Algebra (Period 1)

Introduction to Quadratic Equations: 13- 1

f(x) = ax² + bx + c where a, b, and c are real numbers and a ≠ 0.
Standard form for a quadratic.
Put 4x² + 7x = 5 in standard form
it becomes
4x² + 7x - 5 = 0

Now you can determine a, b, and c
a = 4
b = 7
c = -6

What about
5x² = -4x

in standard form it is:
5x² + 4x = 0

a = 5
b = 4
c = 0


Example 3.

5x² = -4

5x² + 4 = 0

a = 5
b = 0
c = 4

Solving any
ax² + bx = 0

2x² - 15x = 0
Factor.. think GCF...
5x(4x -3) = 0
ZERO PRODUCTS property
5x = 0 and 4x - 3 = 0
so x = 0 and x = 3/4

{0. 3/4}

Any quadratic in this form a
ax² + bx = 0
has 2 roots
One of the roots is always 0 and the other is -b/a

We tried
10x² - 6x = 0
we found by factoring that we did get x = 0 and x = 3/5


Graphing quadratics:
You can graph quadratics exactly the way you graphed lines
by plugging in your choice of an x value and using the equation to find your y value.

Because it's a U shape, you should graph 5 points as follows:
STEP 1: determine Point 1: the vertex -
the minimum value of the smile or
the maximum value of the frown

The x value of the VERTEX = -b/2a
We get the values for a and b from the actual equation
f(x) = ax² + bx + c
just plug in the b and the a value from your equation into -b/2a and you have the x-value of the vertex.
Now to find the y value -- take that x value and PLUG it into the equation


STEP 2: Next, draw the AXIS OF SYMMETRY : x = -b/2a
a line through the vertex parallel to the y axis . Draw this line as a dashed line. REMEMBER: It will be a dashed line parallel to the y-axis


STEP 3: Point 2- Pick an x value to the right or left of the axis and find its y by plugging into the equation.

STEP 4: Point 3- Graph its mirror image on the other side of the axis of symmetry by counting from axis of symmetry

STEP 5: Points 4 and 5- Repeat point 2 and 3 directions with another point even farther from the vertex

JOIN YOUR 5 POINTS IN A "U" SHAPE AND EXTEND LINES WITH ARROWS ON END

Parabolas that are functions have domains that are ALL REAL NUMBERS
Their ranges depend on where the vertex is and also if the a coefficient is positive or negative

EXAMPLE: f(x) = -3x² (or y = -3x²)
the a coefficient is negative so it is a frown face
the x value of the vertex (maximum) is -b/2a or 0/2(-3) = 0
the y value of the vertex is 0
So the vertex is (0, 0)


To graph this function:
1) graph vertex (0, 0)
2) Draw dotted line x = 0 (actually this is the y axis!)
3) Pick x value to the right of axis of symmetry, say x = 1
Plug it in the equation: y = -3(1) = -3
Plot (1, -3)
4) count steps from axis of symmetry and place another point to the LEFT of axis in same place
5) pick another x value to the right of the axis of symmetry, say x = 2
plug it in the equation y = -3(2) = -6
plot ( 2, -6). Count the steps from the axis of symmetry and place another point to the LEFT of the axis in the same place.

The domain is all real numbers.
The range is y is less than or equal to zero

Pre Algebra (Period 2 & 4)

Exponents ( continued) 9

Review the odd/even rule


IF THERE IS A NEGATIVE INSIDE PARENTHESES:

Odd number of negative signs or odd power = negative

Even number of negative signs or even power = positive



EXAMPLES:
(-2)⁵ = -32

(-2)4<.sup>= +16



IF THERE IS A NEGATIVE BUT NO PARENTHESES:
ALWAYS NEGATIVE!

-2⁵ = -32

-2⁴ = -16

Finish CHAPTER 4-8

NEGATIVE POWERS = FRACTIONS

They're in the wrong place in the fraction

m³/m⁵ = m^-2     

   m³/m⁵

= mmm/ mmmmm

= 1/mm



Again, by transitive property of equality:


m³/m⁵ = m^-2 = 1/m²

Remember the rule of powers with (  ) 

When there is a product inside the (  ), then everything inside is to the power!

If there are no (  ), then only the variable/number right next to the power is raised to that power.

3x^-2does not equal (3x)^-2
The first is 3/x² and the second is 1/9x²

RESTATE A FRACTION INTO A NEGATIVE POWER:

1) Restate the denominator into a power

2) Move to the numerator by turning the power negative



EXAMPLE:

1/32

1/(2)⁵

Algebra (Period 1)

Linear Functions 12-3 and Quadratic Functions: 12- 4

Linear Functions

y = mx +b is a linear function.
f(x) = mx + b

graph it-- we always get a straight line

Real Life problems

Rental Truck Co charges $35 a day plus $.21 per mile. Find the cost of renting a truck for a day trip of 340 miles.

f(x) = .21m + 35
f(340) = .21(340) + 35
f(340) = 106.50
$106.50

The cost of renting a chain saw is $5.90 a hour plus $6.50 for a can of gas. Find the cost of using the chain saw for 7.5 hours.

f(h) = 5.90h + 6.50
f(7.5) = 5.90(7.5) = 6.50
f(7.5) = 50.75
$50.75

Quadratic Functions: 12- 4
A QUADRATIC FUNCTION is not y = mx + b
(which is a LINEAR function),
but instead is
y = ax² + bx + c
OR
f(x) = ax² + bx + c
where a, b, and c are all real numbers and
a cannot be equal to zero because
it must have a variable that is squared ( degree of 2)
[If a = 0, then we would end up with y = bx + c which is really y= mx + b]

Quadratics have a squared term, so they have TWO possible solutions also called roots. You already saw this in Chapter 6 when you factored the trinomial and used zero products property!! ( CHAPTER 6-- again)

If the domain is all real numbers, then you will have a PARABOLA which looks like
a smile when the a coefficient is positive or
looks like a frown when the a coefficient is negative.


How is the graph of y = 2x² related to the graph of y = x²?
How is the graph of y = 2x² related to the graph of y =-2x²?

The vertex is the maximum or minimum point of a parabola. It is the maximum point with a quadratic such as y = -x²
and it is a minimum point with a quadratic in the form y = x²
Axis of symmetry- if you fold the graph so the two sides of the parabola coincide, the the fold line is the axis of symmetry.
THe y-axis is the axis of symmetry for all equations of the form y = ax².

For a parabola defined by the equation y = ax² + bx + c
the x coordinate of the vertex is -b/2a
and the line of symmetry is x = -b/2a





Graphing quadratics:
You can graph quadratics exactly the way you graphed lines
by plugging in your choice of an x value and using the equation to find your y value.

Because it's a U shape, you should graph 5 points as follows:
STEP 1: determine Point 1: the vertex -
the minimum value of the smile or
the maximum value of the frown

The x value of the VERTEX = -b/2a
We get the values for a and b from the actual equation
f(x) = ax² + bx + c
just plug in the b and the a value from your equation into -b/2a and you have the x-value of the vertex.
Now to find the y value -- take that x value and PLUG it into the equation


STEP 2: Next, draw the AXIS OF SYMMETRY : x = -b/2a
a line through the vertex parallel to the y axis . Draw this line as a dashed line. REMEMBER: It will be a dashed line parallel to the y-axis


STEP 3: Point 2- Pick an x value to the right or left of the axis and find its y by plugging into the equation.

STEP 4: Point 3- Graph its mirror image on the other side of the axis of symmetry by counting from axis of symmetry

STEP 5: Points 4 and 5- Repeat point 2 and 3 directions with another point even farther from the vertex

JOIN YOUR 5 POINTS IN A "U" SHAPE AND EXTEND LINES WITH ARROWS ON END

Parabolas that are functions have domains that are ALL REAL NUMBERS
Their ranges depend on where the vertex is and also if the a coefficient is positive or negative

EXAMPLE: f(x) = -3x² (or y = -3x²)
the a coefficient is negative so it is a frown face
the x value of the vertex (maximum) is -b/2a or 0/2(-3) = 0
the y value of the vertex is 0
So the vertex is (0, 0)


To graph this function:
1) graph vertex (0, 0)
2) Draw dotted line x = 0 (actually this is the y axis!)
3) Pick x value to the right of axis of symmetry, say x = 1
Plug it in the equation: y = -3(1) = -3
Plot (1, -3)
4) count steps from axis of symmetry and place another point to the LEFT of axis in same place
5) pick another x value to the right of the axis of symmetry, say x = 2
plug it in the equation y = -3(2) = -6
plot ( 2, -6). Count the steps from the axis of symmetry and place another point to the LEFT of the axis in the same place.

The domain is all real numbers.
The range is y is less than or equal to zero

Math 6 Honors (Period 6 and 7)

Ratios 7-6

In our textbook, the example given involves the number of students --at what I called a mythical middle school --as well as the number of teachers. There are 35 teachers and 525 students. We can compare the number of teachers to the number of students by writing a quotient


number of teachers
number of students

35
525

1/15

The quotient of one number divided by a second number is called the ratio of the first number to the second number.

We can write a ratio in the following ways:

1/15 OR 1:15 OR 1 to 15

All of these expressions are read one to fifteen.

If the colon notation is used the first number is divided by the second. A ratio is said to be lowest terms if the two numbers are “relatively prime.”
You do not change an improper fraction to a mixed number if the improper fraction represents a ratio


There are 9 players on a baseball team. Four of these are infielders and 3 are outfielders. Find each ratio in lowest terms.

a. infielders to outfielders

b. outfields to total players

# of infielders
# of outfielders
= 4/3 or 4:3 or 4 to 3



# of outfielders
# total of players
= 3/9 = 1/3 or 1:3 or 1 to 3

Some ratios compare measurements. In these cases we must be sure the measurements are expressed in the same units

It takes Vince (or Robbie) 4 minutes to mix some paint for his science project. It takes her 3 hours to complete painting his science project. What is the ratio of the time it takes Vince (or Robbie) to mix the paint to the time it takes VInce ( or Robbie) to paint his project?
Use minutes as a common unit for measuring time. You must convert the hours to minutes first

3h = 3 • 60min = 180 min

The ratio is :
min. to mix
min. to paint

= 4/180 = 1/45 or 1:45

Some ratios are in the form

40 miles per hour or 5 pencils for a dollar
“ I want my… I want my…. I want my … MPG!!”

These ratios involve quantities of different kinds and are called rates. Rates may be expressed as decimals or mixed numbers. Rates should be simplified to a per unit form. When a rate is expressed in a per unit form, such a rate is often called a unit rate.

Ashlee’s dad’s car went 258 miles on 12 gallons of gas. Express the rate of fuel consumption in miles per gallon.

The rate of fuel consumption is

258 miles
12 gallons

= 21 1/2 miles per gallon

Some of the most common units in which rates are given are the following:
mi/gal or mpg miles per gallon
mi/h or mph miles per hour
km/L kilometers per liter
km/h kilometers per hour

Pre Algebra (Period 2 & 4)

Powers of Products & Quotients 5-9

(4∙2)³ = (4∙2)∙(4∙2)∙(4∙2)
= 4∙4∙4∙2∙2∙2
= (4∙4∙4)∙(2∙2∙2)
=4³∙2³

Raising a Product to a Power

(5∙3) ⁴ = 5³∙3³

or Algebraically:
(ab)^m = a^mb^m

Remember to simplify an expression, you write it with NO like terms or paranthese.

Simplify (4x²)³
Raise each factor to the power 3
= 4³∙x²∙³
Use the rule for Raising a Power to a Power
=4³∙x⁶
simplify
= 64x⁶

WE did several of these in class:
(2p)⁴ = (2∙2∙2∙2)∙(p∙p∙p∙p)
= 16p⁴

(xy²)⁵ = x⁵y¹⁰

(5x³)² = 25x⁶

The location of a negative sign affects the value of an expression. Look at the differences between the following

(-5x² = (-5)²x² = (-5)(-5)x² = 25x²

-(5x)² = - (5)(5)x² = -25x²


(-2y)⁴ = 16y⁴
-(2y)⁴ = -16y⁴

Do you see the subtle differences? Make sure you can determine why one is postive and the other is negative

Finding Powers of Quotients

(4/5)³ = (4/5)(4/5)(4/5)

or
4∙4∙4
5∙5∙5
=
4³
5³
= 64/125

To raise a quotient to a power, raise both the numerator and the denominator to the power

(2/3)⁴ =

2⁴
3⁴
=
16/81

(1/2)³ =
1/8

(-2/3)⁴
16/81

Why is it positive?

(2x²/3)³=

8x⁶/27


∙∙∙

Algebra (Period 1)

Relations & Functions 12-1 and Functions & Graphs 12-2
RELATIONS: Set of ordered pairs where the x values are the DOMAIN and the y values are the RANGE.

FUNCTIONS: Relations where there is just one y value for each x value IN OTHER WORDS----YOU CAN'T HAVE TWO y VALUES for the SAME x value!!!
If you see x repeated twice, it's still a relation, but it's not a function.
In the real world, there are excellent examples....pizza prices.
A restaurant can't have two different prices for the same size cheese pizza.
If you charge $10 and $12 on the same day for the same pizza, you don't have a function.
But, you certainly can charge $10 for a cheese pizza and $12 for a pepperoni pizza.

VERTICAL LINE TEST: When you graph a function, if you draw a vertical line anywhere on the graph, that line will only intersect the function at one point!!!!
If it intersects at 2 or more, it's a relation, but not a function.
So a horizontal line function, y = 4, is a function, but a vertical line function, x = 4 is not.

Any line, y = mx + b, is a function.

INPUTS: x values
OUTPUTS: y values

f(x) means the value of the function at the given x value
You can think of f(x) as the y value

Finding the value of a function: Plug it in, plug it in!
f(x) = 2x + 7
Find f(3)
f(3) = 2(3) + 7 = 13
The function notation gives you more information than using y
If I tell you y = 13 you have no idea what the x value was at that point
But if I tell you f(3) = 13, you know the entire coordinate (3, 13)

Domain of a function = all possible x values (inputs) that keep the solution real
Range of a function = all possible y values (outputs) that result from the domain

EXAMPLE:
f(x) = x + 10 has the domain of all real numbers and the same range because every value will keep the answer f(x) a real number

EXAMPLE:
f(x) = x² has the domain again of all real numbers, BUT the range is greater than or = to zero
because when a number is squared it will never be negative! So f(x) will always be 0 or positive

EXAMPLE:
f(x) = absolute value of x has the domain of all real numbers, but again the range will be greater than or equal to zero because absolute value will never be negative

EXAMPLE:
f(x) = 1/x has a domain of all real numbers EXCEPT FOR ZERO because it would be undefined if zero was in the denominator. The range is all real numbers except zero as well.
This function will approach both axes but never intersect with them.
The axes are called asymptotes which means that they will get very close but never reach them

EXAMPLE:
f(x) = (x - 10)/x + 3

Domain is all real numbers EXCEPT -3 because -3 will turn the denominator into zero (undefined)
What is the range?

Pre Algebra (Period 2 & 4)

EXPONENTS 4-7 & 4-8

MULTIPLYING Powers with LIKE BASES:
Simply ADD THE POWERS

WITH VARIABLES:
m⁵m³ = m⁸
You can check this by EXPANDING:
(mmmmm)(mmm) = m⁸

WITH NUMBERS:
(2⁵)(2³) = 2⁸


DIVIDING Powers with LIKE BASES:
Simply SUBTRACT the POWERS
m⁸ = m³
m⁵
Again, you can check this by EXPANDING:
mmmmmmmm
mmmmm

ZERO POWERS:
Anything to the zero power = 1
(except zero to the zero power is undefined or indeterminate)
Proof of this was given in class:
1 =
mmmmmmmm
mmmmmmmm

=
m⁸ = m^8-8 m⁰ (by power rules for division)
m⁸

: 1 = m⁰

By the transitive property of equality

Math 6 Honors (Period 6 and 7)

Multiplication and Division of Mixed Numbers 7-5

One method of finding the product of two mixed numbers is to first change the mixed numbers into improper fractions and then multiply

6 X 3 1/12 becomes
6/1 X 37/12
Simplify using the methods taught and reviewed this week
6/1 X 37/ 12 = 37/6 = 18 1/2

5 3/4 X 4 2/3
23/4 X 14/3 becomes
23/2 X 7/ 3 = 161/6 = 26 5/6

13 1/3 X 15 3/4
40/3∙ 63/4
simplify ( GCF of 40 and 4 is 4 and GCF of 3 and 63 is 3) to
10/1 ∙ 21/1 = 210



To divide one mixed number by another, we change the mixed numbers into improper fractions and use the methods from the previous lessons

2 2/3 ÷ 10 2/3 first change to improper fractions
8/3 ÷ 32/3 Remember. that dividing by a fraction is the same a multiplying by its reciprocal
8/3 X 3/32
Using the GCF to simplify first
1/1 X 1/4 = 1/4

What about
1 7/15 ÷ 5 1/2
22/15 ÷ 11/2
22/15 X 2/11 = 4/15

or 42 2/3 ÷ 3 5/9

128/3 ÷ 32/9
128/3 ∙ 9/32
simplify and you get
4/1 ∙ 3/1 = 12

hat if we have equations such as the book shows
n X 2 1/3 = 6 5/12
First rewrite it as (2 1/3)n = 6 5/12
Then change the mixed numbers to improper fractions
(7/3)n = 77/12
Now, we know to isolate the variable, we must use the inverse operation and in this case we would multiply 7/3 by its reciprocal 3/7 to both sides
(3/7)(7/3) n = (77/12)(3/7)
n = 11/4
n = 2 3/4