Math 6 Honors ( Periods 1, 2, & 3)

Graphs of Equations 11-9 (cont'd)
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 ( Periods 1, 2, & 3)

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 1, a corresponding value of y is determined
y = (1) + 2 = 3
The ordered pair is (1, 3)
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 = -1
y = (-1) + 2 = 1 ( -1, 1)

I like to remember ordered pairs---> ( ordered, pairs)

We graphed the line on a mini graph stickie.
The line is graphed using a ruler and connecting all the points we plotted. Put arrows at each end (since a line continues with out end) and write the line's equation right above the line.


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


y = x + 2
in the future you will see it written as
f(x) = x + 2
so if x = 2
f(2) = (2) + 2 = 4


We used a three column chart to compute our ordered pairs.
Please refer to the orange sheet glued into your spiral notebook for the examples we completed from the class exercises found on Page 393 -- if you were absent, please come in one morning and I will review that chart with you.

Algebra Honors (Period 6 & 7)

Percents 7-5
You have been doing this since at least 6th grade so this portion should really be review.
The word percent means hundredths or divided by 100

Some examples:
29 percent = 29% = 29/100 = 0.29
2.6 percent = 2.6 % = 2.6/100 = 26/1000 = 0.026
637 percent = 637% = 637/100 = 6 37/100 = 6.37
0.02 percent = 0.02% = 0.02/100 = 2/10000 = 0.0002
1/4 percent = 1/2% = 0.25 % = .25/100 = 25/10000 = 0.0025
33 1/3 percent = 33 1/3 % = 100/3% = 100/3 ÷ 100 = 1/3

Most calculators have a % key which will enable you to check your work!!
Remember in percent problems, the word, OF means Multiply and the word IS means EQUALS.

You can set up your problems either with an equation or with a proportion
Example 1:
15% of 180 is what number?
15/100 × 180 = x or
.15 × 180 = x

or set up a proportion
%/100 = 'is'/'of'
15/100 = x/180

In both cases x = 27

Example 2:
23 is 25% of what number?
23 = .25×n

or as a proportion
25/100 = 23/x

Again in both cases
x = 92

Example 3:

What Percent of 64 is 48?
x% × 64 = 48
or
x/100 × 64 = 48
64x/100 = 48

Set up as a proportion
x/100 = 48/64

x = 75
Remember to look at the question to make sure you have answered it. In this case you need to make sure you answer with 75%

When you solve with decimal coefficients, you can multiply both sides of the equation by a power of 10 (10, 100, 100 and so on) to get an equivalent equation with integral coefficients.

Example 4:
1.2x = 36 + 0.4x
Multiply both sides by 10 because the coefficients are tenths
12x = 360 + 4x
8x = 360
x = 45
{45}

Example 5:

94 = 0.15x + 0.08(1000 - x)
Multiply both sides by 100 because the coefficients are hundredths

9400 = 15x + 8(1000 - x)
9400 = 15x + 8000 - 8x
1400 = 7x
200 = x
{200}

Word Problem Example:
During a sale, a sporting goods store gave a 40% discount on sleeping bags. How much did Ross pay for a sleeping bag with an original price of $75?

Two different methods to find the same solution:
Method 1
Find 40% of $75? 0.40 × 75 = 30
Subtract the amount of discount from the original price
75-30 = $45
Ross paid $45 for the sleeping bag

Method 2
If the sleeping bag was discounted 40%, it then cost 100% -40% or 60% of its original price
Find 60% of 75 0.60 × 75 = 45
Ross paid $45 for the sleeping bag.

Math 6 Honors ( Periods 1, 2, & 3)

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) (+,-)