Algebra Period 3 (Tuesday)

Review of how to graph a quadratic

Find the vertex ( x = -b/2a, then plug in to the equation to find the y coordinate)

Find the axis (line) of symmetry. Just use the x value of the vertex that is
x = -b/2a

Draw the line of symmetry using a dotted or dashed line

Find 2 more points in an x y table (go either left or right of the vertex and use 2 points close to the line of symmetry for the x, then plug into the equation to find the y values)

Find the 2 shadow (mirror) points by counting over from the axis at the exact same y value

Draw a U (not a V) through the points. Extend it past and put arrows at the end
Remember to label the x and y axes and put arrows on the axes.

Function vs. Relations: Functions are special relations where there is a unique x for each y
Therefore, there will never be 2 x’s that will repeat and if you use a vertical line on the graph, it will only hit one point on the graph.

Domain vs. Range: the domain for any quadratic function is ALL REAL NUMBERS
The range depends on where the vertex is and whether the quadratic is a smile or a frown.
Generally, it will be of the form:

y such that y is either greater than or equal to (≥ ) or less than or equal to (≤) the y value of the vertex. {y l y≥ n} or {y l y ≤ n}

f(x) is just a different way of saying y. What is better about it? Without seeing the work before the solution, you can actually tell the x value as well as the y value in the solution.

For example, if the solution is f(-3) = 12 you know that the point is (-3, 12)
Compare that to the solution y = 12. For that solution, you would not know the x value unless you look back in the problem.

Math 6 Honors Periods 6 & 7 (Tuesday )

Areas of Rectangles and Parallelograms 10-1

In Chapter 4 (so long ago) we measured lengths of segments and found perimeters of various polygons. Now we will measure the part of the plane enclosed by a polygon. We call this measure the area of the polygon.

In the metric system the unit area often use is the square centimeter. cm²

Notice that the area of each rectangle is the product of the lengths of the two consecutive sides. These sides are called the length and the width of the rectangle The length names the longer side and the width names the shorter side. These sides are also named the base and height.

Formula

Area of rectangle = length · width

A= lw

The length and the width of the rectangle are called its dimensions

In the case of a parallelogram, we may consider either pair of parallel sides to the bases. (the word base is also used to denote the length of the base) the height is the perpendicular distance between the bases.

Formula

Area of parallelogram = base · height

A= bh

The unit areas used are square meters (m²), square millimeters (mm²) and square centimeters (cm²). For very large regions, such as the State of California, or entire countries (Canada), we could use square kilometers (km²)

Sometimes we work with an unspecified unit of length. Then the unit of area is simply called a square unit. It is vital, however, that you indicate the area is in square units.

Algebra Period 3

Compound Sentences 9-2

Conjunction = and - the graph is an intersection ("yo") and inequality looks like our domains and ranges on our projects EXAMPLE: 5 < x < 10
open dots; between 5 and 10 is colored in

Disjunction = or - graph will go opposite ways ("dorky" dancer) and inequality looks like this:
x < -2 OR x > 4
open dots; one arrow goes right at 4 and the other arrow goes left at -2

Equations and Absolute Value - 1 VARIABLE 9-3
Solve the equation twice - Once with the solution positive and once with it negative in this setting [ ] represents absolute value.
[2x - 4] = 10
Solve it twice:
2x - 4 = 10 or 2x - 4 = -10
x = 7 or x = -3

If there is a term on the same side of the equation as the absolute value, move that to the other side of the equation first (just like we did with radical equations!) Then solve twice.

REMEMBER THAT THE SOLUTION GIVEN CANNOT BE NEGATIVE (the null set)

Inequalities and Absolute Value - 1 VARIABLE 9-4
There are 2 possible types of inequalities - less than and greater than
For less thAND:
These are conjunctions and so you solve it twice and the solution ends us between them
[3x] < 15 is equal to -15<3x<15
so x is greater than -5 and less than 5

For greatOR than:
These are disjunctions and are solved twice with the solution infinitely in different directions
[3x] >15 is equal to 3x < -15 and 3x > 15

INEQUALITIES IN 2 VARIABLES 9-5
You will shade an x y graph to find the side of a linear equation that fits the solution
EXAMPLE: x + y > 5
You graph the line with DOTTED line because it cannot be equal to 5
You pick an easy point on one side of the line and substitute to see if that side is a solution.
If that does not work, pick an easy point on the other side to see if that side checks.
I USUALLY USE (0,0) as my first point!
If there is more than 1 Inequality (a system of inequalities), follow the same procedure and where the 2 shadings overlap each other is called the solution of the system of inequalities.

SYSTEMS OF INEQUALITIES IN TWO VARIABLES 9-6
If there is more than 1 Inequality (a system of inequalities),
1. Follow the same procedure as above for one equation
but you will need to do it for each equation.
2. Use a different type of shading for each so you won't get confused
(Ex: Use slanted lines one way and then slanted lines the other way. Use different colors if possible. Make one set of lines wavy and the other set straight)
3. Where the 2 shadings overlap each other is called the solution of the system of inequalities.
(any point in the overlap should work in BOTH inequalities - make sure you check!!!!!)

Pre Algebra Periods 1, 2, & 4

Graphing Review - make sure to look at your book as you review these notes. Look at the examples

Box and Whisker Plots

Suppose you have a large set of data and you want a display that gives a general idea of how the data clusters together. A box-and-whisker plot displays the median, the quartiles, and outliers of a set of data but does not display any other specific values.

To make a box-and-whisker plot:

Write the data in order from least to greatest

Draw a number lime that can show the data in equal intervals – make sure to have intervals that include the least and the greatest

Find the median – Mark it with a dot below ( or above the number line)

Find the upper quartile ( the median of the numbers above the actual median) Mark it with a dot below ( or above the number line)

Find the lower quartile ( the median of the numbers below the actual median) Mark it with a dot below ( or above the number line)

Mark with a dot the upper extreme – the greatest number

Mark with a dot the lower extreme – the lowest number

Draw a box between the lower quartile and upper quartile. Split the box by drawing a vertical line through the median.

Draw two ‘whiskers’ from the quartiles to the extremes.

50% of all the data will be within the box. 25% will be below and 25% will be above.

Frequency Table

A frequency table is a way to show how often an item, a number or a range of numbers occurs.

number	1	2	3	4
frequency	4	0	5	8

The range is the difference between the highest number and the lowest. In this case 4-1 = 3

Line Plots help show the spread of the data. When you look at a line plot you can easily see the range, the mode, and any outliers in the data,

Draw a horizontal line segment on grid paper

Make a scale of numbers below the line. The numbers should include the greatest value and the least value of the set of data.

For each piece of data, draw an X above the corresponding number.

Stem-And-Leaf Plots

Stem and Leaf Plots allow you to easily see the greatest, least, and median values in a set of data.

For example—from class

As of 1997 the following are the ages, in chronological order, at which US Presidents were inaugurated

57,61, 57, 57, 58, 57, 61, 54, 68, 51, 49, 64, 50, 48, 65, 52, 56, 46, 54, 49, 50, 47, 55, 55, 54, 42, 51, 56, 55, 51, 54, 51, 60, 62, 43, 55, 56, 61, 52, 69, 64, 46

To make a stem-and-leaf plot:

Write the data in order from least to greatest

Find the least and greatest values

Choose stem values that will include the extreme values. For this graph, it makes sense to use tens.

Write the tens vertically from least to greatest. Draw a vertical line to the right of the stem values.

Separate each number into stems (tens – in this case) and leaves (ones- in this case). Write each leaf to the right of its stem in order from least to greatest.

Write a key that explains how to read the steams and leaves.

Scatter Plots

Suppose you want to analyze two sets of data to see how closely they are related. On a scatter plot you plot corresponding numbers from two sets of data as order pairs (x,y). You then decide if they are related by determining how close they come to forming a straight line.

For example, here is a frequency table of the number of hours studied and grades of a student

study hours	1.5	1	3	2.5	1.5	4	3.5
grade on test	75	71	88	86	80	97	92

To make the scatter plot

Decide which set of numbers you will plot on each axis and label the axis. In this case, The study time ( in hours) would be the X-axis and the Grade would be the Y-axis.

Choose a scale for each axis.

Plot corresponding numbers as ordered pairs. For example (1.5, 75) and (1, 71) are the first two from the table above.

The dots on the scatter plot are close to forming a straight line ( going up) so this is a strong positive correlation.

If the line formed was going down—it would be a negative correlation

and if you could not determine any line- it would be no correlation.

Math 6 Honors Periods 6 & 7

Percent and Fractions 9-1

The word “percent” is derived from the Latin “per centum” meaning “per hundred” or “out of one hundred” so 28% means 28 out of 100

A percent is a ratio that compares a number to 100. Therefore you can write a percent as a fraction with a denominator of 100, so 28% is also 28/100

Our book’s example is as follows;

During basketball season, Alice made 17 out of 25 free throws, while Nina made 7 out of 10. To see who did better, we compare the fractions representing each girl’s successful free throws.

17/27 or 7/10 or

In comparing fractions it is often convenient to use the common denominator 100, even if 100 is not the LCD of the fractions.

17. 4 = 68 thithithiett7.10=70
24^t4th100thiethithei10t10t100

Since Alice makes 68 free throws per 100 and Nina makes 70 per hundred, Nina is the better free throw shooter.

the ratio of a number to 100 is called a percent. We write percents by using the symbol %

so

17/25=68% and 7/10= 70%

Rule

To express the fraction a/b as a percent, solve the equation

n/100 = a/b

for the variable n and write n%

Just set up as a ratio and compare your fraction to 100. You know three out of the four numbers. Express 17/40 as a percent

Use the proportion method or just divide 17 by 40 multiply both sides by 100

n= 42½

Therefore, 17/40 = 42.5%

Rule

To express n% as a fraction, write the fraction

n/100 in lowest terms

Express 7 1/2 % as a fraction in lowest terms

7 ½ % = 7.5% = 7.5/100 How do we get rid of the decimal? multiply numerator and denominator by ten

7.5 . 10 = 75 simplify
100 t10 t1000

=3/40

Since a percent is the ratio of a number to 100, we can have percents that are greater than or equal to 100%

100 = 100%
100

165 =165%
100

Write 250% as a mixed number in simple form

250% =

250=
100

2 50/100 = 2 1/2

Percents and Decimals 9-2

By looking at the following examples, you will be able to see a general relationship between decimals and percents

51% = 47/100 = 0.57 113% = 113/100 = 1.13

0.79 = 79/100 = 79% .06 = 6/100 = 6%

Rules

To express a percent as a decimal, move the decimal point two places to the left and remove the percent sign

57% = 0.57 thith 113% = 1.13

To express a decimal as a percent, move the decimal point two places to the right and add a percent sign

0.79 = 79% thithith0.06 = 6%

Express each percent as a decimal ( remember- the decimal is so sad to see the % leave that it runs away – two places to the left—from where it was)

1. 83.5% = 0.835

2. 450% = 4.5

3. .25% = 0.0025

Express each decimal as a percent [careful] ( remember- the decimal is so happy to see the % that it runs ( 2 place to the right) towards the %)

1. 10.5 = 1050.%

2. 0.0062 = 0.62%

3. 0.574= 57.4%

In 9-1 you learned one method of changing a fraction into a percent. Here is an alternative method

Rule

To express a fraction as a percent, first express the fraction as a decimal

and then as a percent

Express 7/8 as a percent

Divide 7 by 8

7/8

= 0.875 = 87.5%

Express 1/3 as a percent to the nearest tenth of a percent

divide 1 by 3

1/3

since this becomes a repeating decimal

.33333333….

or .3 with a vinculum, we write 1/3 as

33 1/3%

Pre Algebra Periods 1, 2, & 4

SQUARE ROOTS & PYTHAGOREAN THEOREM 11-1 & 11-2

Square root undoes squaring!
So you're looking for the number/variable that was squared to get the radicand
1) RADICAL sign: The root sign, which looks like a check mark.
If there is no little number on the radical, you assume it's the square root
But many times there will be a number there and then you are finding the root that the number says.
For example, if there is a 3 in the "check mark," you are finding the cubed root.
One more example: The square root of 64 is 8. The cubed root of 64 is 4. The 6th root of 64 is 2.
2)RADICAND : Whatever is under the RADICAL sign
In the example above, 64 was the radicand in every case.
3) ROOT (the answer): the number/variable that was squared (cubed, raised to a power)
to get the RADICAND (whatever is under the radical sign)
4) SQUARE ROOTS: The number that is squared to get to the radicand. Every POSITIVE number has 2 square roots - one positive and one negative.
Example: The square root of 25 means what number squared = 25
Answer: Either positive 5 squared OR negative 5 squared

You can estimate nonperfect square roots by guess and check
Find the 2 numbers that it is between
Example: Square root of 52
It's between the 2 perfect squares: 49 and 64
So the square root is between 7 and 8
Since 52 is only 3 away from 49 and 12 away from 64, the square root will be closer to 7
Guess: 7.2 Square this: (7.2)(7.2) = 51.84 (adjust your estimate as necessary)

REAL NUMBER SYSTEM
2 PARTS: RATIONAL AND IRRATIONAL (both real)

Review LEAP FROG number systems
I: RATIONAL NUMBERS (definitions of different number systems):
Natural = counting = 1, 2, 3, . . .
Whole = natural + 0 = 0, 1, 2, 3, . . .
Integers = whole + opposites = -3, -2, -1, 0, 1, 2, 3, . . .
Rational = integers and all the fractions/decimals in between - terminating and repeating decimals

II: IRRATIONAL - numbers like pi and square root of 3 - never repeat or terminate - round!

PYTHAGOREAN THEOREM
FOR RIGHT TRIANGLES ONLY!
2 legs - make the right angle - called a and b
(doesn't matter which is which because you will add them and adding is COMMUTATIVE!)
hypotenuse - longest side across from the right angle - called c
You can find the third side of a right triangle as long as you know the other two sides:
a² + b² = c²
After squaring the two sides that you know, you'll need to find the square root of that number to find the length of the missing side (that's why it's in this chapter!)
EASIEST - FIND THE HYPOTENUSE (c)
Example #1 from p. 510
8² + 15² = c²
64 + 225 = c²
289 = c²
c = 17

A LITTLE HARDER - FIND A MISSING LEG (Either a or b)
Example #5 from p. 510
5² + b² = 13²
25 + b² = 169
b² = 169 - 25
b² = 144
b = 12

CONVERSE OF PYTHAGOREAN THEOREM
If you add the squares of the legs and that sum EQUALS the square of the longest side, it's a RIGHT TRIANGLE.
If you add the squares of the 2 smallest sides and that sum is GREATER THAN the square of the longest side, you have an ACUTE TRIANGLE.
If you add the squares of the 2 smallest sides and that sum is LESS THAN the square of the longest side, you have an OBTUSE TRIANGLE.

Algebra Period 3 (Monday)

SOLVING SYSTEMS OF EQUATIONS 8-1 TO 8-3:
(2 equations with 2 variables)
You cannot solve an equation with 2 variables - you can find multiple coordinates that work
TO SOLVE MEANS THE
ONE COORDINATE THAT WORKS FOR BOTH EQUATIONS

There are 3 ways to find that point:

1. Graph both equations: Where the 2 lines intersect is the solution
2. Substitution method: Solve one of the equations for either x or y and plug in to the other equation
3. Addition method: Eliminate one of the variables by multiplying the equations by that magical number that will make one of the variables the ADDITIVE INVERSE of the other

Example solved all 3 ways:
Find the solution to the following system:

2x + 3y = 8 and 5x + 2y = -2

1. GRAPH BOTH LINES: Put both in y = mx + b form and graph
Read the intersection point (you should get (-2, 4))

2. SUBSTITUTION:

Isolate whatever variable seems easiest
I will isolate y in the second equation: 2y = -5x - 2
y = -5/2 x - 1
Plug this -5/2 x - 1 where y is in the other equation
2x + 3(-5/2 x - 1) = 8
2x - 15/2 x - 3 = 8
4/2 x - 15/2 x - 3 = 8
-11/2 x - 3 = 8
-11/2 x = 11
-2/11(-11/2 x) = 11(-2/11)
x = -2
Plug into whichever equation is easiest to find y

3. ADDITION:

Multiply each equation so that one variable will "drop out" (additive inverse)
I will eliminate the x, but could eliminate the y if I wanted to
5(2x + 3y) = (8)5
-2(5x + 2y) = (-2)-2

10x + 15y = 40

-10x - 4y = 4

ADD TO ELIMINATE THE x term:
11y = 44
y = 4
Plug into whichever equation is easiest to find x

NOTICE THAT FOR ALL 3 METHODS, THE SOLUTION IS THE SAME!
THEREFORE, USE WHATEVER METHOD SEEMS EASIEST!!!

Math 6 Honors Periods 6 & 7

Problem Solving: Using Proportion 7-8

Proportions can be used to solve word problems. Use the following steps to help you in solving problems using proportions

Ø Decide which quantity is to be found and represent it by a variable

Ø Determine whether the quantities involved can be compared using ratios (rates)

Ø Equate the ratios in a proportion

Ø Solve the proportion

Some guidelines you can use to determine when it is appropriate to use a proportion to solve a word problem.

Ask the following questions

If one quantity increase does the other quantity also increase? (If one quantity decreases, does the other quantity decrease?) When the number of tires is increase, the cost is also increased.

Does the amount of change (increase or decrease) o one quantity depend upon the amount of change (increase or decrease) of the other quantity? The amount of increase in the cost depends upon the number of additional tires bought.

Does one quantity equal some constant times the other quantity? The total costs equals the cost of one tire times the number of tires. The cost of one tire is constant.

If the answers to all the questions above is YES, then it is appropriate to use a proportion.

Sometimes setting up a table can be useful

Number of tires	Cost
4	$264
5	c

4c = 5(264)

Although the problems in this lesson may be solved with out using proportions, I must insist that you write a proportion for each problem and solve using this method. You may check your work using another other method you know.

Scale Drawing 7-9

Opening your books to page 237, you will notice a drawing of a house. In this drawing of the house, the actual height of 9 meters is represented by a length of 3 centimeters, and the actual length of 21 meters is represented by a length of 7 centimeters.

This means that 1 cm in the drawing represents 3 m in the actual building. Such a drawing in which all lengths are in the same ratio to actual lengths is called a scale drawing.

The relationship of length in the drawing to actual length is called the scale. In the drawing of the house the scale is 1cm: 3m

We can express the scale as a ratio, called the scale ratio, if a common unit of measure is used. Since 3 m = 300 cm, the scale ratio is 1/300

Using the book’s drawing on page 237, find the length and width of the room shown, if the scale of the drawing is 1cm: 1.5 m

Measuring the drawing, we find that it has a length of 4 cm and a width of 3 cm

Algebra Period 3 (Tuesday)

Simplify, multiply, and divide RATIONAL EXPRESSIONS: 10-1, 10-2, & 10-3
Rational Expressions = Expressions in fraction format (division) with a variable in the denominator
You have already been simplifying, multiplying and dividing these throughout this year!

10-1: SIMPLIFY
You will need to FACTOR (Chapter 6) both the numerator and denominator and "cross out" common factors in both (their quotient is 1!)

EXAMPLE:

Simplify

y² + 3y + 2the=the(y + 2)(y + 1)thie= th y + 2
Ss y² - 1 andtheisto(y - 1)(y + 1)thethietiy - 1

10-2: MULTIPLY
Factor if possible, cross cancel if possible, multiply numerators, then denominators, simplify

EXAMPLE:

[(y + 4)³][y² + 4y + 4] THEI = (y + 4)³ (y + 2)² R= R y + 4
[(y + 2)³][y² + 8y + 16] THISTH(y + 2)³ (y + 4)² THEy + 2

10-3: DIVIDE
Same as the previous example, only this time you will need to FLIP THE SECOND FRACTION
then MULTIPLY!!!!!

EXAMPLE:
x + 1 ÷ x + 1 T= ( x + 1) ( x² - 2x + 1) =E( x + 1)( x - 1)(x - 1) =
x² – 1Tx² - 2x + 1 T(x + 1)( x - 1) (x + 1)TH(x + 1)( x - 1)(x + 1) TH

x - 1
x + 1

Algebra Period 3 (Monday)

Radical Equations with Quadratics 13-6

THIS IS A REVIEW OF CHAPTER 11!!!

You square both sides to get rid of the radical sign
For these problems, you'll get a quadratic on one side after you square
BE SURE TO CHECK BOTH ANSWERS TO MAKE SURE THEY BOTH CHECK!

WORD PROBLEMS WITH QUADRATICS 13-7
There are 2 major types: Frame problems, triangle problems

FRAME PROBLEMS:
You know the frame's dimensions and the picture's area inside. You need to find the width of frame.
1)Set x = width of the frame
2) Write the area formula using the dimensions given in the problem minus 2x
as the base and height of the area of the picture.
Why are we subtracting 2x? Because there are two widths, one on the top and one on the bottom, one on the right side and one on the left side
3) Set this product equal to the area of the picture given in the problem
EXAMPLE:
A picture frame is 20 cm by 12 cm. The picture has an area of 84 cm²
(20 - 2x)(12 - 2x) = 84
FOIL
240 - 40x - 24x + 4x² = 84
Combine like terms and bring the 84 over
4x² -64x + 240 - 84 = 84 - 84
4x² - 64x + 156 = 0
Divide by 4 on each side
x² - 16x + 39 = 0
FACTOR or use the QUADRATIC FORMULA
(x - 3)(x - 13 ) = 0
x = 3 cm or x = 13 cm
The 13 cm does not make sense! How can the width of the frame be more than one of its dimensions (the height was only 12 cm!!!)???
Therefore, the only answer is that the frame is 3 cm wide.

The right triangle example in the book on p. 603 uses the Pythagorean Theorem

Pre Algebra Periods 1, 2, & 4

Space Figures 10-4 (solids)
SPACE FIGURES OR SOLIDS OR 3 DIMENSIONAL FIGURES

prisms = 2 congruent parallel bases - all other sides are rectangles
cylinder = 2 congruent circle bases
When you remove one base from a prism, it becomes a pyramid - all other sides are triangles
When you remove one base from a cylinder, it becomes a cone
When you have a set of points in all directions that are equal distance from a central point, you have a sphere
Vertices - the points where edges connect (the corners)
Edges - the line segments that connect the vertices

You should be able to visualize what a figure will look like if you could cut it apart and open it—that is called a net

SURFACE AREA OF CYLINDERS & PRISMS 10-5
Surface area is just the sum of the areas of all the sides

SHORTCUT TO ADDING UP ALL THE SIDES:
Just find the perimeter of ONE base and then multiply that by the height of the prism
Add to that the areas of the TWO bases, and you'll have the surface area
S.A. = area of 2 bases + (perimeter of one base)(height of prism)

According to our textbook:

the perimeter of one base X height = LATERAL AREA (L.A.)

the area of the base (base X height) = B

so

S.A. = L.A. + 2B
For a cylinder, do the same exact thing except now it's the CIRCUMFERENCE of the base
S.A. = area of the 2 bases + (circumference of one base)(height of cylinder)

S.A. OF PYRAMIDS, CONES, AND SPHERES
PYRAMIDS 10-6
For pyramids, you can simply add up the areas of all the sides
S.A. = area of 1 base + lateral areas (sides)
Lateral areas are all triangles in a pyramid and A = 1/2 bh
This time the height means the height of the triangle that is a side. It has a special name = slant height
The slant height is denoted as a cursive l

so S.A. = area of base + (1/2 bl)(# of sides)

CONES
For cones, you have one circle base plus a sort of triangular piece with the base being rounded
The cursive l now stands for the length of the slant side of this piece

S.A. = area of the circle + (pi)(radius of base)(slant height)
S.A. = (pi)r² + (pi)rl [pronounce this pie roll!)

SPHERES
S.A. + 4(pi)r²
(I always thought of this as you need 4 circles to wrap the basketball!)

Pre Algebra Periods 1, 2, & 4

AREA Sections 10-1, 10-2 & 10-3

All these formulas are related to the basic concept of A = bh
Where the perimeter/circumference fenced in my puppy, the area of the yard will tell me how much sod (grass) I should buy to stop the puppy's paws from getting muddy!

Area of rectangles and parallelograms 10-1

A = (base)(height)
In a parallelogram, whether it's a rectangle, rhombus, square or other parallelogram
A = bh with the height being a line perpendicular to both bases (not the slanted side!)
You have learned the area of a rectangle as A = lw, but the l = b and the w = h
You may have learned the area of a square as A = s² , but that's because the b = h


Area of triangles 10-2
A = (1/2)(base)(height or altitude)
Any parallelogram can be split into 2 triangles using a diagonal.
Because of this, the area of a triangle is half that of a parallelogram.

A = ½bh

Area of trapezoid = (average of the 2 bases)(height)
A = ½(b₁+ b₂)h

Just add the two bases together and divide by 2—that’s the average!!

Then multiply by the height

A trapezoid has 2 bases that ARE NOT EQUAL. So which base is THE base?
If you use the smaller base, you won't have enough sod for your yard and the puppy's paws are still getting muddy.
If you use the larger base, you'll have too much sod for your yard and the extra will rot.
Sooooooooo..... you actually need to take the average of the two bases times the height
A = (average of the 2 bases)(height)
A = (b₁ + b₂) h
theitheis2

Area of Circles 10-3
Area of circle = (Pi)(r²)

If you are given the diameter remember to take half of the diameter to find the radius and make sure to square the radius!!

Use 3.14 or 22/7 as a good approximation of pi. Remember pi is irrational—never ending and never repeating

Algebra Period 3 (Friday)

THE QUADRATIC FORMULA 13-4

-b plus or minus the square root of b squared minus 4ac all over 2a

Notice how the first part is the x value of the vertex -b/2a
The plus or minus square root of b squared minus 4ac represents
how far away the two x intercepts (or roots) are from the vertex!!!!

Very few real world quadratics can be solved by factoring or square rooting each side.
And completing the square always works, but it long and cumbersome!

All quadratics can be solved by using the QUADRATIC FORMULA.

(you will find out that some quadratics have NO REAL solutions, which means that there are no x intercepts - the parabola does not cross the x axis! Think about what kinds of parabolas would do this....ones that are smiles that have a vertex above the x or ones that are frowns that have a vertex below the x axis. You will find out in Algebra II that these parabolas have IMAGINARY roots)

So now you know 5 ways that you know to find the roots:

1. graph
2. factor if possible
3. square root each side
4. complete the square - that's what the quadratic formula is based on!
5. plug and chug in the Quadratic Formula -
This method always works if there's a REAL solution!

DON'T FORGET TO PUT THE QUADRATIC IN STANDARD FORM BEFORE PLUGGING THE VALUES INTO THE QUADRATIC FORMULA!

ax² + bx + c = 0

DISCRIMINANTS - a part of the Quadratic Formula that helps you to understand the graph of the parabola even before you graph it!
the discriminant is b² - 4ac

(the radicand in the Quadratic Formula, but without the SQRT)

Depending on the value of the radicand, you will know
HOW MANY REAL ROOTS IT HAS

1) Some quadratics have 2 real roots (x intercepts or solutions) - Graph crosses x axis twice
2) Some have 1 real root (x intercept or solution) - Vertex is sitting on the x axis
3) Some have NO real roots (no x intercepts or solutions) - vertex either is above the x axis and is a smiley face (a coefficient is positive) or
the vertex is below the x axis and is a frown face (a coefficient is negative)

In both of these cases, the parabola will NEVER CROSS (intercept) the x axis!

b² -4ac > 0 That is, if it's positive, then there are 2 roots
b² -4ac = 0 That is, if it's zero , then there is 1 root
b² -4ac <>if it's negative , there are no real roots

Pre- Algebra Periods: 1, 2, & 4

Transforming geometry, 9-8, 9-9, & 9-10

Rigid Motions:
Translations (slides)
Rotations (turns)
Reflections (flips)

Translations 9-8

You can move pattern blocks by sliding them, flipping them, or turning them. Each of these moves is a transformation.
Sometimes called a slide. Moves a figure - Doesn't change the shape or size
A translation is a transformation that moves points in the same distance and in the same direction. A figure and its translated image are congruent. You can see examples of translations in wallpaper, fabric and wrapping paper.
The figure you get after a transformation is called the image.
To name the image of a point, you use the prime notation, Point A and its image Point A^’
After the figure is translated (moved), it is then called the image and you use ' (the prime notation)
So triangle ABC once it's moved up 2 spaces is now named ABC'

You can describe a transformation using arrow notation, which describes the mapping of a figure onto its image. A --> A^’
You can also use arrow notation to write a general rule that describes a transformation.

So... If point A is (3,5)----> A' is (5,1)
(A prime)
Write an equation.
Ask yourself, what do you have to do to X as 3 to get X' as 5?
And what do you have to do to Y as 5 to get Y' as 1

(X,Y)---> (X + 2, Y - 4) then you have A' as (5,1) and you proved it.


Symmetry 9-9

Symmetry means that if you divide a figure in half, both halves look exactly the same
Line of symmetry: where you can "fold" the figure and have it exactly the same
Figures can have 0 or more lines of symmetry
A random drawing would probably have no lines of symmetry
A butterfly has one line of symmetry
A rectangle has 2 lines of symmetry
A square has 4 lines of symmetry
A circle has infinite lines of symmetry (you can rotate it infinitely and divide it in half)

A figure has reflective symmetry when one half is a mirror image of the other half. A line of symmetry divides a figure with reflectional symmetry into two congruent halves.

Again, Reflections (flips)

Line of symmetry---> Fold Figure
Reflection Reflection is sometimes called a flip of the figure
The figure flips over a LINE OF REFLECTION
For example, if a triangle is in Quadrant I, it may flip over the y axis into Quadrant II
(the right side of the triangle is now the left side)
Or it may flip over the x axis into Quadrant IV
(the top of the triangle is now the bottom)
Or it may flip over any other line like y = x. This is more complicated to understand!





Rotation 9-10

This transformation takes a figure and TURNS IT
about a point called the CENTER OF ROTATION.
The angle of turning is called the ANGLE OF ROTATION.
The two most common angles: 90 degrees and 180 degrees

If you are going to rotate 90 degrees the rule is this

The Value of X Becomes the opposite of the Y Value & the Y value becomes what the value of x was.

EX: (X,Y)--->(-Y, X)

(5,4) rotated 90 degrees would be (-4,5)
(5,4)---> (-4,5)

If you rotate 90 degrees a figure in quadrant I it would rotate into quadrant II
I-->II
II-->III
III-->IV
IV-->I

The rotation is one quadrant over-- counterclockwise

If you are going to rotate 180 degrees the rule is this
(X,Y)--->(-X,-Y)
180 degrees: ends up in the opposite quadrant (I goes to III , II goes to IV and so on)
The image is: (x, y) goes to (-x, -y) SAME NUMBERS...OPPOSITE SIGNS!



submitted by Lorenzo