Transcript Thinking Mathematically - Marquette University High School

Thinking Mathematically
Algebra 1
By: A.J. Mueller
Properties
Proprieties

Addition Property (of Equality)

4+5=9
Multiplication Property (of Equality)
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5●8=40
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Reflexive Property (of Equality)
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12=12
Symmetric Property (of Equality)
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If a=b then b=a
Proprieties
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Transitive Property (of Equality)
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Associative Property of Addition
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(0.6+5.3)+4.7=0.6+(5.3+4.7)
Associative Property of Multiplication
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If a=b and b=c then a=c
(-5●7) 3=-5(7●3)
Commutative Property of Addition
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2+x=x+2
Proprieties

Communicative Property of Multiplication
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Distributive Property
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5(2x+7)= 10x+35
Prop. of Opposites or Inverse Property of
Addition
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b3a2=a2b3
a+(-a)=0 and (-a)+a=0
Prop. of Reciprocals or Inverse Prop. of
Multiplication
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x2/7•7/x2=1
Proprieties
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Identity Property of Addition
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Identity Property of Multiplication
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x●1=x
Multiplicative Property of Zero
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-5+0=-5
5●0=0
Closure Property of Addition
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For real a and b, a+b is a real number
Proprieties
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Closure Property of Multiplication
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Product of Powers Property
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x3+x4=x7
Power of a Product Property
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ab = ba
(pq)7=p7q7
Power of a Power Property
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(n2) 3
Proprieties

Quotient of Powers Property
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Power of a Quotient Property
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(a/b) 2
Zero Power Property
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X5/x3=x2
(9ab)0=1
Negative Power Property
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h-2=1/h2
Proprieties

Zero Product Property
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Product of Roots Property
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ab=0, then a=0 or b=0
√20= √4•√5
Power of a Root Property

(√7) 2=7
Solving 1st Power Inequalities
in One Variable
Solving 1st Power Inequalities
in One Variable
With only one inequality sign
x > -5
Solution Set: {x: x > -5}
Graph of the Solution:
-5
Conjunctions
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Open endpoint for these symbols: > <
Closed endpoint for these symbols: ≥
or ≤
Conjunction must satisfy both
conditions
Conjunction = “AND”
{x: -4 < x ≤ 9}
-4
9
Disjunctions

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Open endpoint for these symbols: > <
Closed endpoint for these symbols: ≥ or ≤
Disjunction must satisfy either one or both
of the conditions
Disjunction = “OR”
{x: x < -4 or x ≥ 7}
-4
7
Special Cases That = {All Reals}

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Watch for special cases
No solutions that work: Answer is Ø
Every number works: Answer is {reals}
Disjunction in same direction: answer is one
arrow
{x: x > -5 or x ≥ 1}
-5
1
Special Cases That = 
{x: -x < -2 and -5x ≥ 15}
Ø
Linear equations in two
variables
Linear equations in two variables

Lots to cover here: slopes of all types of
lines; equations of all types of lines,
standard/general form, point-slope form, how
to graph, how to find intercepts, how and
when to use the point-slope formula, etc.
Remember you can make lovely graphs in
Geometer's Sketchpad and copy and paste
them into PPT.
Important Formulas
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Slope-
rise
run
Standard/General form- ax+bx=c
y y
Point-slope form- x  x
Use point-slope formula when you know
4 points on 2 lines.
b

Vertex- 2a
X-intercepts- set f(x) to 0 then solve
Y-intercepts- set the x in the f(x) to 0
and then solve
1
2
1
2
Examples of Linear Equations


Example 1
y=-3/4x-1
4
2
-5
5
-2
-4
Examples of Linear Equations

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Example 2
3x-2y=6 (Put into standard form)
2y=-3x+6 (Divide by 2)
y=-3/2x+6 (Then graph)
4
2
-5
5
-2
-4
Linear Systems
Substitution Method

Goal: replace one variable
with an equal expression
Step 1: Look for a variable with a
coefficient of one.
Step 2: Isolate that variable
Equation A now becomes: y = 3x
+1
Step 3: SUBSTITUTE this expression into
that variable in Equation B
Equation B now becomes 7x – 2(
3x + 1 ) = - 4
Step 4: Solve for the remaining variable
Step 5: Back-substitute this coordinate
into Step 2 to find the other coordinate.
(Or plug into any equation but step 2 is
easiest!)


 3x  y  1
7 x  2 y  4
Addition/ Subtraction
(Elimination) Method

Goal: Combine equations
to cancel out one variable.
Step 1: Look for the LCM of the coefficients on either x or y. (Opposite signs are
recommended to avoid errors.)
Here: -3y and +2y could be turned into -6y and +6y
Step 2: Multiply each equation by the necessary factor.
Equation A now becomes: 10x – 6y = 10
Equation B now becomes: 9x + 6y = -48
Step 3: ADD the two equations if using opposite signs (if not, subtract)
Step 4:
Solve for the remaining variable
Step 5:
Back-substitute this coordinate into any equation to find the other coordinate.
(Look for easiest coefficients to work with.)
Factoring
Types of Factoring
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Greatest Common Factor (GFC)
Difference of Squares
Sun and Difference of Cubes
Reverse FOIL
Perfect Square Trinomial
Factoring by Grouping (3x1 and 2x2)
GFC


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To find the GCF, you just look for the
variable or number each of the numbers
have in common.
Example 1
x+25x+15

x(25+15)
Difference of Squares

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Example 1
27x4+75y4

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3(9x4+25y4)
3(3x2+5y2)(3x2-5y2)
Example 2
45x6-81y4

9(5x4-9y4)
Sun and Difference of Cubes
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Example 1
3
(8x +27)
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(2x+3)
(4x2-6x+9)
Example 2
(p3-q3)

(p-q) (p2+pq+q2)
Reverse FOIL
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Example 1
2
x -19x-32
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(x+8)(x-4)
Example 2
2
6y -15y+12
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(3y-4)(2y-3)
Perfect Square Trinomial
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Example 1
2
4y +30y+25
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(2y+5) 2
Example 2
2
x -10x+25

(x-5) 2
Factoring By Grouping
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3x1
Example 1
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a2+4a+4-b2
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(a+4a+4)-(b2)
(a+2)-(b2)
(a+2-b)(a+2+b)
Factoring by Grouping
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2x2
Example 1
2x+y2+4x+4y

[x+y][2+y]+4[x+y]
Quadratic Equations
Factoring Method
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Set equal to zero
Factor
Use the Zero Product Property to solve.
Each variable equal to zero.
Factoring Method Examples
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Any # of terms- look for GCF first
Example 1
2x2=8x (subtract 8x to set equation equal to zero)
2
2x -8x=0 (now factor out the GCF)
2x(x-4)=0
Factoring Method Examples
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Set 2x=0, divide 2 on both sides and
x=0
Set x-4=0, add 4 to both sides and x=4
x is equal to 0 or 4
The answer is {0,4}
Factoring Method- Binomials
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Binomials – Look for Difference of
Squares
Example 1
2
x =81 (subtract 81 from both sides)
x2-81=0 (factoring equation into conjugates)
(x+9)(x-9)=0
x+9=0 or x-9=0
Factoring Method- Binomials
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x+9=0 (subtract 9 from both sides)
x=-9
x-9=0 (add 9 to both sides)
x=9
The answer is {-9,9}
Factoring Method-Trinomials
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Trinomials – Look for PST
Example 1
x2-9x=-18 (add 18 to both sides)
2
2
x -9x+18=0 (x -9x+18 is a PST)
(x-9)(x-9)=0
x-9=0 (add 9 to both sides) x=9
The answer is {9d.r.} d.r.- double root
Square Roots of Both Sides
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Reorder terms IF needed
Works whenever form is (glob)2 = c
Take square roots of both sides
Simplify the square root if needed
Solve for x, or in other words isolate x.
Square Roots Of Both Sides

Example 1
2
 2 x  12 x  4  0 (Factor

2(x2-6x-2)=0
out the GCF)
2
(You can get rid of the 2
because it does not play a role in this type of
equation)

x2-6-2x=0 (Add the 2 to both sides)
x2-6x__=2__ (Take half of the middle

x2-6x+9=2+9
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number which right now is 6)
(Simplify)
Square Roots Of Both Sides
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(x-3)=11 (Then take the square root of both sides)
(x-3)=  11 (Continue to simplifying)
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( x  3)  11 (Add the 3 to both sides)
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x  3  11 (Final Answer)
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Completing the Square
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Example 1
2x2-12x-4=0 (Factor out the GCF)
2(x2-6x-2)=0 (You can get rid of the 2 because
it does not play a role in this type of equation)

x2-6-2x=0 (Add the 2 to both sides)
x2-6x__=2__ (Take half of the middle number

x2-6x+9=2+9
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which right now is 6)
(Simplify)
Completing the Square
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(x-3)=11
(Then take the square root of both
sides)
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√(x-3)= +/-√11 (Continue to simplifying)
(x-3)=+/- √11 (Add the 3 to both sides)
x=3+/- √11 (Final Answer)
Quadratic Formula
• This is a formula you will need to
memorize!
• Works to solve all quadratic equations
• Rewrite in standard form in order to
identify the values of a, b and c.
• Plug a, b & c into the formula and
simplify!
2 - 4ac
-b±
b
• QUADRATIC FORMULA: x =
2a
Quadratic Formula Examples
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Example 1
3x2-6=x2+12x
Put this in standard form: 2x2-12x6=0
Put into quadratic formula
-(-12)± (-12)2 - 4(2)(-6)
x =
2(2)
Quadratic Formula Examples
12±
144+48
12±
192
x =

4
4
12±
64
 3
x =
4
12±8
3
x =
3 ± 2 3
4
The Discriminant –
Making Predictions
b2-4ac2 is called the discriminant
 Four Cases
1. b2 – 4ac positive non-square two
irrational roots
2. b2 – 4ac positive square two rational
roots
3. b2 – 4ac zero one rational double root

4. b2 – 4ac negative no real roots
The Discriminant –
Making Predictions
Use the discriminant to predict how many “roots”
each equation will have.
1. x2 – 7x – 2 = 0 49–4(1)(-2)=57 2 irrational
roots
2. 0 = 2x2– 3x + 1 9–4(2)(1)=1  2 rational
roots
3. 0 = 5x2 – 2x + 3 4–4(5)(3)=-56  no real
4.
roots
x2 – 10x + 25=0 100–4(1)(25)=0  1 rational
double root
The Discriminant –
Making Predictions
The “zeros” of a function are the x-intercepts on it’s graph.
Use the discriminant to predict how many x-intercepts
each parabola will have and where the vertex is located.
1. y = 2x2 – x - 6 1–4(2)(-6)=49  2 rational
zeros
opens up/vertex below x-axis/2 xintercepts
2. f(x) = 2x2 – x + 6 1–4(2)(6)=-47  no real
zeros
opens up/vertex above x-axis/No xintercepts
The Discriminant –
Making Predictions
3. y = -2x2– 9x + 6 81–4(-2)(6)=129 2 irrational
zeros
opens down/vertex above x-axis/2 xintercepts
4. f(x) = x2 – 6x + 9 36–4(1)(9)=0  one rational
zero
I (A.J. Mueller) got
these last four slides
from Ms. Hardtke’s
Power Point of the
Quadratic Methods.
opens up/vertex ON the x-axis/1 xintercept
Functions
About Functions
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Think of f(x) like y=, they are really the
same thing.
The domain is the x line of the graph
The Range is the y line of the graph
Functions
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f(x)= x 2 -2x-8
First find the vertex.
b

( 2a ) The vertex of this equation is (1,-9)
Find the x-intercepts by setting f(x) to 0.
The x-intercepts are {-2,4}
Find the y-intercept by setting the x in
the f(x) to 0. You would get -8.
The graph the equation.
Simplifying expressions with
exponents
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This site will example how to simplify
expressions with exponents very well.
http://www.purplemath.com/modules/si
mpexpo.htm
Radicals
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Example 1
12  50  18 (Simplify)
3 4  2 25  2 9
2 3  5 2  3 2 (Now you can cancel the √2s)
Radicals
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Example 2
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2
2
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2

2
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(Multiply by
2
2
That equals
2 2
2
2
2
)
2 2
4
Cancel out the 2s and the final answer
is 2
Radicals
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Example 3
x16
Take the square root
of that.
4
Final answer is x
Word Problems
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Example 1
If Tom weighs 180 on the 3th day of his
diet and 166 on the 21st day of his diet,
write an equation you could use to
predict his weight on any future day.
(day, weight)
(3,180)
21,166)
Word Problems
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Point Slope: m=166-180/21-31
That can be simplified to -14/18 and
then -7/9.
4-180=-7/9(x-3)
4-180=-7/9+21/9
Answer: y=-7/9x+182 1/3
Word Problems


Click to open the hyperlink. Then try
out this quadratic word problem, it will
walk you through the process of finding
the answer.
http://www.algebra.com/algebra/home
work/quadratic/word/02-quadratic.wpm
Word Problems

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Here is another link to a word problem
about time and travel.
http://www.algebra.com/algebra/home
work/word/travel/07-cockroach.wpm
Word Problems

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This word problem is about geometry.
http://www.algebra.com/algebra/home
work/word/geometry/02-rectangle.wpm
This site is good study tool for word
problems.
Line of Best Fit

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The Line of Best Fit is your guess where
the middle of all the points are.
http://illuminations.nctm.org/ActivityDet
ail.aspx?id=146
This URL is a good site to example Line
of Best Fit. Plot your points, guess your
line of best fit, then the computer will
give the real line of best fit.
Line of Best Fit

Your can use a Texas Instruments TI-84
to graph your line of best fit and also all
other types of graphs.