Transcript Thinking Mathematically

Thinking Mathematically
William E. Blommer
James M. Lueken
Equivalence Properties of Equality
-All are quite easy to understand, but are very similar. Make sure to
know the differences of these properties.
A. Reflexive Property (of Equality)
Example: The property that a = a.
Hint: It’s your reflex to think a=a
B. Symmetric Property (of Equality)
Example: If a = b then b = a.
Hint: To be symmetric two things must be the same. In math two
numbers must be the same for it to be the symmetric property
C. Transitive Property (of Equality)
Example: If a = b and b = c then a = c.
Hint: Transitive starts with a “T” as well does three. There must be three
number that equal the same thing to be the transitive property.
Associative Property of Adding
and Multiplication
D.
E.
Associative Property of Adding
Example: (3 + 4) + 5 = 3 + (4 + 5)
Associative Property of Multiplication
Example: a(bc)=(ab)c
Hint: Think associative as grouping. In
this property you are grouping numbers
together with parentheses.
Commutative Property of Addition
and Multiplication
-This property is simply moving numbers
around in a number problem
F.
Commutative Property of Addition
Example: 5 + 6 = 6 + 5
G.
Commutative Property of Multiplication
Example: 5*8=8*5
Hint: You are commuting the numbers to
different places in the problem.
Distributive Property of
Multiplication over Adding
I.
Property of Opposites or Inverse
Property of Addition
Examples: a(b + c) = ab + ac
(b + c)a = ba + ca
Hint: You are distributing a to b and c.
It’s an easy property to grasp.
Property of Opposites or Inverse
Property of Addition
I.
Property of Opposites or Inverse
Property of Addition
Examples: a3+(-a3)=0
6+(-6)=0
Hint: This is quite a simple property. It
is basically taking the opposite of the
number either it being negative or
positive to make the problem equal
zero
Identity Property of Addition
J.
Identity Property of Addition
Example: x+0=x, a2+0=a2
Hint: An extremely easy property to
remember. Simply adding zero to the
problem will make the problem an
identity property of addition.
Identity Property of Multiplication
K.
Identity Property of Multiplication
Examples: x*1=x,
6*1=6
Hint: Multiplying by one is the only way
an identity multiplication problem can
work
Multiplicative Property of Zero
L.
Multiplicative Property of Zero
Examples:18*0=0
x*0=0
Hint: Your goal in this problem is to get
zero. Anything multiplied by zero will
equal zero
Closure Property of Addition and
Multiplication
M.
N.
Closure Property of Addition
Explained: Closure property of real number addition
states that the sum of any two real numbers equals
another real number.
Example: 5,7 are both real numbers
5+7=13, 13 is a real number
Closure Property of Multiplication
Explained: Closure property of real number multiplication
states that the product of any two real numbers equals
another real number.
Example: 2, 3 are both real numbers
2*3=6, 6 is a real number
Hint: Real numbers are rational numbers. A rational
number is a number that can be written as a simple
fraction i.e. 3= 3/1.
Zero Product Property
O.
Zero Product Property
Explained: If two numbers equal zero
one number in the problem must be
zero
Example: ab=0, a=0 b=0
Product of Roots Property
P.
Product of Roots Property
Explained: For all positive real numbers
a and b
Example:
Hint: That is, the square root of the
product is the same as the product of
the square roots.
Quotient of Roots Property
Q.
Quotient of Roots Property
Explained:For all positive real numbers
a and b, b ≠ 0
Example:
Hint: The square root of the quotient is
the same as the quotient of the square
roots.
Root of a Power Property
S.
Root of a Power Property
Addition Property and Multiplication
Property
Addition
If the same number is added to both sides of an equation,
the two sides remain equal. That is, if x = y, then x + z =
y + z.
Multiplication
For all real numbers a and b , and for c ≠ 0 ,
a = b is equivalent to ac = bc .
Power of a Product Property
Explained: When a number is repeatedly
multiplied by itself, we get the powers of
that number
Examples: x5 = xxxxx
62=(6)(6)=36
Power of a Power Property
Explained: A power of a power is a
problem that involves multiplying two
powers together. An example will explain
this better
Examples: (32)2=34
(x7)2=x14
Hint: If you see a power within a
parentheses and one outside you simply
multiply the exponents
Zero Power Property
Explained: If an exponent of a number is 0
then it automatically becomes one. This
does trick people into thinking it is 0, but it
is really only one.
Example: 134342340=1
x0=1
Power of a Quotient
POWER OF A QUOTIENT
Explained: The law of exponents for a power of an indicated quotient may be
developed from the following
Problem
•
Therefore,
•
•
The law is stated as follows: The power of a quotient is equal to the quotient
obtained when the dividend and divisor are each raised to the indicated power
separately, before the division is performed.
Property of Reciprocals or Inverse
Property of Multiplication
Explained: A reciprocal is the number you
have to multiply a given number by to get
one.
Example: 1/2x=1
1/2(2)x=1(2)
x=2
The reciprocal of 1/2 is 2 or 2/1
Hint: Another name for "reciprocal" is "multiplicative
inverse."
Power of a Root Property
Negative Power Product
Explained: A negative exponent just means that the base is on the wrong
side of the fraction line, so you need to flip the base to the other side. For
instance, "x–2" (x to the minus two) just means "x2, but underneath, as in
1/(x2)".
Examples:
Write x–4 using only positive exponents.
•
Write x2 / x–3 using only positive exponents.
•
Write 2x–1 using only positive exponents.
•
Note that the "2" above does not move with the variable; the exponent is only on the "x".
Write (3x)–2 using only positive exponents.
•
Property Quiz
n
a
o
d
g
b
i
h
j
k
h
l
m
q
p
r
8+7=7+38
a*b*c=c*b*a
a(b+7)= ab+ 7a
a = b then b = a
5,7 are both real numbers 5+7=13
x*0=0
6*1=6
2, 3 are both real numbers 2*3=6
x+0=x
6+(-6)=0
a = b and b = c then a = c
a+(b+c)= (a+b)+c
a(bc)=(ab)c
a. Commutative Property of Multiplication
b. Zero Property Product
c. Product of Roots
d. Symmetric Property
e. Reflexive Property
f. Transitive Property
g. Closure Property of Addition
h. Closure Property of Multiplication
i. Identity Property of Multiplication
j. Identity Property of Addition
k. Property of Opposites or Inverse Property
of Addition
l. Associative Property of Addition
m. Associative Property of Multiplication
n. Commutative Property of Addition
o. Distributive Property
p. Quotient of Roots Property
q. Product of Roots Property
r. Negative Power Property
Solving 1st power inequalities in
one variable
Solving linear inequalities is very similar to solving linear equations, except for
one small but important detail: you flip the inequality sign whenever you
multiply or divide the inequality by a negative. The easiest way to show this is
with some examples:
Solve 6x − 3 < 2x + 5
Solve 6x > 12
6x > 12
6
6
x>2
<,> are greater than
or equal to.
To finish this problem simply
divide by four on each side.
The final product should be
x<2.
x>0
x <0
This is a special case.
This symbol means
absolute value.
Absolute value means
the distance the
number is from zero so
the number can not be
negative. This answer
would be null set or 0
This is another special case.
Any number put into this will
work meaning it is all real
numbers or {all reals}
Inequalities Continued
T. Conjunctions
Explained: A conjunction is a mathematical operator that
returns an output of true if and only if all of its answers
are true.
Example: -2 < x < 4
Answer Graphed:
Hint: When graphing the solution set it must be
connected for it to be a conjunction. If it is not then it
can not be a conjunction. Also, If it is <,> there is an
open circle. If it is <,> the circle is shaded
Inequalities Continued
U.
Disjunctions
Explained: A disjunction of two statements is formed
by connecting them with the word "or." A disjunction
is true when one or both statements are true. The
solution set of a disjunction is the union of the two
graphs.
Example: -2x-6>4 or x+5>4 (You would solve this
problem like the previous slides)
Answer Graphed:
Hint: Disjunctions can be graphed no matter what. If
they over lap they are still disjunctions.
Linear Equations
Understanding the Y-intercept Form
A linear equation in this form is called the y-intercept form or slope-intercept form.
In this form, you can just LOOK at the equation and pick out the important information
that you need to graph the line. This is the reason why the y-intercept form is
preferred over the standard form!
In this format, "b" is where the line crosses the y-axis and "m" is the slope of the
line.
The place where the line crosses the y-axis is called the y-intercept.
General Form
Ax + By + C = 0
General Form is can be put into Y-intercept
form by simplifying.
Example:
6x+2y=4
2y=-6x+4
2= 2
y=-3x+2
Linear Equations Cont.
Use to slope formula to help complete a linear equation.
The slope m of the line through the points (x1, y1) and (x
2, y 2) is given by
If you can not find the y intercept of the line use the point
slope formula. Simply plug in the numbers to find it.
Linear Equations Cont.
Analyzing Slope
m=slope in y intercept form
m<0 The slope is negative making the line fall like this.
m>0 The slope is positive making the slope rise like this.
Linear Equations Cont.
How to graph a line:
Linear Equations in Two Variables
Explained: A system of linear equations is two or more linear equations
that are being solved simultaneously. In general, a solution of a
system in two variables is an ordered pair that makes BOTH
equations true. In other words, it is where the two graphs
intersect, what they have in common. So if an ordered pair is a
solution to one equation, but not the other, then it is NOT a
solution to the system.
•
Vocab
-A consistent system is a system that has at least
one solution.
-An inconsistent system is a system that has no
solution.
-The equations of a system are dependent if ALL the
solutions of one equation are also solutions of the other
equation. In other words, they end up being the same line.
-The equations of a system are independent if they do not
share ALL solutions. They can have one point in common,
just not all of them.
Linear Equations in Two Variables Cont.
One Solution Explained:
If the system in two variables has one solution, it is an ordered pair that is a solution to
BOTH equations. In other words, when you plug in the values of the ordered pair it makes BOTH
equations TRUE.
If you do get one solution for your final answer, is this system consistent or inconsistent?
Answer: Consistent
If you do get one solution for your final answer, would the equations be dependent or
independent?
Answer: Inconsistent
The graph to the side illustrates a system of two
equations and two unknowns that has one solution:
Linear equations in Two Variables Cont.
No Solution Explained
If the two lines are parallel to each other, they will never intersect. This means they do not
have any points in common. In this situation, you would have no solution.
If you get no solution for your final answer, is this system consistent or inconsistent?
Answer: Inconsistent
If you get no solution for your final answer, would the equations be dependent or
independent?
Answer: Independent
The graph to the side illustrates a system
of two equations and two unknowns that
has no solution:
Hint: Perpendicular lines never have solutions
Linear equations in Two Variables Cont.
Infinite Solutions Explained:
If the two lines end up lying on top of each other, then there is an infinite number of
solutions. In this situation, they would end up being the same line, so any solution that would
work in one equation is going to work in the other.
If you get an infinite number of solutions for your final answer, is this system consistent or
inconsistent?
Answer: Consistent
If you get an infinite number of solutions for your final answer, would the equations be dependent
or independent?
Answer: Dependent
The graph to the side illustrates a system
of two equations and two unknowns that
has an infinite number of solutions:
Linear Systems
Solving By Substitution Explained:
The method of solving "by substitution" works by solving one of the equations (you
choose which one) for one of the variables (you choose which one), and then
plugging this back into the other equation, "substituting" for the chosen variable and
solving for the other. Then you back-solve for the first variable. Here is how it works.
2x – 3y = –2
4x + y = 24
y = –4x + 24
4x + y = 24
Here you are
isolating “y” so you
can substitute it for
“y” in the other
system.
2x – 3(–4x + 24) = –2
2x + 12x – 72 = –2
14x = 70
x=5
Now simply plug “y” in for x
and solve.
Now you can plug this x-value back into either equation, and solve for y. But since there already exists an
expression for "y =", it will be simplest to just plug into this:
y = –4(5) + 24 = –20 + 24 = 4
Then the solution is (x, y) = (5, 4).
Linear Systems Cont.
Solving by Addition / Elimination Explained
The addition method of solving systems of equations is also called the method of
elimination. This method is similar to the method you probably learned for solving
simple equations. If you had the equation "x + 6 = 11", you would write "–6" under
either side of the equation, and then you'd "add down" to get "x = 5" as the solution.
Solve the following system using addition.
2x + y = 9
3x – y = 16
Hint: Note that the y's will cancel out. So draw an "equals" bar under the
system, and add the like terms:
2x + y = 9
3x – y = 16
5x
= 25
Now you can divide through to solve for x = 5, and then back-solve, using either of the original
equations, to find the value of y. The first equation has smaller numbers, so we will backsolve in that one:
2(5) + y = 9
10 + y = 9
y = –1
Then the solution is (x, y) = (5, –1).
It doesn't matter which equation you use for the back solving; you'll get the same answer
either way. If I'd used the second equation, I'd have gotten:
3(5) – y = 16
15 – y = 16
–y = 1
y = –1
Factoring
Factoring out the Greatest Common Factor (GCF) is perhaps the most used type of factoring because
it occurs as part of the process of factoring other types of products. Before you can factor trinomials,
for example, you should check for any GCF.
#1: Factor the following problem completely 2x-14
Look for the greatest factor common to every term
Answer: 2
Factor out the GCF by dividing it into each term
Answer 2(x-7)
Oftentimes when there is no factor common to all terms of a polynomial there will be factors common
to some of the terms. A second technique of factoring called grouping is illustrated in the following
examples.
#2. Factor the following problem completely 3ax+6ay+4x+8y
Factor out 3a from the first 2 terms and 4 from the last 2 terms.
Answer:
Notice that the terms inside each set of parentheses are the same. Those terms have now become the
GCF. The answer may be checked by multiplying the factored form back out to see if you get the
original polynomial.
Final Answer: (3a+4)(x+2y)
Hint: Grouping is only effective if there is a GCF between factors like in this problem
(x+2y)
Factoring Cont.
A difference in two perfect squares by definition states that there must be two terms, the sign
between the two terms is a minus sign, and each of the two terms contain perfect squares. The
answer after factoring the difference in two squares includes two binomials. One of the binomials
contains the sum of two terms and the other contains the difference of two terms. In general, we
say
#6: Factor the following problem completely
a. Examine the problem for a GCF. There is none.
2.
To factor a difference in two squares, use two sets of parentheses.
3.
Take the square root of each term. The square root of a variable’s exponent will be half of the
exponent.
and
Use the square roots to fill in the parentheses. Be sure to check that neither factor will factor again.
What is the final answer?
Final Answer: (3x+4y)(3x-4y)
4.
Factoring Cont.
Factoring the sum or difference in two perfect cubes is our next technique. As with squares, the difference in two cubes
means that there will be two terms and each will contain perfect cubes and the sign between the two terms will be
negative. The sum of two cubes would, of course, contain a plus sign between the two perfect cube terms. The follow
formulas are helpful for factoring cubes: Sum:
Difference: Notice that the sum and the difference are exactly the same except for the signs in the factors. Many
students have found the acronym SOAP extremely helpful for remembering the arrangement of the signs. S
represents the fact that the sign between the two terms in the binomial portion of the answer will always be the same
as the sign in the given problem.
O implies that the sign between the first two terms of the trinomial portion of the answer will be the opposite of the
sign in the problem.
AP states that the sign between the final two terms in the trinomial will be always positive.
Factor the following problem completely
This is a difference in two cubes, so begin with two sets of parentheses.
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In the first set, there will be a binomial containing the cube root of each term. In this problem, x and 3.
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In the second set there will be a trinomial. The first term of the trinomial is the square of the first term in the binomial.
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The last term is the square of the last term in the binomial.
•
The middle term is the product of the two terms in the binomial.
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You will be finished when you insert the appropriate sign between each of the terms.
•
(x-3)(x2+3x+9)
Factoring Cont.
Before factoring a trinomial, examine the trinomial to be sure that terms are
arranged in descending order. Most of the time trinomials factor to two
binomials in product form.
Factor the following problem completely.
The three terms are arranged in descending order. There is not a GCF.
Therefore the factoring process is begun by opening two sets of parentheses.
•
Place the factors for the first term of the trinomial in the front of each set of
parentheses.
•
Then, because the sign of the last term is positive, factor the last term of the
trinomial to factors that multiply to give 12 and add to give 7.
•
Finally, because the sign of the last term is positive, the sign of the 4 and the
sign of the 3 will each have the same sign. Because the sign of the 7 is
positive, the sign of the 4 and the sign of the 3 will each be a positive sign.
Check the answer using multiplication.
•
(x+4)(x+3)
Factoring Cont.
A general trinomial is one whose first term has a coefficient that can not be factored out as a GCF. The method
of trial and error will be used to mentally determine the factors that satisfy the trinomial. We will show
you the steps to factor each of the following general trinomials completely.
Factor the following problem completely.
Factor out the GCF.
•
In factoring the general trinomial, begin with the factors of 12. These include the following: 1, 12, 2, 6, 3, 4. As a
general rule, the set of factors closest together on a number line should be tried first as possible
factors for the trinomial.
•
The only factors of the last term of the trinomial are 1 and 3, so there are not other choices to try. Because the
last term is negative the signs of the factors 1 and 3 must be opposite.
•
This is the first trial. The answer must be checked by multiplication, as follows:
•
Rational Expressions
Rational Expressions Cont.
Solve Problems
First Factor
Once Factored simply
cancel like terms
Final Answer
Rational Expressions Cont.
Addition and Subtraction of Rational Functions
Explained: To add and subtract rational functions, we follow the same method as fractions.
Step 1 Factor everything and find the least common denominator.
Step 2 Multiply the numerators and the denominators by the appropriate denominator so that the denominator
becomes the least common Denominator.
Step 3 Add the numerators together.
Step 4 Factor the numerator.
Step 5 Cancel any common factors.
Multiplication of Rational Functions
Recall that when we multiply fractions we first cross cancel. When we multiply rational expressions we follow
the same approach. First we factor then we cross cancel.
x2 - 2x + 1
x2 + 4x + 3
First Factor
x +1
x-1
(x - 1)2
(x + 3)(x + 1)
Cancel the x + 1 and the x - 1 and the x - 1
x+1
x-1
= (x - 1)(x + 3)
Rational Functions Cont.
Division of Rational Functions
Explained: Essentially division of rational
functions is the same as multiplication.
Instead you flip the term behind the
division side.
Example:
Quadratic Equations in One Variable
Quadratic Equations in One Variable
Explained: A quadratic equation in x is any equation that may be
written in the form
ax2 + bx + c = 0, where a, b, and c are coefficients and a ≠ 0.
Note that if a=0, then the equation would simply be a linear
equation, not quadratic.
Examples
x2 + 2x = 4 is a quadratic since it may be rewritten in the form
ax2 + bx + c = 0 by applying the Addition Property of Equality
and subtracting 4 from both sides of =.
(2 + x)(3 – x) = 0 is a quadratic since it may be rewritten in the
form ax2 + bx + c = 0 by applying the Distributive Property to
multiply out all terms and then combining like terms.
x2 - 3 = 0 is a quadratic since it has the form ax2 + bx + c = 0
with b=0 in this case.
Solving Quadratic Equations – Method 1 - Factoring
The easiest way to solve a quadratic equation is to solve by factoring, if possible.
Here are the steps to solve a quadratic by factoring:
1. Write your equation in the form ax2 + bx + c = 0 by applying the Distributive
Property, Combine Like Terms, and apply the Addition Property of Equality to
move terms to one side of =.
2. Factor your equation by using the Distributive Property and the appropriate
factoring technique. Note: Any type of factoring relies on the Distributive
Property.
3. Let each factor = 0 and solve. This is possible because of the Zero Product
Law.
Example: Solve (3x + 4)x = 7
(3x + 4)x = 7 Given
3x2 + 4x = 7 by the Distributive Property
3x2 + 4x – 7 = 0 by the Addition Property of Equality
Now, factor 3x2 + 4x – 7 = 0
This factors as (3x + ?)(x - ?) = 0 or (3x - ?)(x + ?) = 0 where the two unknown
numbers multiply to -7 when we use the Distributive Property to multiply out.
Also the first two terms must multiply out to 3x2. The middle products must
add
up to 4x.
(3x + 7)(x - 1) = 0 gives us middle products 7x and –3x adding up to 4x.
Solving Quadratic Equations – Method 2 – Extracting Square Roots
Extracting square roots is a very easy way to solve quadratics, provided the
equation is in the correct form. Basically, Extracting Square Roots allows you
to rewrite x2 = k as x = ±√k, where k is some real number. Algebraically, we
are taking square roots of both sides of the equation as shown below and
inserting the ± to account for both a positive and negative
case. Note that the squared quantity must be isolated on one side of = before
you can extract the square roots.
Example: Solve x2 = 9 by extracting square roots
Example: Solve (2x – 5)2 + 5 = 3
(2x – 5)2 + 5 = 3 Given
(2x – 5)2 = -2 Addition Property of Equality used to add –5 to both sides
√ (2x – 5)2 = ±√(-2) Extract Square Roots
2x – 5 = ± i√2 Simplify Radicals and Apply Definition of “i”
2x = 5 ± i√2 Addition Property of Equality
x = (5 ± i√2) / 2 Division Property of Equality
Solving Quadratic Equations – Method 3 – Completing The Square
This method of solving quadratic equations is straightforward, but requires a specific sequence of
steps. Here is the procedure:
Example: Solve 3x2 + 4x – 7 = 0 By Completing The Square
1. Isolate the x2 and x-terms on one side of = by applying the Addition Property of
Equality.
3x2 + 4x = 7
2. Apply the Division Property of Equality to divide all terms on both sides by the
coefficient on x2.
(3x2)/3 + (4x)/3 = 7/3
x2 + (4/3)x = 7/3
(3x2)/3 + (4x)/3 = 7/3
x2 + (4/3)x = 7/3 Note: Steps 1 and 2 may be done in either order.
3. Take ½ of the coefficient on x. Square this product. Add this square to both sides
using the Addition Property of Equality. In this case, we take ½ of 4/3 which is (1/2)•(4/3)
= 4/6. Square 4/6 to get (4/6) •(4/6) = 16/36 = 4/9 when reduced. Add 4/9 to both sides to
get
x2 + (4/3)x + 4/9 = 7/3 + 4/9
x2 + (4/3)x + 4/9 = 21/9 + 4/9 multiply 7/3 by 3/3 to get common denominator
x2 + (4/3)x + 4/9 = 25/9 add fractions
4. Factor the left side. Note: It will always factor as (x ± the square root of what you
added)
2(x + 2/3)2 = 25/9
5. Solve by extracting square roots.
√ (x + 2/3)2 = ±√(25/9) Extract Square Roots
x + 2/3 = ±5/3 Simplify Radicals
x = -2/3 ± 5/3 Addition Property of Equality
Solving Quadratic Equations – Method 4 – Using The Quadratic Formula
Solving a quadratic equation that is in the form ax2 + bx + c = 0 only involves
plugging a, b, and c into the formula
Example: Solve (x + 3)2 = x – 2
(x + 3)2 = x – 2 Given
(x + 3)(x + 3) = x – 2 Rewrite
x2 + 6x + 9 = x – 2 Multiply out with Distributive Property, Combine Like
Terms
x2 + 5x + 11 = 0 Addition Property of Equality - add 2, add –x to both sides
Plug a=1, b=5, c =11 from 1x2 + 5x + 11 = 0 into the Quadratic Formula to
get
which simplifies to
after we simplify the radical and rewrite √(-19) as (√19) • i by applying the
definition of i.
Quadratic Discriminant
The discriminant is a number that can be calculated from any quadratic equation A quadratic
equation is an equation that can be written as
ax ² + bx + c where a ≠ 0
The discriminant in a quadratic equation is found by the following formula and the discriminant
provides critical information regarding the nature of the roots/solutions of any quadratic equation.
discriminant= b² − 4ac
Positive Discriminant
b² − 4ac > 0
Two Real Solutions If the
discriminant is a perfect
square the roots are rational.
Otherwise, they are
irrational.
Discriminant of Zero
b2-4ac=0
There will be one solution
Negative Discriminant
b2-4ac<0
There are no real solutions
Functions
F of x
f(x) means the function of x. It's a short hand way of saying the values that make up a
line which are found from some function (or equation) based on x.
Are all relations functions?
Functions are relations only when every input has a distinct output, so no, not all
relations are functions but all functions are relations.
State the domain and range of the following relation. Is the relation a function?
{(2, –3), (4, 6), (3, –1), (6, 6), (2, 3)}
The above list of points, being a relationship between certain x's and certain y's, is a
relation. The domain is all the x-values, and the range is all the y-values. To give the
domain and the range, I just list the values without duplication:
domain: {2, 3, 4, 6}
ange: {–3, –1, 3, 6}
Functions Cont.
Given to order pairs of data, find a linear function that contains those points.
This question can be easily done in a few steps
1.
Find the Slope with the slope formula
2.
Use the slope-point formula to find y intercept
These steps would look like this in a problem
{(1,2)(2,6)}
1.
6-2 = 4 M=4
Here you are using the slope formula
2-1 = 1
2.
y-2=4(x-1)
Here you are using the slope-point formula
=y-2=4x-2
to find the y intercept.
=y=4x
Graphing Parabolas
Graphing the Parabola y = ax2 + bx + c
1. Determine whether the parabola opens upward or downward.
a. If a > 0, it opens upward.
b. If a < 0, it opens downward.
2. Determine the vertex.
a. The x-coordinate is .
b. The y-coordinate is found by substituting the x-coordinate, from
step 2a, in the equation y = ax2 + bx + c.
3. Determine the y-intercept by setting x = 0.
4. Determine the x-intercepts (if any) by setting y = 0, i.e., solving the
equation
ax2 + bx + c = 0.
5. Determine two or three other points if there are no x-intercepts.
Graphing Parabolas Cont.
Graph of y = x 2
Simplifying Expressions with Radicals
When presented with a problem like
, we don’t have too much difficulty
saying that the answer 2 (since ). Even a problem like
is easy once we
realize
. Our trouble usually occurs when we either can’t easily see the
answer or if the number under our radical sign is not a perfect square or a perfect
cube.
A problem like
may look difficult because there are no two numbers that
multiply together to give 24. However, the problem can be simplified. So even though
24 is not a perfect square, it can be broken down into smaller pieces where one of
those pieces might be perfect square. So now we have .
Simplifying a radical expression can also involve variables as well as numbers. Just
as you were able to break down a number into its smaller pieces, you can do the
same with variables. When the radical is a square root, you should try to have terms
raised to an even power (2, 4, 6, 8, etc). When the radical is a cube root, you should
try to have terms raised to a power of three (3, 6, 9, 12, etc.). For example,
These types of simplifications with variables will be helpful when doing operations
with radical expressions. Let's apply these rule to simplifying the
Simplifying Expressions with Radicals
Examples:
Line of Best Fit or Regression Line
A regression line is a line drawn through a scatter plot of two
variables. The line is chosen so that it comes as close to the points
as possible. A graphing calculator is very helpful for finding this.
Below is a scatter plot with a regression line in it.
Word Problems
If Sally can paint a house in 4 hours, and
John can paint the same house in 6
hours, how long will it take for both of
them to paint the house together?
a. 2 hours and 24 minutes
b. 3 hours and 12 minutes
c. 3 hours and 44 minutes
d. 4 hours and 33 minutes
Solution
If Sally can paint the house in 4 hours, then in 1 hour she can
paint 1/4 of the house.
If John can paint the house in 6 hours, then in 1 hour he can
paint 1/6 of the house.
Let x = number of hours it would take them together to paint the
house, then working together, in 1 hour they can paint 1/x of
the house.
The equation is this:
Multiply both sides by the LCD which is 12x:
= 2.4 hours or A
Word Problem
If 2 pens and 3 notebook cost 4.55 and 3
pens and 2 note books cost 3.70, find the
price of each pen
a. 1.25
b. .50
c. .40
d. 1.00
e. .67
Solution
P: Pens
N: Notebooks
-2(2p+3n=4.55)
3(3p+2n=3.70)
p=.40 or C
Use elimination method.
Word Problem
The Sales price of a car is 12,590, which is
20% off the original price. What is the
original price?
a. 14,000
b. 14,670.30
c. 17,894
d. 15,737.50
Solution
Multiply .2 by 12,950 and then add that
number to 12,950 to get 15,737.50 or D
Word Problem
Employees of an appliance store receive an
additional 20% off of the lowest price on
an item. If an employee purchases a
dishwasher 15% off sale, how much will
he pay if the dish washer orginally cost
450?
a. 287.96
b. 333.39
c. 306
Solution
Simply find how much 15% of 450 is
subtract that from 450. Take that
subtracted number and multiply it by 20%
and subtract that from that number to get
306
Citations
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http://www.tutorvista.com/content/math/algebra/linear-two-variable/linear-equationstwo-variable.php
http://hotmath.com/search/hotmath-search.jsp?term=disjunction
http://www.chacha.com/topic/properties-of-square-roots
http://philosophy.lander.edu/logic/conjunct.html
http://rachel5nj.tripod.com/NOTC/ssoewog2.html
http://www.tutorvista.com/content/math/algebra/linear-two-variable/linear-equationstwo-variable.php
http://honolulu.hawaii.edu/distance/sci122/SciLab/L6/bestfit.html
http://www.algebralab.org/lessons/lesson.aspx?file=Algebra_radical_simplify.xml
http://a-s.clayton.edu/garrison/math%200099/parabola.htm
http://www.purplemath.com/modules/rtnldefs.htm
http://www.mathnstuff.com/math/spoken/here/2class/320/quadequ.htm
http://www.tpub.com/math1/17.htm
http://www.sosmath.com/diffeq/system/linear/basicdef/basicdef.html
FIN