Math 6 Honors (Period 6 and 7)

The following equations create curves that are called PARABOLAS!! Notice the difference in these equations from our previous equations
y = x² +1
when we create your three column table using integers from -2 to 2
we notice
y = (-2)² +1 = 4 + 1 = 5 ordered pair (-2, 5)
y = (-1)² +1 = 1 + 1 = 2 ordered pair (-1, 2)
y = (0)² +1 = 0 + 1 = 1 ordered pair (0, 1)
y = (1)² +1 = 1 + 1 = 2 ordered pair (1, 2)
y = (2)² +1 = 4 + 1 = 5 ordered pair (-2, 5)

When you graph this... you get a "U" shaped graph.

Remember linear equations LINEar equations are lines!1
and look like y = x + 2

PARABOLAS have the form y = x² or y = -x²

Let's try
y = 2 -x²
With our 3 column table
for values of x from -2 to 2
we find
y = 2 -(-2)² = 2 -(4) = -2 and the ordered pair is (-2,-2)
y = 2 -(-1)² = 2 - (1) = 1 and the ordered pair is ( -1, 1)
y = 2 -(0)² = 2 - 0 = 2 and the ordered pair is (0, 2)
y = 2 -(1)² = 2 -1 = 1 and the ordered pair is (1, 1)
y = 2 -(2)² = 2 - (4) = -2 and the ordered pair is (2, -2)

When you graph these ordered points you find you have an upside down U
hmmm... y = -x² results in a sad face parabola
and y = x² results in a happy face parabola!!
Graphing Inequalities

You will need to look at the graphs in your textbook. .. page 397

Whenever we graph relations that are inequalities we must be aware of all the facts that can influence your work. You need to ask yourself, "What kind of numbers is the solution supposed to be?"
When you graphed inequalities such as
-3 < x < 2 where x was an integer we used a point on the number line for each integer that could be a solution to that inequality. To show every number in x < 2 we would use a number line and place an Open Dot at 2 indicating that 2 was NOT part of the solution and then draw a darkened ray away from 2 indicating 1, 0, -1, -2... were all part of the solution.
To show that this line has infinite solutions in that direction, you MUST place an arrow at the end of that darkened ray.

If the inequality was a " less than or equal to" " ≤" you would use a Closed Dot at 2 to indicate that 2 was part of the solution.

We can now graph inequalities such as y ≥ x + 2
first you find the BOUNDARY LINE which is just y = x + 2 and you can use the 3 column table as we have done before or use a T chart as shown in class.

Remember you only need 2 points to determine a line---> but 3 points will help you make sure you have 3 correct points on the line!!

I am going to try to set up a T chart using "I" to separate the x and y
X I Y
-2 I 0
-1 I 1
0 I 2
1 I 3
2 I 4

(Note: it doesn't line up well here.. but hopefully you get the idea)

Plot those points on the graph and you have what appears to be a straight line. Since we are graphing y ≥ x + 2 we ARE including the line so we draw a solid line.

But.. what points are included?
Well, we know that (-2,0) works but we also see if we plug into our inequality that (-2,1) and (-2,2) work as well.

We need to shade the part above the line to indicate all those points are part of the solution as well.

Three set method for graphing an inequality

(1) Determine the boundary line. Draw it--
use a solid line if the boundary line is part of the graph (≤ or ≥)
use a dashed line if the boundary line is NOT part of the graph (< or >)

(2) Shaded either the part above the boundary line or the part below the boundary line.
If the inequality reads y > or y ≥ shade ABOVE the line.
If the inequality reads y < or y ≤ shade BELOW the line

(3) Always CHECK- choose a point you think works within the shaded region and see if it does work.. or use (0,0) and determine if it is part of the solution or not!!

Math 6 Honors (Period 6 and 7)

Graphs of Equations 11-9

An equation in two variables y = x + 2
produces an infinite number of ordered pairs
If we give x the value of 3, a corresponding value of y is determined
y = (3) + 2 = 5
The ordered pair is (3, 5)
If we let x = 4
y = (4) + 2 = 6
and we get the ordered pair (4, 6)
What happens if x = 0
y = (0) + 2 = 2 ( 0, 2)
or x = -2
y = (-2) + 2 = 0 ( -2, 0)
For each value of x there is EXACTLY 1 value of y.
set of ordered pairs in which no two ordered pairs have the same x is called a FUNCTION
I like to remember ordered pairs---> ( ordered, pairs)

y = 2x -3
in the future you will see it written as
f(x) = 2x -3
so if x = 3
f(3) = 2(3) -3 = 6-3 = 3
so f(3) = 3
if x = 5
f(5)= 2(5) - 3 = 10 -3 = 7
so f(5) = 7

We used a three column chart to compute our ordered pairs.
Please refer to the blue sheet glued into your spiral notebook for the examples we completed in class-- if you were absent, please come in one morning and I will review that chart with you.

Pre Algebra (Period 2 & 4)

Proportions 6-2

A proportion = 2 equal ratios (2 equivalent fractions)

Solve using equivalent fractions or

Cross multiplication and then a one-step equation

(see if you can simplify the fractions before multiplying)

Example: Solve the proportion for y:


4/3 = y
/21

EQUIVALENT FRACTION APPROACH:
Multiply both top and bottom by 7, y = 28



CROSS PRODUCTS APPROACH:

You'll get 3y = (21)(4)
Now divide each side by 3.
Do this before multiplying on the right side!
Why? Because a lot of the time you'll be able to simplify and keep the numbers smaller!


3y/3 = (21)(4) /3

See how the 3 cross cancels into the 21?
so y = 28


ALWAYS SIMPLIFY THE FRACTIONS FIRST!

WORD PROBLEMS WITH PROPORTIONS:


It's all about setting up the LABELS first!


label A _____________ = ____________ label A

label B xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxlabel B

Math 6 Honors (Period 6 and 7)

Graphs of Equations 11-9

An equation in two variables y = x + 2
produces an infinite number of ordered pairs
If we give x the value of 3, a corresponding value of y is determined
y = (3) + 2 = 5
The ordered pair is (3, 5)
If we let x = 4
y = (4) + 2 = 6
and we get the ordered pair (4, 6)
What happens if x = 0
y = (0) + 2 = 2 ( 0, 2)
or x = -2
y = (-2) + 2 = 0 ( -2, 0)
For each value of x there is EXACTLY 1 value of y.
set of ordered pairs in which no two ordered pairs have the same x is called a FUNCTION
I like to remember ordered pairs---> ( ordered, pairs)

y = 2x -3
in the future you will see it written as
f(x) = 2x -3
so if x = 3
f(3) = 2(3) -3 = 6-3 = 3
so f(3) = 3
if x = 5
f(5)= 2(5) - 3 = 10 -3 = 7
so f(5) = 7

We used a three column chart to compute our ordered pairs.
Please refer to the blue sheet glued into your spiral notebook for the examples we completed in class-- if you were absent, please come in one morning and I will review that chart with you.


The following equations create curves that are called PARABOLAS!! Notice the difference in these equations from our previous equations
y = x² +1
when we create your three column table using integers from -2 to 2
we notice
y = (-2)² +1 = 4 + 1 = 5 ordered pair (-2, 5)
y = (-1)² +1 = 1 + 1 = 2 ordered pair (-1, 2)
y = (0)² +1 = 0 + 1 = 1 ordered pair (0, 1)
y = (1)² +1 = 1 + 1 = 2 ordered pair (1, 2)
y = (2)² +1 = 4 + 1 = 5 ordered pair (-2, 5)

When you graph this... you get a "U" shaped graph.

Remember linear equations LINEar equations are lines!1
and look like y = x + 2

PARABOLAS have the form y = x² or y = -x²

Let's try
y = 2 -x²
With our 3 column table
for values of x from -2 to 2
we find
y = 2 -(-2)² = 2 -(4) = -2 and the ordered pair is (-2,-2)
y = 2 -(-1)² = 2 - (1) = 1 and the ordered pair is ( -1, 1)
y = 2 -(0)² = 2 - 0 = 2 and the ordered pair is (0, 2)
y = 2 -(1)² = 2 -1 = 1 and the ordered pair is (1, 1)
y = 2 -(2)² = 2 - (4) = -2 and the ordered pair is (2, -2)

When you graph these ordered points you find you have an upside down U
hmmm... y = -x² results in a sad face parabola
and y = x² results in a happy face parabola!!

Math 6 Honors (Period 6 and 7)

Graphs of Ordered Pairs 11-8

A PAIR of numbers whose ORDER is important is called an
ordered pair!!
(ordered, pair)

(2,3) is not the same as (3,2)

The two perpendicular lines are called axes.
The x-axis deals with the 1st number of the ordered pair and the y-axis deals with the 2nd number of the ordered pair.
The AXES meet at a point called the Origin (0,0)
The plane is called the coordinate plane

There are 4 quadrants, Use Roman Numerals to name them!!
Quadrant I ---> both the x and y coordinates are positive
(x,y) (+,+)

Quadrant II --> the x coordinate is negative but the y is positive
(-x,y) (-,+)

Quadrant III --. both the x and y coordinates are negative
(-x,-y) (-,-)

Quadrant IV --> the x coordinate is positive but the y coordinate is negative
(x,-y) (+,-)