Transcript Example

Tony Hren
Algebra 1 Review
May 14, 2010
Addition Property (of Equality)
If the same number is added to both sides of an equation, the two sides
remain equal.
Example: If a=b, then a+c = b+c
http://www.icoachmath.com/sitemap/Addition_Property_of
_Equality_and_Inequality.html
Multiplication Property (of Equality)
You can multiply both sides of an equation by the same nonzero
number,
and this won't change the truth of the equation.
Example: If a=b, then ac=bc
http://www.onemathematicalcat.org/algebra_book/online_problems/mult_prop_eq.htm
Reflexive Property (of Equality)
Example: a=a
Symmetric Property (of Equality)
Example: If a=b, then b=a
Transitive Property (of Equality)
Example: If a=b, and b=c, then a=c
Associative Property of Addition
The property which states that for all real numbers a, b, and c, their sum is always
the same, regardless of their grouping.
Example:
(a + b) + c = a + (b + c)
http://www.harcourtschool.com/glossary/math_advantage/definitions/associative_add7.html
Associative Property of
Multiplication
The property which states that for all real numbers a, b, and c, their product is always the
same, regardless of their grouping.
Example:
(a x b) x c = a x (b x c)
http://www.harcourtschool.com/glossary/math_advantage/definitions/associative_mul7.html
Commutative Property of Addition
The Commutative Property of Addition states that changing
the order of addends does not change the sum
Example: if a and b are two real numbers, then a + b = b + a.
http://www.icoachmath.c
om/SiteMap/Commutativ
ePropertyofAddition.html
Commutative Property of
Multiplication
The Commutative Property of Multiplication states that
changing the order of the factors does not change the
product.
Example: if a and b are two real numbers, then a × b = b × a.
Distributive Property (of
Multiplication over Addition)
The property which states that multiplying a sum by a number gives the
same result as multiplying each
addend by the number and then adding the products.
Example:
a(b + c) = a X b + a X c
http://www.harcourtschool.com/glossary/math_advantage/definitions/distributive_p7.html
Property of Opposites or Inverse
Property of Addition
The property that states the sum of a number and its opposite is always
zero.
Example: a+(-a)=0
http://www.washoe.k12.nv.us/ecollab/washoemath/dic
tionary/vmd/full/a/dditionpropertyofopposites.htm
Prop of Reciprocals or Inverse
Prop. of Multiplication
For every non-zero real number a there is a unique real number 1/a such
that:
1
Example: a( )=1
a
http://everyonehatesmath.com/property-of-reciprocals/
Identity Property of Addition
Identity property of addition states that the sum of zero and any number or
variable is the number or variable itself.
Example: a+0=a
http://www.northstarmath.com/Sitemap/IdentityPropertiesofAdditionandMultiplication.html
Identity Property of Multiplication
Identity property of multiplication states that the product of 1 and any
number or variable is the number or variable itself.
Example: 4 x 1=4
Multiplicative Property of Zero
Any number multiplied by zero equals zero.
Example: a(0)=0
Closure Property of Addition
Closure property of addition states that the sum of any two real
numbers equals another real number.
Example: If a and b are real numbers, then a +
b equals a real number.
Closure Property of Multiplication
Closure property of multiplication states that the product of any two real
numbers equals another real number.
Example: If a and b are real numbers,
then a x b is equal to a real number.
Product of Powers Property
This property states that to multiply powers having the same base, add
the exponents.
Example: for a real number non-zero a and two integers m and n,
http://www.ic
oachmath.co
m/sitemap/P
ower_Proper
ties.html
am × an = am+n.
Power of a Product Property
This property states that a product of a power can be obtained by finding
the powers of each property and multiplying them.
Example: (ab)m = am × bm
Power of a Power Property
This property states that the power of a power can be found by multiplying
the exponents.
Example: (am)n
= amn
Quotient of Powers Property
This property states that to divide powers having the same
base, subtract the exponents.
Example:
http://www.icoachmath.com/sitemap/Power_Properties.html
.
Power of a Quotient Property
This property states that the power of a quotient can be
obtained by finding the powers of numerator and denominator
and dividing them.
Example:
Zero Power Property
When a number is raised to the zero
power, it is always equal to 1.
Example: a0=1
Negative Power Property
To solve for negative exponents,
write the reciprocal of the expression, and change
the negative to the positive power.
http://www.mathsisfun.
com/algebra/negativeexponents.html
Example:
Zero Product Property
The Zero Product Property simply states that if ab =
0, then either a = 0 or b = 0 (or both). A product of
factors is zero if and only if one or more of the factors
is zero.
Example: ab=0. If this is true, a, b, or a
and be must be equal to zero.
Product of Roots Property
The factors of the square root of a number are
equal to the square root of one factor of the
original number multiplied by the square root of
another.
=
Example:
X
Quotient of Roots Property
The roots of a square root of a quotient are equal to the
square root of the numerator written over/divided by the
http://www.tutorvista.
square root of the denominator.
com/math/squareroot-propertycalculator
Example:
=
Root of a Power Property
Example:
a4 = a2 x a2 = (a)(a)(a)(a)
http://hotmath.
com/hotmath_
help/topics/pro
perties-ofexponents.html
Power of a Root Property
Example:
PROPERTY QUIZ!
Click when you’re ready for the answers.
1.) (am)n = amn
8.) (a x b) x c = a x (b x c)
Power of a Power Property
Associative Poperty of Multiplication
2.) If a=b, then b=a
9.) If a and b are two real
numbers, then a × b = b × a.
Symmetric Property (of Equality)
3.) If a=b, then a+c = b+c
Addition Property (of Equality)
4.) a+(-a)=0
Commutative Property of Multiplication
10.) If a and b are two real
numbers, then a + b = b + a.
Property of Opposites/Inverse Operation of Addition
5.) If a=b, then ac=bc
Multiplication Property of Equality
6.) If a=b, then ac=bc
11.) a(b + c) = a X b + a X c
Dist. Prop. (of Multiplication over Division)
Multiplicative Property of Zero
7.) If a=b, and b=c, then a=c
Transitive Property of Equality
Commutative Property
of Addition
12.) a0=1
Zero Power Property
Solving 1st Power Inequalities in
One Variable
-With only one inequality sign
Example: 6x > 24
A linear equation has only one value for the solution that holds true. For
example, the linear equation 6x = 24 is a true statement only when x = 4.
However, the linear inequality 6x > 24 is satisfied when x > 4. So, there
are many values of x which will satisfy the inequality 6x > 24, which is the
same thing as x > 4, which is the answer.
As shown below, x > 4. This is the answer because after dividing
each side by 6, we are left with x > 4.
http://www.mathsteacher.com.au/year10/ch02_linear_equations/07_subtract/solve.htm
Another Example
The solution set consists of all numbers
less than or equal to –2, as shown on
the following number line.
http://www.mathsteacher.com.au/year10/ch02_linear_equations/07_subtract/solve.htm
Special Case: Division and
Multiplication of Negative Numbers
• With only one inequality sign
• if an inequality is multiplied (or divided) by
the same negative number, then:
Example: -3x > 27
[Divide each side by (-3)]
Answer: x < -9
http://www.mathsteach
er.com.au/year10/ch0
2_linear_equations/07
_subtract/solve.htm
Conjunctions
• Must satisfy both conditions of the
inequality
RESULT:
Examples:
Graph A: 4 ≤ x < 9
Graph B: 4 < x ≤ 9
Graph C: 4 < x < 9
Graph D: 4 ≤ x ≤ 9
http://image.tutorvista.com/Qimages/QD/28881.gif
Disjunctions
• Must satisfy either one or both of the
conditions
< AND > have
Example:
open endpoints,
while ≥ and ≤
x ≤ -3
have closed
OR
endpoints
x>2
http://hotmath.com/images/gt/lessons/genericalg1/f-352-21-ex-1.gif
Special Cases
• If there are no solutions that work, the
answer is Ø
• If every number works, the answer is
{reals}
• If there is a disjunction in the same
direction, only one arrow is needed
Linear Equations with Two Variables
Slopes and Equations of Lines
Positive: Lines that rise as you move from left to right.
Negative: Lines that fall as you move from left to right.
Rising and Falling lines are associated with rise/run (rise over run)
which is the same as the difference between ycoordinates/difference between x-coordinates.
Horizontal Lines: Each of these has a slope of 0, and refers to a
straight line from left to right.
Vertical Lines: Each of these has no slope, and refers to a straight
line running up and down.
Positive
Negative
Vertical
Horizontal
http://www.google.com/imgres?imgurl=http://
www.learningwave.com/lwonline/algebra_sec
tion2/graphics/typesslope2.gif&imgrefurl=http:
//www.learningwave.com/lwonline/algebra_se
ction2/slope2.html&usg=__fF7XXBj0ZQWoD
V0dCqXQObS45x8=&h=192&w=203&sz=6&
hl=en&start=5&um=1&itbs=1&tbnid=NLy_ChI
j2lDfM:&tbnh=99&tbnw=105&prev=/images%3
Fq%3Dthe%2B4%2Btypes%2Bof%2Bslopes
%26um%3D1%26hl%3Den%26sa%3DN%26
ndsp%3D20%26tbs%3Disch:1
Point-Slope
formula:
How to
Graph and
find
Intercepts?
In the slope-intercept
formula, the equation is
y=mx+b., whereas ‘m’ is
equal to the slope, ‘b’ is the
y-intercept and the ‘y’ and
‘x’ are the coordinates.
http://www.saskschools.ca/curr_content/matha30rev1/lesson3-4/ponitslopeformula.jpg
Start graphing by placing a
point on the y-intercept.
Standard Form
Then follow the slope to
Ax + By = C
make your line. The xA, B, C are integers (positive or negative whole numbers)
intercept is where on the xNo fractions nor decimals in standard form.
axis the line passes
Traditionally the "Ax" term is positive.
through.
http://www.algebralab.org/studyaids/studyaid.aspx?file=algebra1_5-5.xml
Linear Systems
Substitution Method: First solve one equation for one of the
variables. Substitute this expression in the other equation and
solve for the other variable next. Then, substitute this value in
the equation of the first step and solve. Example: z=4y-4
Addition/Subtraction Method
(Elimination Method): First add the
similar terms of the two equations to find
the x. Then solve the resulting equation.
Substitute that answer for the other
variable to find y. Then check your final
answers in both equations.
Terms Quiz
Choices: dependent, inconsistent, consistent
1.) A system in whichthe solution id all points on the line
dependent
2.) A system in which the lines cross on one point.
consistent
3.) A system which is false or null set because it is parallel
inconsistent
Factoring
GCF: For any number of terms, factor first the common
factors of the expressions, and then fill in what is left
over.
Difference of Squares: For binomials, first find the GCF,
then find the squares and simplify what is left over,
hopefully finding conjugates.
Sum or Difference of Cubes: For binomials, simply find the
square or cube root from each side.
PST: For trinomials, If 1st & 3rd terms are squares and the
middle term is twice the product of their square roots,
use this method by reverse foil.
Reverse Foil: for trinomials, do the opposite of F O+I L.
Factor By Grouping: Usually do this with 4 or more terms,
and take the roots of the algebraic expressions.
Rational Expressions
• To simplify a rational expression:
• Completely factor numerators and
denominators.
• Reduce common factors.
• Example : Simplify
Examples:
Addition: 3/5 + 1/5 = 4/5.
Subtraction: 11/12 – 4/12 = 7/12.
Division: ¾ / 2 = ¾ x ½ =3/8.
Multiplication: ¾ x ¾ =9/4 = 2 and ¼.
http://www.cliffsnotes.com/study_guide/Simplifying-Rational-Expressions.topicArticleId-38949,articleId-38901.html
Quadratic Equations in one
Variable
Factoring: x2+5x+6=0 = (x+2)(x+3)=0
(-2, -3)
Taking Square Root of each side:
x2= 4
x=2.
Completing the Square:
(x – 4)2 = 5
x – 4 = ± sqrt(5)
x = 4 ± sqrt(5)
x = 4 – sqrt(5) and x = 4 + sqrt(5)
http://www.purplemath.com/modules/sqrquad.htm
Quadratic Formula
http://www.sosmat
h.com/algebra/qua
draticeq/quadrafor
mula/quadraformul
a.html
The discriminant tells you how many X-axis intercepts a polynomial function has.
Functions
F(x) means the same thing as “y” but gives more information. The expression
"f(x)" means "plug a value for x into a formula f "; You solve it the same way
you would for a “y”. Not all relations are function.
y = √(x + 4)
The domain of the function is x ≥ −4,
because x cannot take values less than
-4. the range for this function is y ≥ 0,
because There is no value of x that we
can find such that we will get a
negative value of y. In order to find a
linear equation when given two pairs of
data, follow the rule:
y2-y1/x2-x1. For example if you had the
ordered pairs (2,3) and (1,5), you
would first do 5-3, which is equal to 2,
over 1-2, which is equal to -1, which is
the slope. Then substitute zero for one
of the values and solve for x and y.
http://www.intmath.com/Functions-and-graphs/2a_Domain-and-range.php
•
•
•
•
•
•
Parabolas
WRITE YOUR EQUATION ON PAPER. IF NECESSARY, TRY TO REARRANGE
THE EQUATION INTO THE FORM OF A PARABOLA, Y - K = A (X - H)^2. (OUR
EXAMPLE IS Y - 3 = - 1/6 (X + 6)^2, WHERE ^ DENOTES AN EXPONENT.)
FIND THE VERTEX OF THE PARABOLA. THE VERTEX IS THE EXACT CENTER
OF THE PARABOLA, THE KEY COMPONENT. USING THE FORMULA FOR A
PARABOLA, Y - K = A (X - H)^2, THE VERTEX X-COORDINATE (HORIZONTAL)
IS "H" AND THE Y-COORDINATE (VERTICAL) IS "K." FIND THESE TWO VALUES
IN YOUR ACTUAL EQUATION. (OUR EXAMPLE IS H = -6 AND K = 3.)
FIND THE Y-INTERCEPT BY SOLVING THE EQUATION FOR "Y." SET "X" TO "0"
AND SOLVE FOR "Y." (OUR EXAMPLE IS Y = -3.)
FIND THE X-INTERCEPT BY SOLVING THE EQUATION FOR "X." SET "Y" TO "0"
AND SOLVE FOR "X." WHEN TAKING THE SQUARE ROOT OF BOTH SIDES,
THE SINGLE NUMBER SIDE OF THE EQUATION BECOMES BOTH POSITIVE
AND NEGATIVE (+/-), RESULTING IN TWO SEPARATE SOLUTIONS, ONE USING
THE POSITIVE AND ONE USING THE NEGATIVE.
DRAW A BLANK LINE GRAPH ON GRAPH PAPER. DETERMINE THE SIZE
ANDAREA OF THE GRAPH. A PARABOLA GOES TO INFINITY, SO THE GRAPH
http://www.eh
IS ONLY A SMALL PORTION NEAR THE VERTEX, WHICH IS THE TOP OR
ow.com/how_
BOTTOM OF THE PARABOLA. THE GRAPH NEEDS TO BE DRAWN IN
4546044_grap
PROXIMITY TO THE VERTEX. THE X AND Y-INTERCEPTS TELL THE ACTUALhPOINTS THAT APPEAR ON THE GRAPH. DRAW A STRAIGHT HORIZONTAL parabola.html
LINE AND A STRAIGHT VERTICAL LINE INTERCEPTING AND PASSING
THROUGH THE HORIZONTAL LINE. DRAW AN ARROW AT BOTH ENDS OF
BOTH LINES TO REPRESENT INFINITY. MARK SMALL TICK LINES ON EACH
LINE AT EQUAL INTERVALS REPRESENTING NUMERAL INCREMENTS IN THE
VICINITY OF THE SIZE OF THE COORDINATES. MAKE THE GRAPH A FEW
TICKS LARGER THAN THESE COORDINATES.
PLOT THE PARABOLA ON THE LINE GRAPH. PLOT THE VERTEX, XINTERCEPT, AND Y-INTERCEPTS POINTS ON THE GRAPH WITH LARGE DOTS.
CONNECT THE DOTS WITH ONE CONTINUOUS U-SHAPED LINE AND
CONTINUE THE LINES TO NEAR THE END OF THE GRAPH. DRAW AN ARROW
AT BOTH ENDS OF THE PARABOLA LINE TO REPRESENT INFINITY.
Simplifying Expressions with Exponents and Radicals
Simplify (–46x2y3z)0
This is simple enough: anything to the zero power is just 1.
(-46x2y3z)0 =1
x6 × x5 = (x6)(x5)
= (xxxxxx)(xxxxx)
= xxxxxxxxxxx
= x11
(6 times, and then 5 times)
(11 times)
=
Radical Expressions Examples
http://www.algebralab.org/lessons/lesson.aspx?file=algebra_radical_simplify.xml
Word Problems
Many single-variable algebra word problems have to do with the relations between
different people's ages. For example: Al's father is 45. He is 15 years older than
twice Al's age. How old is Al?We can begin by assigning a variable to what we're
asked to find. Here this is Al's age, so let Al's age = x.
We also know from the information given in the problem that 45 is 15 more than
twice Al's age. How can we translate this from words into mathematical symbols?
What is twice Al's age? Al's age is x, so twice Al's age is 2x, and 15 more than twice
Al's age is 15 + 2x. That equals 45. Now we have an equation in terms of one
variable that we can solve for x: 45 = 15 + 2x.
original statement of the problem:
45 = 15 + 2x
subtract 15 from each side:
30 = 2x
divide both sides by 2:
15 = x
Since x is Al's age and x = 15, this means that Al is 15 years old
Word Problems (cont.)
Ned got a 12% discount when he bought his new
jacket. If the original price, before the discount, was
$50, how much was the discount?
Word problems tend to be even wordier than this one.
The solution process involves making the problem
simpler and simpler, until it's a math problem with no
words.
Step 1. Identify what they're asking for, and call it x.
x = amount of the discount.
Step 2. Use the information given to write an equation
that relates the quantities involved.
12% of 50 dollars = the amount of the discount (x).
Step 3. Translate into Math:
(12/100) * 50 = x.
Step 4. Solve for x:
6 = x.
This means that Ned's 12% off amounted to a $6
discount.
http://www.algebra.com/algebra/homework/Percentage-and-ratio-word-problems/Percentage-Word-Problems-(discount).lesson
Word Problems (cont.)
Adding To The Solution
Mixture Problems: Example 1:
John has 20 ounces of a 20% of salt solution, How much
salt should he add to make it a 25% solution?
Solution:
Step 1: Set up a table for salt.
origina
l
added
result
concentratio
n
Step 2: Fill in the table with
information given in the
question.
John has 20 ounces of a 20% of
salt solution. How much salt
should he add to make it a 25%
solution?
The salt added is 100% salt,
which is 1 in decimal.
Change all the percent to
decimals
Let x = amount of salt added.
The result would be 20 + x.
origina
l
added
result
concentratio
n
0.2
1
0.25
amount
20
x
20 + x
amount
Step 3: Multiply down each column.
origina
l
added
result
concentratio
n
0.2
1
0.25
amount
20
x
20 + x
multiply
0.2 ×
20
1×x
0.25(20 + x)
http://www.onlinemathlearning.com/mixture-problems.html#add
Step 4:original + added = result
0.2 × 20 + 1 × x = 0.25(20 + x)
4 + x = 5 + 0.25x
Isolate variable x
x – 0.25x = 5 – 4
0.75x = 1
Answer: He should add
ounces of salt.
Word Problems (Cont.)
You put $1000 into an investment yielding 6% annual interest; you
left the money in for two years. How much interest do you get at
the end of those two years?
In this case, P = $1000, r = 0.06 (because I have to convert the
percent to decimal form), and the time is t = 2. Substituting, I get:
I = (1000)(0.06)(2) = 120
I will get $120 in interest.
You invested $500 and received $650 after three years. What had
been the interest rate?
For this exercise, I first need to find the amount of the interest. Since
interest is added to the principal, and since P = $500, then I = $650 –
500 = $150. The time is t = 3. Substituting all of these values into the
simple-interest formula, I get:
150 = (500)(r)(3)
150 = 1500r
150/1500 = r = 0.10
Of course, I need to remember to convert this decimal to a percentage.
I was getting 10% interest
http://www.purplemath.com/modules/percntof.htm
Line of Best Fit
A line of best fit is a straight line that best
represents the data on a scatter plot.
This line may pass through some of the
points, none of the points, or all of the points.
A graphing calculator helps because if
you enter information correctly, it will
draw one for you.
Here's an example.
Suppose you want to find out whether
more hours spent studying will have
an affect on a person's mark.
You set up an experiment with some
people, recording how many hours
they spent studying and then
recording what happened to their
mark.
You can see the data in the table at
the right.
It's difficult to see any pattern in the
table, although it's clear that different
things happened to different people.
One person studied for 1 hour and
had their mark go up 2%, while
another person who also studied for 1
hour saw a drop of 1%!
I HOPE YOU
ENJOYED THIS
BEAUTIFUL ALGEBRA
1 PRESENTATION!
The end.