Algebra Period 3 (Friday)

Parallel and Perpendicular Lines 7-8

2 SPECIAL LINES AND THEIR SLOPES

PARALLEL LINES:
2 lines that are parallel to each have the SAME SLOPE!
y = 2x - 10 and y = 2x + 3/4 are parallel because they both have a slope of 2

PERPENDICULAR LINES:
2 lines that are perpendicular to each other have SLOPES that are:
OPPOSITE SIGNS
and
RECIPROCALS
y = 2x - 10 is perpendicular to y = -1/2 x + 3/4


THIS IS ANOTHER TWIST TO OUR MYSTERY LINE PUZZLE!!!
If you know that the mystery line is parallel or perpendicular to another given line
then you know the mystery line's slope!!!


EXAMPLE:
Your mystery line has a point of (2, -5) and
is PARALLEL to line y = 2x + 3/4
So you know the mystery line's slope because it is the same as the given line
( m = 2)
Substitute the slope and the point given on the mystery line and solve for b.


EXAMPLE:
Your mystery line has a point of (2, -5) and
is PERPENDICULAR to line y = 2x + 3/4
So you know the mystery line's slope because it is the opposite sign reciprocal of the given line

Lines that parallel to the y axis in the form x = c, ( such as x = 3) have an "undefined slope"
or NO SLOPE because if you find their slope using
y₂ - y₁ / x₂ - x₁
you will discover
a number/0
which is undefined

On the other had lines parallel to the x axis, in the form y = b ( such as y = -5) have a zero slope becuase using the same
y₂ - y₁ / x₂ - x₁
you will discover
0/ a number
which = 0

Algebra Period 3

Finding the Equation of a Line 7-6

Always remember that y = mx + b and you can substitute and solve for whatever you're missing!
It helps to memorize this little rhyme:
Oh mystery line,
What can you be?
If I could only find you,
y = mx + b

So first I find m
Then I find b.
Now put it all together
And you've found me!
y = mx + b

The rhyme has 3 steps and usually you will have 3 steps or questions to ask yourself:
1) Do I have the slope (m)? If not, find it by using the slope formula
2) Do I have the y intercept (b)? If not, find it by plugging in a point and the slope
3) Don't forget to put it all together in one equation at the end.

HERE ARE THE 5 CASES THAT THE BOOK INCLUDES:

First case:You're given the slope and the y intercept (easiest case)
m = 3/2 b = -7/5
Just plug in to the generic slope intercept equation: y = 3/2 x - 7/5

Second case: You're given a point and and the slope and need to find the intercept (b)
(3, 1) m = 2
Plug in the point and the slope and solve for b
1 = 2(3) + b
1 = 6 + b
b = -5
Now put it altogether with the given slope and the intercept you just found:
y = 2x -5

Third case: You're given a point and and the y intercept and need to find the slope
(3, 1) b = 2
Plug in the point and the y intercept and solve for slope
1 = 3m + 2
-1 = 3m
m = -1/3
Now put it altogether with the given intercept and the slope you just found:
y = -1/3 x + 2

Fourth case: You're given 2 points and need to find the slope and the intercept
(1 , 3) and (-2 , -3)
You need to first find the slope:
m = change in y / change in x = 3 - (-3)/ 1 - (-2) = 6/3 = 2
Now plug the slope in with one of the points and find the intercept b
3 = 2(1) + b
3 = 2 + b
b = 1
Finally, put it all together:
y = 2x + 1

Fifth case: You have a graph of a line and need to determine the equation
Look at the graph and find 2 easy points to use to find the slope (make sure they are integers!)
(If the y intercept is not an integer, then follow fourth case completely)
Put the information together in y = mx + b form


Another way to find the equation with 1 point and slope-
Point Slope Form of the equation

You know one point and the slope. This is the same case a the SECOND CASE, but there is a ANOTHER WAY to solve it other using slope intercept form. This is not the most common way. Most people use the slope intercept form for all cases.

This one is very specific, but many students love it! (It is MY personal favorite!!)

Point-slope form of a line: You know one point and the slope. Use the following formula:
y - y₁ = m (x - x₁)

Using the same example from the second case above: (3 , 1) and m = 2
y - 1 = 2 (x - 3)
Simplify: y - 1 = 2x - 6
y = 2x - 5

If you're trying to link the slope intercept form to the point slope form of the same line:

The point slope version eliminates one step from using the slope intercept form. In the slope intercept form, you plug in the point and slope, solve for b, and then rewrite the equation using the intercept that you found. In point slope form, once you plug in the point and slope, you just simplify and the equation is already done!

Pre Algebra Periods 1, 2, & 4

Probability and Odds 6-4

Outcomes-- possible results of any action

an event-- any outcome or group of outcomes.

Probability of an even is expressed either as a ratio or as a percent (%)

P(event) = # of favorable outcomes/ # of possible outcomes
The Probability of finding an 's' in Mississippi
P(s) = 4/11 = .363636... = 36 4/11%

Odds

favorable outcomes/ unfavorable outcomes

DO NOT WRITE AS A PERCENT

odd of getting a 's' in Mississippi becomes 4/7 or 4:7 or 4 to 7


The odds of UCLA winning the NCAA 3:1 3 favorable/1 unfavorable

changing that to probability it becomes 3/4 or 75%


probability -- the denominator is TOTAL outcomes

odds-- the denominator is unfavorable outcomes.

Some Astro Camp Photos!!

Pre Algebra Periods 1, 2, & 4

Similar Figures & Scale Drawings 6-3

Proportions are used a lot in Geometry

Two figures are similar if they are proportional

(1) they are exactly the same shape but different sizes

(2) each corresponding side of one of the figures is the same ratio of every other corresponding side.

(3) every corresponding angle is exactly the same measurement ( congruent)

All circles and all squares are similar.

Every square has 90° angles and each side will be in the same ratio to any other square.

Every circle has a measurement of 360° and every circle has the same ratio between its circumference and its diameter. That is C/d = π.

This is actually true for every regular polygon.

What is a regular polygon? All sides of the polygon are equal and all angles are equal!! Or.. a polygon with congruent sides and congruent angles. For example:

Equilateral triangle

Square

A good example of a regular octagon would be a stop sign.

When you name similar figures, you should go in the same order of their corresponding parts.
The symbol for SIMILAR looks like 1 of the 2 wavy lines that means approximately equal The symbol ~ means “is similar to”

SCALE DRAWING PROBLEMS ARE PERFECT TO SOLVE WITH PROPORTIONS!

For example a map says that 3 cm: 1000 miles
You measure that you need to go 5.4 cm on the map
How many miles is that?

Set up a proportion with the labels

cm/mi = cm/mi

3/100 = 5.4/x

Use cross products to solve

3x = 1000(5.4) but don’t multiply out yet

divide both sides by 3 so

x = 1000(5.4)/3 use your divisibility rules to simplify first

x = 1000(1.8) because 5.4/3 = 1.8

x = 1800 miles. Make sure your answer seems reasonable. That is, you know if 3cm is 1000 miles , then 6 cm would represent 2000 miles so your answer should be less than 2000 miles.

Indirect measurement

Person 5 ft tall casts a 4 ft shadow. A tree casts a 10 foot shadow. How tall is the tree. Set up proportions but make sure your labels match

5/4 = x/ 10 5(10) = 4x again don’t multiply out first

divide both sides by 4 so 5(10)/4 = x 25/2 = x so the tree must be 12.5 ft tall

set up

actual height of person/shadow = actual height of tree/shadow

or

person shadow/ tree shadow = person’s height/ tree’s height

You will arrive at the same answer!!