Date post: | 25-Dec-2015 |
Category: |
Documents |
Upload: | shanon-lewis |
View: | 224 times |
Download: | 1 times |
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt1
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics§1.1 §1.1
ExpressionsExpressions& Real No.s& Real No.s
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt2
Bruce Mayer, PE Chabot College Mathematics
Basic TerminologyBasic Terminology
A LETTER that can be any one of various numbers is called a VARIABLE.
If a LETTER always represents a particular number that NEVER CHANGES, it is called a CONSTANT
A & B are CONSTANTS
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt3
Bruce Mayer, PE Chabot College Mathematics
Algebraic ExpressionsAlgebraic Expressions
An ALGEBRAIC EXPRESSION consists of variables, numbers, and Math-Operation signs.• Some
Examples , 2 2 , .4
yt l w m x b
When an equal sign is placed between two expressions, an EQUATION is formed →
374
2 ty
cmEbxmy
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt4
Bruce Mayer, PE Chabot College Mathematics
Translate: English → AlgebraTranslate: English → Algebra
“Word Problems” must be stated in ALGEBRAIC form using Key Words
per of less than more than
ratio twicedecreased byincreased by
quotient of times minus plus
divided byproduct ofdifference of sum of
divide multiply subtract add
DivisionMultiplicationSubtractionAddition
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt5
Bruce Mayer, PE Chabot College Mathematics
Example Example Translation Translation
Translate this Expression:
Eight more than twice the product of 5 and an Unknown number
SOLUTION• LET n ≡ the UNknown Number
8 2 5 n
Eight more than twice the product of 5 and a number.
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt6
Bruce Mayer, PE Chabot College Mathematics
Evaluate Algebraic ExpressionsEvaluate Algebraic Expressions
When we REPLACE A VARIABLE with a number, we are SUBSTITUTING for the variable. The calculation that follows is
called EVALUATING the expression• Note: the normal result for a “Evaluation”
is usually a SINGLE NUMBER with NO LETTERS
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt7
Bruce Mayer, PE Chabot College Mathematics
Example Example Evaluating Evaluating
Evaluate
8 for 2, 7, and 3.xz y x y z
SOLUTION
8xz − y = 8·2·3 − 7
= 41
= 48 − 7
Substituting
Multiplying
Subtracting
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt8
Bruce Mayer, PE Chabot College Mathematics
Exponential NotationExponential Notation
The expression an, in which n is a counting number (1, 2, 3, etc.) means
n factors
In an, a is called the base and n is called the exponent, or power. When no exponent appears, it is assumed to be 1. Thus a1 = a.
a a a a a
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt9
Bruce Mayer, PE Chabot College Mathematics
Order of Operations (PEMDAS)Order of Operations (PEMDAS)
Perform operations in this order:1. Grouping symbols: parentheses ( ),
brackets [ ], braces { }, absolute value | |, and radicals √
2. Exponents from left to right, in order as they occur.
3. Multiplication/Division from left to right, in order as they occur.
4. Addition/Subtraction from left to right, in order as they occur.
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt10
Bruce Mayer, PE Chabot College Mathematics
Example Example Order of Ops Order of Ops
Evaluate 2 22 3 12 for 2.x x x
SOLUTION
2(x + 3)2 – 12 x2
Substituting
Simplifying 52 and 22
Multiplying and Dividing
Subtracting
= 2(2 + 3)2 – 12 22
2 22 5 12 2 2 25 12 4
Working within parentheses
50 3 47
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt11
Bruce Mayer, PE Chabot College Mathematics
FormulasFormulas
A FORMULA is an equation that uses letters to represent a RELATIONSHIP between two or more quantities.
Example The area, A, of a circle of radius r is given by the formula:
2rA r
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt12
Bruce Mayer, PE Chabot College Mathematics
Example Example Temp Conversion Temp Conversion
The formula for converting the temperature in degrees Celsius (C) to degrees Fahrenheit (F)
F 9
5C 32.
Use the formula to convert 86F to the Celsius form
Substitute 86 for F 86 9
5C 32
Solve for C 5 86 59
5C 32
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt13
Bruce Mayer, PE Chabot College Mathematics
Temperature Conversion cont.Temperature Conversion cont.
Solve for C. 5 86 59
5C 32
430 9C 160
270 9C
270
9C
30 C
Thus 30 °C is the Equivalent of 86 °F
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt14
Bruce Mayer, PE Chabot College Mathematics
Mathematical ModelingMathematical Modeling
A mathematical model can be a formula, or set of formulas, developed to represent a real-world situation.
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt15
Bruce Mayer, PE Chabot College Mathematics
Example Example Math Model Math Model
Mei-Li is 5ft 7in tall with a Body Mass Index (BMI) of approximately 23.5. What is her weight?
SOLUTION1. Familiarize. The body mass index I
depends on a person’s height and weight. The BMI formula:
2
704.5WI
H
– where W is the weight in pounds and H is the height in inches.
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt16
Bruce Mayer, PE Chabot College Mathematics
BMI Example contBMI Example cont
2. Translate. Solve the formula for W:
2
704.5WI
H
2 704.5IH W2
704.5
IHW
3. Carry Out
5 ft 7 in. = 67 in.
2 223.5(67)
704.5 704.5
IHW
150W
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt17
Bruce Mayer, PE Chabot College Mathematics
BMI Example contBMI Example cont
4. Check 5. State Answer
2
704.5WI
H
2
704.5(150)
67I
23.5I
Mei-Li weighs about 150 Pounds
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt18
Bruce Mayer, PE Chabot College Mathematics
Set of Real NumbersSet of Real Numbers
SET ≡ a Collection of Objects
Braces are used to indicate a set. For example, the set containing the numbers 1, 2, 3, and 4 can be written {1, 2, 3, 4}.• The numbers 1, 2, 3, and 4 are called the
members or elements of this set.
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt19
Bruce Mayer, PE Chabot College Mathematics
Set NotationSet Notation
Roster notation: {2, 4, 6, 8}
Set-builder notation: {x | x is an even number between 1 and 9}
“The set of all x such that x is an even number between 1 and 9”
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt20
Bruce Mayer, PE Chabot College Mathematics
Sets of Real NumbersSets of Real Numbers
Natural Numbers (Counting Numbers)• Numbers used for counting: {1, 2, 3,…}
Whole Numbers• The set of natural numbers with 0 included:
{0, 1, 2, 3,…}
Integers• The set of all whole numbers AND their
opposites: {…,−3, −2, −1, 0, 1, 2, 3,…}
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt21
Bruce Mayer, PE Chabot College Mathematics
Sets of Real Numbers, Sets of Real Numbers, xx
Rational Numbers (Integer Fractions)• Maybe expressed as a FRACTION of
two INTEGERS– Terminating Decimals (e.g.; 7/16)
– Repeating NonTerminating Decimals (e.g., 2/7)
Irrational Numbers• Can NOT be expressed as a Fraction of
two Integers– NONterminating, NONreapeating decimals
(e.g., π = 3.1459………………………)
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt22
Bruce Mayer, PE Chabot College Mathematics
4 77 85, ,0, ,9.45, 8.734
Irrational numbers:
10,
3,
,
15,
7.161161116...
Integers: 5 192
8 3 7, , ,3.4
19, 3, 1
Zero: 0
Real Numbers:
2319, 10,0, , ,17.8
Rational Numbers:
19, 1,0,5,23
Rational numbers that are not integers:
NegativeIntegers:
Whole numbers:0, 1, 2, 3, 29
Positive integers or natural numbers:1, 2, 3, 29
Real N
o. F
amily T
reeR
eal No
. Fam
ily Tree
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt23
Bruce Mayer, PE Chabot College Mathematics
Real Number NestReal Number Nest
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt24
Bruce Mayer, PE Chabot College Mathematics
Real Number LineReal Number Line
An Infinite line whose points have been assigned number-coordinates is called the real number line
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt25
Bruce Mayer, PE Chabot College Mathematics
3 Sectors of the Number Line3 Sectors of the Number Line
1. The negative real numbers are the CoOrds to the left of the origin O
2. The real number zero is the CoOrd of the origin O
3. The positive real numbers are the CoOrds to the right of the origin O
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt26
Bruce Mayer, PE Chabot College Mathematics
InEqualitiesInEqualities
< means “is less than” ≤ means “is less than or equal to” > means “is greater than” ≥ means “is greater than or equal to” For any two numbers
on a number line, the one to the left is said to be less than the one to the right.
–1 0 1 2 3
–1 < 3 since –1 is to the left of 3
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt27
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
§1.2 §1.2 Operations Operations with Real with Real
No.sNo.s
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt28
Bruce Mayer, PE Chabot College Mathematics
Absolute ValueAbsolute Value
The ABSOLUTE VALUE of a number is its distance from zero on a number line.
The symbol |x| to represents the absolute value of a number x.
Example |5| = |−5| = 5
5 units from 0 5 units from 0
• “Distance” is ALWAYS Positive
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt29
Bruce Mayer, PE Chabot College Mathematics
Find Absolute ValueFind Absolute Value Finding Absolute Value
• If a number is negative, make it positive
• If a number is positive or zero, leave it alone
Example Find the absolute value of each number.
• a) |−4.5| b) |0| c) |−3|
Solution
• a) |−4.5| The dist of −4.5 from 0 is 4.5, so |−4.5| = 4.5
• b) |0| The distance of 0 from 0 is 0, so |0| = 0.
• c) |−3| The distance of –3 from 0 is 3, so |−3| = 3
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt30
Bruce Mayer, PE Chabot College Mathematics
Real No. Addition RulesReal No. Addition Rules1. Positive numbers: Add the numbers.
The result is positive.2. Negative numbers: Add absolute values.
Make the answer negative.3. A positive and a negative number: Subtract the
smaller absolute value from the larger. Then:a) If the positive number has the greater absolute value,
make the answer positive.b) If the negative number has the greater absolute
value, make the answer negative.c) If the numbers have the same absolute value,
the answer is 0.
4. One number is zero: The sum is the other number
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt31
Bruce Mayer, PE Chabot College Mathematics
Inverse Property of AdditionInverse Property of Addition
For any real number a, the opposite, or additive inverse, of a, (which is −a) is such that
a + (−a) = −a + a = 0 Example Find the opposite, or
additive inverse: a) 8 b) −13 Solution
• a) 8 −(8) = −8
• b) −13 −(−13) = 13
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt32
Bruce Mayer, PE Chabot College Mathematics
SubtractionSubtraction
Subtraction ≡ The difference a − b is the unique number c for which a = b + c. • That is, a − b = c if c is a number
such that a = b + c
Subtracting by Adding the Opposite• For any real numbers a and b
a − b = a + (−b) – We can subtract by adding the opposite
(additive inverse) of the number being subtracted (the Subtrahend)
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt33
Bruce Mayer, PE Chabot College Mathematics
Real No. MultiplicationReal No. Multiplication
The Product of a Positive and a Negative Number• To multiply a positive number and a
negative number, multiply their absolute values. The answer is negative: 3(−2) = −6
The Product of Two Negative Numbers• To multiply two negative numbers, multiply
their absolute values. The answer is positive: (−13)(−11) = +143
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt34
Bruce Mayer, PE Chabot College Mathematics
Real No. DivisionReal No. Division
The quotient a b or a/b where b ≠ 0, is that unique real number c for which a = b • c.
Example Divide 8
64
5
45
)) ba
SOLUTION
95
45
)a Because −5 · (−9 ) = 45
Because −8 · (8) = −6488
64
)b
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt35
Bruce Mayer, PE Chabot College Mathematics
Multiply & Divide RulesMultiply & Divide Rules To multiply or divide two real numbers:
1. Multiply or divide the absolute values.2. If the signs are the same, then the
answer is positive.3. If the signs are different, then the answer
is negative. DIVISION by ZERO
• NEVER divide by ZERO. If asked to divide a nonzero number by zero, we say that the answer is UNDEFINED. If asked to divide 0 by 0, we say that the answer is INDETERMINATE.
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt36
Bruce Mayer, PE Chabot College Mathematics
Real Number PropertiesReal Number Properties
COMMUTATIVE property of addition and multiplication
a + b = b + a and ab = ba ASSOCIATIVE property
a + (b + c) = (a + b) + c and a(bc) = (ab)c
DISTRIBUTIVE property
a(b + c) = ab + ac
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt37
Bruce Mayer, PE Chabot College Mathematics
Real Number PropertiesReal Number Properties ADDITIVE IDENTITY property
a + 0 = 0 + a = a ADDITIVE INVERSE property
−a + a = a +(−a) = 0 MULTIPLICATIVE IDENTITY property
a • 1 = 1 • a = a MULTIPLICATIVE INVERSE property
a •(1/a) = (1/a) • a = 1 (a ≠ 0)
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt38
Bruce Mayer, PE Chabot College Mathematics
Simplify ExpressionsSimplify Expressions A TERM is a number, a variable, a
product of numbers and/or variables, or a product or quotient of two numbers and/or variables.
Terms are SEPARATED by ADDITION signs. If there are SUBTRACTION signs, we can find an equivalent expression that uses addition signs.
COLLECTING LIKE TERMS is based on the DISTRIBUTIVE law
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt39
Bruce Mayer, PE Chabot College Mathematics
Like (or Similar) TermsLike (or Similar) Terms Terms in which the variable factors are
exactlyexactly the same, such as 9x and −5x, are called like, or similar terms.
Like Terms UNlike Terms
7x and 8x 8y and 9y2
3xy and 9xy 5ab and 4ab2
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt40
Bruce Mayer, PE Chabot College Mathematics
Example Example Combine Terms Combine Terms
a) 7x + 3x b) 4a + 5b + 2 + a − 6 − 5b
SOLUTIONa) 7x + 3x = (7 + 3)x = 10x
b) 4a + 5b + 2 + a − 6 − 5b
= 4a + 5b + 2 + a + (−6) + (−5b)
= 4a + a + 5b + (−5b) + 2 + (−6)
= [4a + a] + [5b + (−5b)] + [2 + (−6)]
= 5a + 0 + (−4)
= 5a − 4
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt41
Bruce Mayer, PE Chabot College Mathematics
Example Example Simplify Simplify
Remove parentheses and simplify
[6(m + 3) – 5m] – [4(n + 5) – 8(n – 4)]
SOLUTION[6(m + 3) – 5m] – [4(n + 5) – 8(n – 4)]
= [6m + 18 – 5m] – [4n + 20 – 8n + 32] Distribute
= [m + 18] – [–4n + 52] Collect like terms within brackets
= m + 18 + 4n – 52 Removing brackets
= m + 4n – 34 Collecting like terms
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt42
Bruce Mayer, PE Chabot College Mathematics
TERMSTERMS ≠ ≠ factorsfactors
Factors are the “pieces” of a Multiplication Chain; e.g., if
• Then y has four factors: 7, u, v, w
TERMS are the pieces of an ADDITION Chain
• Then z has Three TERMS: 7a, 3b, −5c
cbacbaz 537537
wvuyuvwy 7or7
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt43
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
§1.3 §1.3 GraphingGraphingEquationsEquations
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt44
Bruce Mayer, PE Chabot College Mathematics
Points and Ordered-PairsPoints and Ordered-Pairs
To graph, or plot, points we use two perpendicular number lines called axes. The point at which the axes cross is called the origin. Arrows on the axes indicate the positive directions
Consider the pair (2, 3). The numbers in such a pair are called the coordinates. The first coordinate, x, in this case is 2 and the second, y, coordinate is 3.
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt45
Bruce Mayer, PE Chabot College Mathematics
Plot-Pt using Ordered PairPlot-Pt using Ordered Pair
To plot the point (2, 3) we start at the origin, move horizontally to the 2, move up vertically 3 units, and then make a “dot”• x = 2
• y = 3
(2, 3)
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt46
Bruce Mayer, PE Chabot College Mathematics
Example Example Plot the point ( Plot the point (−−4,3)4,3)
Starting at the origin, we move 4 units in the negative horizontal direction. The second number, 3, is positive, so we move 3 units in the positive vertical direction (up)• x = −4; y = 3
4 units left
3 u
nit
s u
p
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt47
Bruce Mayer, PE Chabot College Mathematics
Example Example Read XY-Plot Read XY-Plot Find the coordinates of pts A, B, C, D, E, F, G
AB
C
D
E
F
G
• Solution: Point A is 5 units to the right of the origin and 3 units above the origin. Its coordinates are (5, 3). The other coordinates are as follows: – B: (–2, 4)
– C: (–3, –4)
– D: (3, –2)
– E: (2, 3)
– F: (–3, 0)
– G: (0, 2)
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt48
Bruce Mayer, PE Chabot College Mathematics
XY QuadrantsXY Quadrants
The horizontal and vertical axes divide the plotting plane into four regions, or quadrants
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt49
Bruce Mayer, PE Chabot College Mathematics
Graphing EquationsGraphing Equations Definitions
• An ordered pair (a, b) is said to satisfy an equation with variables a and b if, when a is substituted for x and b is substituted for y in the equation, the resulting statement is true.
• An ordered pair that satisfies an equation is called a solution of the equation
• Frequently, the numerical values of the variable y can be determined by assigning appropriate values to the variable x. For this reason, y is sometimes referred to as the dependent variable and x as the independent variable.
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt50
Bruce Mayer, PE Chabot College Mathematics
Eqn Graph Eqn Graph Bottom Line Bottom Line
ANY and ALL points (ordered pairs) on a Math Graph are SOLUTIONS to the Equation that generated the Graph
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt51
Bruce Mayer, PE Chabot College Mathematics
Graph of an EquationGraph of an Equation
The graph of an equation in two variables, such as x and y, is the set of all ordered pairs (a, b) in the coordinate plane that satisfy the equation
y = 2x + 6 −2x + y = 6
2x − y + 6 = 0
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt52
Bruce Mayer, PE Chabot College Mathematics
Graphing by Plotting PointsGraphing by Plotting Points
Graph y = x2 – 3 SOLUTION → Make “T-table”
(–3, 6)y = (–3)2 – 3 = 9 – 3 = 6–3
(x, y)y = x2 – 3x
X col
Pick x
Calc y Ordered Pair
(–3, 6) is a solutionto y = x2 – 3
[ 6 = 32 – 3 ]
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt53
Bruce Mayer, PE Chabot College Mathematics
Graph by Pt-Plot Graph by Pt-Plot yy = = xx22 – 3 – 3
Pick “enough” x’s for T-table
(3, 6)y = 32 – 3 = 9 – 3 = 63
(2, 1)y = 22 – 3 = 4 – 3 = 12
(1, –2)y = 12 – 3 = 1 – 3 = –21
(0, –3)y = 02 – 3 = 0 – 3 = –30
(–1, –2)y = (–1)2 – 3 = 1 – 3 = –2–1
(–2, –1)y = (–2)2 – 3 = 4 – 3 = 1–2
(–3, 6)y = (–3)2 – 3 = 9 – 3 = 6–3
(x, y)y = x2 – 3x
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt54
Bruce Mayer, PE Chabot College Mathematics
Graph by Pt-Plot Graph by Pt-Plot yy = = xx22 – 3 – 3
Plot (x,y) Points listed in T-table and connect the dots to complete the plot• Note that most
Graphs are “curves” so connect dots with curved lines
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt55
Bruce Mayer, PE Chabot College Mathematics
Graph by Pt-Plot: y = Graph by Pt-Plot: y = xx22 – 2 – 2xx – 6 – 6 Construct T-table
x y (x, y)
3 9 (3, 9)
2 2 (2, 2)
0 6 (0, 6)
1 7 (1, 7)
2 6 (2, 6)
3 3 (3, 3)
4 2 (4, 2)
5 9 (5, 9)
Plot-Pts & Connect-Dots
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt56
Bruce Mayer, PE Chabot College Mathematics
GRAPH BY PLOTTING POINTSGRAPH BY PLOTTING POINTS Step1. Make a representative
T-table of solutions of the equation.
Step 2. Plot the solutions as ordered pairs in the Cartesian coordinate plane.
Step 3. Connect the solutions in Step 2 by a smooth curve
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt57
Bruce Mayer, PE Chabot College Mathematics
Domain & Range by GraphingDomain & Range by Graphing
Graph y = x2. Then State the Domain & Range of the equation
Select integers for x, starting with −2 and ending with +2. The T-table:
x 2xy Ordered Pair yx,
2 42 2 y 4,2
1 11 2 y 1,1
0 002 y 0,0
1 112 y 1,1
2 422 y 4,2
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt58
Bruce Mayer, PE Chabot College Mathematics
Example Example Domain & Range Domain & Range
Now Plot the Five Points and connect them with a smooth Curve
-2
-1
0
1
2
3
4
5
6
-4 -3 -2 -1 0 1 2 3 4
M55_§JBerland_Graphs_0806.xls
x
y
(−2,4) (2,4)
(−1,1) (1,1)
(0,0)
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt59
Bruce Mayer, PE Chabot College Mathematics
Example Example Domain & Range Domain & Range
The DOMAIN of a function is the set of ALL first (or “x”) components of the Ordered Pairs that appear on the Graph
Projecting on the x-axis ALL the x-components of ALL POSSIBLE ordered pairs displays the DOMAIN of the function just plotted
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt60
Bruce Mayer, PE Chabot College Mathematics
Example Example Domain & Range Domain & Range
Domain of y = x2 Graphically
-2
-1
0
1
2
3
4
5
6
-4 -3 -2 -1 0 1 2 3 4
M55_§JBerland_Graphs_0806.xls
x
y
This Projection Pattern Reveals a Domain of
number real a isxx
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt61
Bruce Mayer, PE Chabot College Mathematics
Example Example Domain & Range Domain & Range
The RANGE of a function is the set of all second (or “y”) components of the ordered pairs. The projection of the graph onto the y-axis shows the range -2
-1
0
1
2
3
4
5
6
-4 -3 -2 -1 0 1 2 3 4
M55_§JBerland_Graphs_0806.xls
x
y
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt62
Bruce Mayer, PE Chabot College Mathematics
Example Example Domain & Range Domain & Range
The projection of the graph onto the y-axis is the interval of the y-axis at the origin or higher, so the range is
-2
-1
0
1
2
3
4
5
6
-4 -3 -2 -1 0 1 2 3 4
M55_§JBerland_Graphs_0806.xls
x
y
0yy
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt63
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
§1.4 Solve§1.4 SolveLinear Linear EqnsEqns
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt64
Bruce Mayer, PE Chabot College Mathematics
Solution For an EquationSolution For an Equation
Any NUMBER-REPLACEMENT for the VARIABLE that makes an equation true is called a SOLUTION of the equation. To Solve an equation means to find
ALL of its Solutions.
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt65
Bruce Mayer, PE Chabot College Mathematics
Equivalence and Addition Principle Equivalence and Addition Principle
Equivalent Equations• Equations with the SAME
SOLUTIONS are called EQUIVALENT equations
Addition Principle• For any real numbers u, v, and w,
u = v is equivalent to u + w = v + w– e.g.; u = 3, v = 3, w = 7 →
3 = 3 and 3+7 = 3+7
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt66
Bruce Mayer, PE Chabot College Mathematics
Multiplication PrincipleMultiplication Principle
The Multiplication Principle: For any real numbers r, s, and t with t ≠ 0, r = s is equivalent to r•t = s•t
Example• Solve for x
154
3x
Solution
3
415
4
3
3
4x Multiply Both
Sides by 4/3
203
4351
x
20x
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt67
Bruce Mayer, PE Chabot College Mathematics
Linear EquationsLinear Equations
A linear equation in one variable, such as x, is an equation that can be written in the standard form
where a and b are real numbers with a ≠ 0
0bax
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt68
Bruce Mayer, PE Chabot College Mathematics
Solution to Linear EquationsSolution to Linear Equations Procedure for solving linear equations
in one variable
1. Eliminate Fractions: if Needed, Clear Fractions by Multiplying both sides of the equation by the least common denominator (LCD) of all the fractions
2. Simplify: Simplify both sides of the equation by removing parentheses and other grouping symbols (if any) and combining like terms
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt69
Bruce Mayer, PE Chabot College Mathematics
Solution to Linear EquationsSolution to Linear Equations
3. Isolate the Variable Term: Add appropriate expressions to both sides, so that when both sides are simplified, the terms containing the VARIABLE are on ONE SIDE and all constant terms are on the other side.
4. Combine Terms: Combine terms containing the variable to obtain one term that contains the variable as a factor.
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt70
Bruce Mayer, PE Chabot College Mathematics
Solution to Linear EquationsSolution to Linear Equations
5. Isolate the Variable: Divide both sides by the coefficient of the variable to obtain the solution.
6. Check the Solution: Substitute the solution into the original equation
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt71
Bruce Mayer, PE Chabot College Mathematics
Example Example Solve Linear Eqn Solve Linear Eqn
Solve for x: 6x 3x 2 x 2 11.
SOLUTION (No Fractions to clear)
6x 3x 2 x 2 11
6x 3x 2x 4 11
6x 3x 2x 4 11
5x 4 11
Step 2
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt72
Bruce Mayer, PE Chabot College Mathematics
Example Example Solve Linear Eqn Solve Linear Eqn
Solve for x: 6x 3x 2 x 2 11.
SOLUTION
Step 3 5x 4 4 11 4
Step 4 5x 15
5x
5
15
5x 3
Step 5
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt73
Bruce Mayer, PE Chabot College Mathematics
Example Example Solve Linear Eqn Solve Linear Eqn
Solve for x: 6x 3x 2 x 2 11.
SOLUTION
6 3 3 3 2 3 2 11
18 9 2 11
18 7 11
Step 6 Check x = 3:
State The Solution is x = 3
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt74
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
§1.5 §1.5 Problem Problem SolvingSolving
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt75
Bruce Mayer, PE Chabot College Mathematics
Mathematical ModelMathematical Model A mathematical model is an
equation or inequality that describes a real situation. Models for many APPLIED
(or “Word”) problems already exist and are called FORMULAS A FORMULA is a mathematical
equation in which variables are used to describe a relationship
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt76
Bruce Mayer, PE Chabot College Mathematics
Formula Describes RelationshipFormula Describes Relationship
Relationship Mathematical Formula
Perimeter of a triangle:
a
b
ch
Area of a triangle:
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt77
Bruce Mayer, PE Chabot College Mathematics
Example Example Geometry of Cone Geometry of Cone
Relationship Mathematical Formulae
h
r
Volume of a cone:
Surface area of a cone:
(slant-sides + bottom)
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt78
Bruce Mayer, PE Chabot College Mathematics
Example Example °F ↔ °C °F ↔ °C
Relationship Mathematical Formulae
Celsius to Fahrenheit:
Fahrenheit to Celsius:
Celsius Fahrenheit
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt79
Bruce Mayer, PE Chabot College Mathematics
Example Example Mixtures Mixtures
Relationship Mathematical Formula
Percent Acid, P:
Base
Acid
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt80
Bruce Mayer, PE Chabot College Mathematics
Solving a FormulaSolving a Formula
Sometimes the formula is solved for a Different variable than the one we need
Example A mathematical model tell us that voltage, V, in a circuit is equal to current, I, times resistance, R: V = I R
To determine the amount of resistance in a circuit, it would help to first solve the formula for R.
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt81
Bruce Mayer, PE Chabot College Mathematics
Example Example Solve Solve VV = = IRIR for for RR
We solve this formula for R by • treating V and I as CONSTANTS
(having fixed values)
• treating R as the only variable.
Begin by writing the formula so that the variable for which we are solving, R, is on the left side.
I R = V
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt82
Bruce Mayer, PE Chabot College Mathematics
Example Example Solve Solve VV = = IRIR for for RR
Finally, use Algebra properties to isolate the variable R.
Divide both sides by I.
VIR
I
V
I
IR
I
VR Isolated R → Done
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt83
Bruce Mayer, PE Chabot College Mathematics
Example Example Trapezoid Base, B Trapezoid Base, B
B
b
h
This formula gives the relationship between the height, h, and two bases, B and b, of a trapezoid and its area, A.
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt84
Bruce Mayer, PE Chabot College Mathematics
Example Example Trapezoid Base, B Trapezoid Base, B
Mult. Prop. of Equality.
Assoc. Prop.
Inverse Prop.
Identity Prop.
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt85
Bruce Mayer, PE Chabot College Mathematics
Example Example Trapezoid Base, B Trapezoid Base, B
Add. Prop. Of Equality
Divide by h.
Distributive Prop.
Inverse Prop.
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt86
Bruce Mayer, PE Chabot College Mathematics
Example Example Prismatic Volume Prismatic Volume
h
b
l
The volume of a triangular prism is given by:
If the volume of a triangular cylinder is 880 cm3, the base is 10 cm, and the length is 22 cm, then find the height.
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt87
Bruce Mayer, PE Chabot College Mathematics
Example Example Prismatic Volume Prismatic Volume
First, solve the equation for h.
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt88
Bruce Mayer, PE Chabot College Mathematics
Example Example Prismatic Volume Prismatic Volume
Second, find the height, h, by substituting the given values of V, b, and l into this formula:
State The height of the triangular prism is 8 cm
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt89
Bruce Mayer, PE Chabot College Mathematics
Example Example % from a Pie Chart % from a Pie Chart
The pie chart shown at right represents the distribution of grades in MTH2 (Calculus) last year. Use the information in the chart to estimate how many B’s will be given in a new class of size 70 students.
F 8%
A 16%
B 28%
C 36%
D 12%
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt90
Bruce Mayer, PE Chabot College Mathematics
Example Example % from a Pie Chart % from a Pie Chart According to the chart, 28%
OF the students should get a grade of B. Let x represent the number of students getting a B.
F 8%
A 16%
B 28%
C 36%
D 12%
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt91
Bruce Mayer, PE Chabot College Mathematics
WhiteBoard WorkWhiteBoard Work
Problems From §1.n Exercise Sets• NONE Today → Lecture PPT took Entire
Class Time
Identity Symbol • Difference of Two Squares Identity
srsrsr 22
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt92
Bruce Mayer, PE Chabot College Mathematics
All Done for TodayAll Done for Today
The“Defined as”
Symbol
C)( 12-Carbon of
12g EXACTLYin
Atoms ofNumber
10023.6
12
23
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt93
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
AppendiAppendixx
–
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt94
Bruce Mayer, PE Chabot College Mathematics
Set Membership NotationSet Membership Notation
BA
BBA
BA
BABA
AaAa
AaAa
in not is ofelement oneleast at that meanswhich
,"set ofsubset anot isA set " read is
in is in element every meanswhich
,"set ofsubset a is set " read is
"set ofmember anot is " read is
set ofmember a is " read is
"
[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt95
Bruce Mayer, PE Chabot College Mathematics
Tool For XY GraphingTool For XY Graphing Called “ Engineering
Computation Pad”• Light Green
Backgound
• Tremendous Help with Graphing and Sketching
• Available in Chabot College Book Store
• I use it for ALL my Hand-Work
Graph on this side!