Transcript Document

An Explosion of Math!!!!
By: Matt and Nick
Quick 1st Power Equation
Example: 4x=12
Answer: x=3
Special Cases of These
Equations
• A. x3-7x2=-6x
-6x=-6x= (All real #’s)
• B. 5x/3 + 7/2 = 4
6*5x/3 + 6*7/2 = 6*4
10x+21 = 24
10x = 24-21
10x = 3
x = 3/10
• C. 4/x=12 x=3
Addition Property (of Equality)
Example:
If x = y, then x + z = y + z.
If a+2=7, then a+2+-2=7+-2
Multiplication Property (of Equality)
Example:
If a = b, then a * c = b * c
Reflexive Property (of Equality)
Example:
3m=3m
Symmetric Property (of Equality)
Example:
If m=n, then n=m
Transitive Property (of Equality)
Example:
If m=n and n=p, then m=p
Associative Property of Addition
Example:
(7+1/4)+3/4=7+(1/4+3/4)
Associative Property of
Multiplication
Example:
a(bc) = (ab)c
Commutative Property of
Addition
Example:
1/4+7+3/4=1/4+3/4+7
Commutative Property of
Multiplication
Example:
ab = ba
Distributive Property (of
Multiplication over Addition)
Example:
If -3(x-2)=1, then -3x+6=1
Prop of Opposites or Inverse
Property of Addition
Example:
a+(-a)=0
Prop of Reciprocals or Inverse
Prop. of Multiplication
Example:
-3/x*-x/3=1
Identity Property of Addition
Example:
0+a=a=a+0
Identity Property of Multiplication
Example:
1*a=a=a*1
Multiplicative Property of Zero
Example:
a×0=0
Closure Property of Addition
Example:
If x and y are real numbers, then x+y is a real
number.
Closure Property of Multiplication
Example:
If x and y are real numbers, then x*y is a real
number.
Product of Powers Property
Example:
ab × ac = a(b + c)
Power of a Product Property
Example:
(ab)m = am · bm
Power of a Power Property
Example:
(ab)c = abc
Quotient of Powers Property
Example:
Power of a Quotient Property
Example:
Zero Power Property
Example:
170 = 1
Negative Power Property
Example:
x-3=1/x3
Zero Product Property
Example:
If ab = 0, then either a = 0 or b = 0 (or
both).
Product of Roots Property
Example:
5  5  25
Quotient of Roots Property
Example:
x
x
x


9
3
9
Root of a Power Property
Example:
3
x x
3
Power of a Root Property
Example:
 x
2
x
Quiz Time!!!
***You will see an example problem
and you will click to see the answer!
There are 10 Problems so it should
only take a few minutes to complete.
Have Fun!
x9*x3=x12
Product of Powers Property
(xy)3= x3y3
Power of Product Property
x3=x3
Reflexive Property of Equality
x3*0=0
Multiplicative Property of Zero
If x-3=9, then x-3+3=9+3
Addition Property of Equality
If x and y are real numbers,
then x+y is a real number.
Closure Property of Addition
x3*1=x3
Identity Property of Multiplication
(x9)3=x27
Power of a Power Property
9(x-y)=9x-9y
Distributive Property
y3x=xy3
Commutative Property of
Multiplication
First Power Inequalities
***In the following slides you will
see how to solve first power
inequalities.
One Inequality Sign
X+3<6
Answer:
X<3
***To answer this, you would subtract 3 from both sides and end
up isolating the variable on the left side and 3 on the other. The
inequality sign would stay the same because you are not
multiplying/dividing by a negative number.
Conjunction
-2<x and x<3
Answer:
-2<X<3
***To solve a conjunction of two open sentences in a
given variable, you find the values of the variable for
which both sentences are true.
Disjunction
y-2<-5 or y-2>5
Answer:
y<-3 or y>7
***To solve a disjunction of two open sentences, you
find the values of the variable for which at least one
of the sentences is true.
All Real #’s
n+5
Answer:
n+5
{All real Numbers}
***As you can see, the inequalities cancel out
to leave a technically true statement leaving
the answer to be “All real numbers”
No Solution
x + 5 > 10 and x -2 < 1
Answer:
No Solution
***Two inequalities have no solution when both of
them must be true and they result in mutually
exclusive conditions. Thus, there is no number that
is both greater than 5 and less than 3, therefore
there is no solution.
How To Do Linear Equations
•
•
•
•
•
Slopes of All Lines:
Rising line-positive slope
Falling line-negative slope
Vertical line- undefined
Horizontal line- 0
•
•
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•
Equations of All Lines
Horizontal- y=c
Vertical- x=c
Diagonal- y=mx+b and Ax+By=C
Linear Equations Cont.
• Standard/general form: Ax+By=C
• Point-slope form: y-y1=m(x-x1)
• Slope intercept form: y=mx+b
• How to Graph: Video from Math TV
• Click here to Graph y=3x-1
Linear Equations Cont.
• How to Find Intercepts
1.Put the equation into Slope-Intercept form
2.Y=mx+b
3.The “b” in the equation is your Y-intercept
Linear Systems
Substitution Method
1. Solve the first equation for y
2. Substitute this expression for y in the
other equation, and solve for x.
3. Substitute the value of x in the equation
in Step 1, and solve for y.
***P.417 in your book has great examples!
Elimination Method
1. Add similar terms of the two equations
2. Solve the resulting equation
3. Substitute what you got for x and plug it
into either of the equations and solve for y
***P.426 in your book has great examples!
Systems of Equations
• Independent- two distinct non-parallel lines
that cross at exactly one point (solution is
always some x,y-point)
• Dependant- two lines that intersect at
every point (solution is the whole line)
• Inconsistent- shows two distinct lines that
are parallel (never intersect), has no
solution
• ***Graphs of these terms are on following
slide!
Graphs from www.purplemath.com
Independent system:
one solution and
one intersection point
Inconsistent system:
no solution and
no intersection point
Dependent system:
the solution is the
whole line
Factoring
• Grouping (2x2 and 3x1)- You use this when you
have 4 or more terms
• GCF- You use this when you have any number
of terms
• Difference of Squares- Use this with Binomials
• Sum and Difference of Cubes- Use with
Binomials
• PST- Trinomials
• Reverse FOIL-Trinomials
Rational Expressions
Factor and Cancel
*Factor first!
*Common factor in
both the numerator
and the
denominator and so
we can cancel the
x-4 from both
Answer
Rational Expressions
Addition and Subtraction of Rational
Expressions
*Common
denominator is:
6x5
*Multiply each
term by an
appropriate
quantity to get
this in the
denominator
and then do
the addition
and
subtraction
Answer
Rational Expressions*The first thing
Multiplication of Rational
Expressions
that we should
always do in the
multiplication is to
factor everything
in sight as much
as possible
*Cancel as
much as we
can and then
do the
multiplication
to get the
answer
Answer
Rational Expressions
Division of rational expressions
*Divide
first!
*Once we’ve done the
division we have a
multiplication problem
and we factor as much as
possible, cancel
everything that can be
canceled and finally do
the multiplication.
Answer
Quadratic Equations in one
Variable
• Quadratic –second power
• Use the discriminant to predict how many
x-intercepts each parabola will have.
Solve by Factoring
So the first thing to do is factor:
x2 + 5x + 6 = (x + 2)(x + 3)
Set this equal to zero:
(x + 2)(x + 3) = 0
Solve each factor:
x + 2 = 0 or x + 3 = 0
x = –2 or x = – 3
Answer: x2 + 5x + 6 = 0 is x = –3, –2
Quadratic Equations in one
Variable Cont.
Taking the square root of each side:
m2=49
m= √49
m= 7
Method of Completing the Square
1.
2.
3.
4.
For: x2+bx+?
Find half the coefficient of x: b/2
Square the result of step 1: (b/2)2
Add the result of step 2 to x2+bx:
x2+bx+(b/2)2
You have completed the square:
x2+bx+(b/2)2=(x+b/2)2
Quadratic Equations in one
Variable Cont.
Complete the square:
X2+14x=?
X2+14x+49= (x+7)2
(14/2)2=49
Quadratic Equations in one
Variable Cont.
Quadratic Formula:
x=-b+√b2-4ac/2a
Line of Best Fit or Regression Line
• We use this to find an equation for a
scatter plot.
• Your calculator will help you find the best
fit line
• The calculator will find an exact regression
line
Line of Best Fit or Regression Line
Cont.
Write an equation of a line that has a slope of -4
and x-intercept of 3.
1. Substitute -4 for m in y=mx+b
2. To find b, substitute 3 for x and 0 for y in y=4x+b
y=-4x+b
0=-4(3)+b
0=-12+b
12=b
Final answer: y=-4x+12
Functions
• A. f(x) means "y“ and not all functions are relations
• B. A function can only use each x-value once
– Domain- set of all x-coordinates (independent)
– Range- set of all y-coordinates (dependant)
• Find the range given f(x)=5x-3 and Domain={-2,0,7}.
f(-2)=5(-2)-3=-13
f(0)=5(0)-3=-3
f(7)=5(7)-3=32
Range={-13,-3,32}
• C. We will show hot to do a Parabola on the next slide
How to Graph a Parabola
1. The easiest way to graph a parabola is to start by finding
the x-coordinate of the vertex, or the turning point of the
function. Given a parabola with a general equation of
y=ax²+bx+c, the x-coordinate of the vertex can be found by
using x=-b/2a, which is the equation of the axis of
symmetry for the parabola. The axis of symmetry runs
through the vertex, and therefore shares a common point.
2. Substitute the x value of the vertex back into the function to
find the y value of the vertex.
3. After you've found your turning point, you can select two xvalues to the right of the turning point and two values to the
left of the turning point.
4. Plot all points
Simplifying Expressions with
Exponents
1. x6 × x5 = (x6)(x5)
= (xxxxxx)(xxxxx) (6 times, and then 5 times)
= xxxxxxxxxxx
(11 times)
= x11
2. Simplify (–46x2y3z)0 This is simple enough: anything to the zero
power is just 1.
(–46x2y3z) =1
3. 6
x 2
The "minus" on the 2 says to move the variable; the "minus" on the 6
says that the 6 is negative. Warning: These two "minus" signs mean
entirely different things, and should not be confused. I have to move
the variable; I should not move the 6.
*** The answer is -6x2
Simplifying Expressions with
Radicals
54
1. Simplify
54  9  6  3 6
2. Simplify
3
3
320
320  3 64  5  4 3 5
3. Simplify
4 2
ab
a b a b
4 2
2
Word Problems
1. Problem: The sum of twice a number plus 13 is 75. Find
the number.
Hint: The word is means equals. The word and means plus.
Therefore, you can rewrite the problem like the
following: The sum of twice a number and 13 equals 75.
Solution:
2N + 13 = 75
N=31
Word Problems Cont.
2. At the same moment, two trains leave Chicago and New York. They
move towards each other with constant speeds. The train from
Chicago is moving at speed of 40 miles per hour, and the train from
New York is moving at speed of 60 miles per hour. The distance
between Chicago and New York is 1000 miles. How long after their
departure will they meet?
x/40 = (1000-x)/60, we can simplify it as x/40 + x/60 = 1000/60, or
x = 40 * 1000/(40 + 60) = 400
*The time that it takes the train from Chicago to travel 400 miles, is x
divided by the speed of the Chicago train, which is t = x / 40 = 10.
***So, the answer is: 10 hours.
Word Problems Cont.
3. The total receipts for a hockey game are $1400 for 788 tickets sold. Adults
paid $2.50 for admission and students paid $1.25. How many of each kind
of tickets were sold?
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•
•
•
•
•
Alright, let's denote the quantity of ADULT tickets sold as x. Since the total number of
tickets is The quantity of student tickets was 788, the number of student tickets is 788-x.
What we need to do now is write the total $$ figure for the revenue. Since every adult ticket
fetched 2.50, adults collectively have paid x*2.50. The students paid 1.25, and since we had
788-x of them, they paid the sum of (788-x)*1.25. So, we have total revenue = x*2.50 + (788x)*1.25
What we know is that the total revenue for the basketball game was 1400. This gives us the
equation
1400 = x*2.50 + (788-x)*1.25
Rewriting, we get
x*(2.50-1.25) = 1400 - 788*1.25 or x*1.25 = 415 or x = 415/1.25 or x = 332
That's the number of adult tickets. The number of student tickets is, therefore, 788-x, or
456. That's it!
***The
answer is 332 adult tickets and 456 student tickets!
Word Problems Cont.
• Alright. So, we have 4(x-4) = 3x-4 or 4x - 4*4 = 3x-4 or, moving
everything x-related to the left and numbers to the right, 4x - 3x =
4*4 - 4 x(4 - 3) = 4*4 - 4 or, dividing by 4 - 3: x = (4*4 - 4) / ( 4 - 3 )
or, calculating x, x = 12
***Bob's age is 12. His father is 36 years old.
The End!
Hope you enjoyed the show and
have a great summer vacation!