

____________ Midpoint Between Two Points (𝒙𝟏, 𝒚𝟏) 𝒂𝒏𝒅 (𝒙𝟐, 𝒚𝟐)

𝑴 = (𝒙𝟏 + 𝒙𝟐

𝟐,𝒚𝟏 + 𝒚𝟐

𝟐)

Equation of a ___________ in Standard Form

(𝒙 − 𝒉)𝟐 + (𝒚 − 𝒌)𝟐 = 𝒓𝟐

𝒄𝒆𝒏𝒕𝒆𝒓 (𝒉, 𝒌) 𝒓𝒂𝒅𝒊𝒖𝒔 = 𝒓


Students will learn the Distance and Midpoint formulas and will be able

to graph and write equations of a circle on examinations, quizzes, and

homework problems.



𝑭𝒊𝒏𝒅 𝒕𝒉𝒆 𝒅𝒊𝒔𝒕𝒂𝒏𝒄𝒆 𝒃𝒆𝒕𝒘𝒆𝒆𝒏 (𝟑, 𝟒)𝒂𝒏𝒅 (𝟕, 𝟏𝟎)

Steps for solving:

1. Use the distance formula D = √(𝒙𝟐 − 𝒙𝟏)𝟐 + (𝒚𝟐 − 𝒚𝟏)𝟐

2. √(_____ − _____)𝟐 + (_____ − _____)𝟐

3. Simplify the above expression

4. Simplify the radical: ___√_____

𝑭𝒊𝒏𝒅 𝒕𝒉𝒆 𝒅𝒊𝒔𝒕𝒂𝒏𝒄𝒆 𝒃𝒆𝒕𝒘𝒆𝒆𝒏 (−𝟖, −𝟓)𝒂𝒏𝒅 (𝟐, −𝟑)

1. 𝒅 = √(_____ − _____)𝟐 + (_____ − _____)𝟐

2. Simplify your answer


𝑭𝒊𝒏𝒅 𝒕𝒉𝒆 𝑴𝒊𝒅𝒑𝒐𝒊𝒏𝒕 𝒃𝒆𝒕𝒘𝒆𝒆𝒏 (𝟐, 𝟑)𝒂𝒏𝒅 (𝟒, 𝟏𝟎)

Steps for solving:

1. Recall the midpoint formula; 𝑴 = (𝒙𝟏+𝒙𝟐

𝟐,

𝒚𝟏+𝒚𝟐

𝟐)

2. 𝑴 = (____+____

𝟐,

____+____

𝟐)

3. (________

𝟐,

________

𝟐)

4. Simplify M = (___,___)

Recall the equation of a circle; (𝒙 − 𝒉)𝟐 + (𝒚 − 𝒌)𝟐 = 𝒓𝟐

Center = (h,k) = (___,___) Radius = r = ____

(𝒙 − ___)𝟐 + (𝒚 − ___)𝟐 = ____𝟐

Simplify the answer


(𝒙 − 𝒉)𝟐 + (𝒚 − 𝒌)𝟐 = 𝒓𝟐

Center = (h,k) = (___,___) Radius = r = ____

(𝒙 − (___))𝟐 + (𝒚 − (___))𝟐 = ___𝟐

Simplify the answer


(𝒙 − 𝒉)𝟐 + (𝒚 − 𝒌)𝟐 = 𝒓𝟐

Center = (0,0)

𝒓𝟐 = 𝟗 → √𝒓𝟐 = √𝟗 → 𝒓 = ___

⇔ 𝑫𝒐𝒎𝒂𝒊𝒏: [_____, _____]

⇕ 𝑹𝒂𝒏𝒈𝒆: [_____, _____]


𝑻𝒉𝒆 𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝒐𝒇 𝒂 𝒄𝒊𝒓𝒄𝒍𝒆: (𝒙 − ___)𝟐 + (𝒚 − ___)𝟐 = ___𝟐

Center = (_____,_____)

𝒓𝟐 = 𝟏𝟔 → √𝒓𝟐 = √𝟏𝟔 → 𝒓 = ___

⇔ 𝑫𝒐𝒎𝒂𝒊𝒏: [_____, _____]

⇕ 𝑹𝒂𝒏𝒈𝒆: [_____, _____]


𝒙𝟐 + 𝟒𝒙 +

( )𝟐

+ 𝒚𝟐 +

(− )𝟐

= 𝟕𝟑 + _____ + _____

(𝒙 + ___)𝟐 + (𝒚 − ___)𝟐 = _____

(𝒙 − 𝒉)𝟐 + (𝒚 − 𝒌)𝟐 = 𝒓𝟐

Center = (_____,_____)

𝒓𝟐 = 𝟖𝟏 → √𝒓𝟐 = √𝟖𝟏 → 𝒓 = ___

⇔ 𝑫𝒐𝒎𝒂𝒊𝒏: [_____, _____]

⇕ 𝑹𝒂𝒏𝒈𝒆: [_____, _____]



Students will be able to solve applied problems using formulas and

functions as well as models on examinations, quizzes, and homework

problems.

(E-text Sections 1.1, 1.2, 1.3, 1.5, 2.8)


𝒑 + 𝒙

𝟐= 𝟑𝟗

𝒙 = 𝟐𝟎𝟏𝟎 − 𝟏𝟗𝟕𝟎 = _____

𝒑 + _____

𝟐= 𝟑𝟗

→ 𝒑 + _____ = 𝟑𝟗

-_____ -_____

𝒑 = _____

𝒑 + 𝒙

𝟐= 𝟑𝟗 𝒙 = 𝟔𝟔 𝒚𝒆𝒂𝒓𝒔 𝒂𝒇𝒕𝒆𝒓 𝟏𝟗𝟕𝟎

→ ____ + 𝒙

𝟐= 𝟑𝟗 𝟏𝟗𝟕𝟎 + 𝟔𝟔 = 𝟐𝟎𝟑𝟔

- ____ - ____

𝒙

𝟐= 𝟑𝟑

X = __________


Transform words into an equation:

𝒕𝒘𝒊𝒄𝒆 𝒂 𝒏𝒖𝒎𝒃𝒆𝒓 𝒊𝒔 𝒅𝒆𝒄𝒓𝒆𝒂𝒔𝒆𝒅 𝒃𝒚 𝟖 𝒕𝒉𝒆 𝒓𝒆𝒔𝒖𝒍𝒕 𝒊𝒔 𝟐𝟖

→ __________ − _____________ = _____________

Solve for x:

x = ______________


Transform words into an equation:

𝒂 𝒏𝒖𝒎𝒃𝒆𝒓 𝒊𝒔 𝒅𝒆𝒄𝒓𝒆𝒂𝒔𝒆𝒅 𝒃𝒚 𝟑𝟎% 𝒐𝒇 𝒊𝒕𝒔𝒆𝒍𝒇, 𝒕𝒉𝒆 𝒓𝒆𝒔𝒖𝒍𝒕 𝒊𝒔 𝟗𝟖

→ 𝒙 − 𝟑𝟎

𝟏𝟎𝟎(𝒙) = 𝟗𝟖

Combine like terms:

Solve for x:

X= ______________


Let M = Master’s and B = Bachelor’s

Transform words into equations:

____ + ____ = 117 and M = ____ - _____

Substitute M in the second equation into the first equation:

_____ - _____ + _____ = 117

Combine like terms:

Solve for B:

Solve for M:


Recall that profit is _____ (+) and loss is _____ (-)

Complete the table:

Amount invested Interest (rate) Amount earned/lost

______________ ___________ _________________

______________ ___________ _________________

Total _____________ ____________ __________________

→ . 𝟏𝟎𝒙−. 𝟎𝟒𝒙(𝟏𝟓, 𝟎𝟎𝟎 − 𝒙) = 𝟑𝟖𝟎

Distribute and combine like terms

X= ____________ (10% gain) and ___________ (4% loss)


Plug in N = 66

Given 𝑵 = 𝒕𝟐−𝒕

𝟐

→ 𝟔𝟔 = 𝒕𝟐−𝒕

𝟐

Multiply each side by ______

→ (____) 𝟔𝟔 = ( 𝒕𝟐−𝒕

𝟐) (_____)

→ _____ = 𝒕𝟐 − 𝒕

-____ -______

→ _____ = 𝒕𝟐 − 𝒕 - _______

Factor:

→ ( 𝒕 + _______)(𝒕 − ______) = _______

Set each side equal to ________

Solve for t:

→ 𝒕 = _____ and 𝒕 ≠ _____


Formula for volume:

𝒗 = (𝒍)(𝒘)(𝒉)

→ 𝒗 = (𝒙)(𝒙)(𝟑) = _________

→ 𝟑𝒙𝟐 = _______

𝒅𝒊𝒗𝒊𝒅𝒆 𝒃𝒐𝒕𝒉 𝒔𝒊𝒅𝒆𝒔 𝒃𝒚 𝟑

→ __________ = ___________

𝒔𝒒𝒖𝒂𝒓𝒆 𝒓𝒐𝒐𝒕 𝒃𝒐𝒕𝒉 𝒔𝒊𝒅𝒆𝒔

→ 𝒙 = _______


Plot the points on the graph

Distance Formula:

D = √(𝒙𝟐 − 𝒙𝟏)𝟐 + (𝒚𝟐 − 𝒚𝟏)𝟐

D = √(___ − (___))𝟐

+ (___ − (___))𝟐

Simplify:

Answer: _________


Recall the standard form of a circle:

(𝒙 − ____)𝟐 + (𝒚 − ___)𝟐 = ___𝟐

Plug in the values:

2 2 2( (___)) ( (___)) ___x y

Simplify your answer:

2 2 2( ___) ( ___) ___x y


s z pg

___ ____

___s pg

Divide both sides by ______

___ ___

s z pg

__ __

__p


1( )

4c f r z

Multiply both sides by ____

1(___) (___) ( )

4c f r z

Distribute

___ ___ ___c r z

Subtract _____ from both sides

___ ___ ___c r z

_____ _____

___ ___ ___c z r

Divide both sides by ____

___ ___ ___

___ ___

c z r

___ ___

___

c zr


hj

r p

Multiply by sides by (_______)

(___ ___) (___ ___)h

jr p

Distribute

___ ___j j h

Add ____ to both sides ___j ___j

___ ___j h j

Divide both sides by _____

___

___

h jr

STUDENT NAME CLASS DAYS/TIME - ProfessorTrimble.com · RADICALS x 3x 4x 5x (8 x3 64 64 12 2....

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