Math 6H ( Periods 3, 6, & 7)

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!!

Algebra Period 4

GRAPHING LINEAR EQUATIONS: 7-2

How do you determine whether a given number is a solution?

Plug it in, plug it in, plug it in!



How do you find a solution to an equation yourself?


Plug in for x and find y!


You can use ANY number for x


Then plug in your number and find y



How can you graph a linear equation?


Make an x/y table of values and then graph the coordinates.


You only need 3 coordinates to make a good line!


(The 3rd coordinate serves as a "check" for the other two...in case you made a mistake!)


I always try x = zero and y = zero first because it's usually easy.

Then pick another easy x value!


If this doesn't work well (you get a fraction as an answer and that's not easy to graph),
then try setting x equal to 1, then 2, then 3



LINEAR EQUATIONS: 7-3

What do they look like (what is not a linear equation?)


The variable is to the 1 power -> like x, or y, or a, or b


What is not a linear equation? the variable is not to the 1 power - like x2, x3, etc, or 1/x (x-1)



Two ways to graph:


1) 3 points using a table (like Ch 7-2)


EXAMPLE: 2x - 3y = -6


x y


0 2


3 4


-3 0




2) 2 points using the y and x intercepts (where the line intersects the y and x axis)


Standard form of a linear equation:

Ax +By = C
 where A, B and C should not be fractions


A should be positive (y will be positive or negative)


We won't be using this form to look at the slope of the line!


This is a good format for finding the x and y intercepts!



If it's in standard form, this way works great if both the x and y coefficients are factors of the constant on the other side of the equal sign.



EXAMPLE: 2x - 3y = -6


If x = 0, y = 2


If y = 0, x = -3



Special linear equations:


Ones that are parallel to either the x or the y axis:


Lines parallel to the y axis are vertical lines:


They end up as the form x = with no y variable in the equation at all!


EXAMPLE: x = 4 ends up as a vertical line at x = 4


Still don't get this???


Pick of few points with the x value of 4:
(4, 0) (4, 2) (4, -3)


Graph those and join them in a line.
 What do you get???


A vertical line!




Lines parallel to the x axis are horizontal lines:


They end up as the form y = with no x variable in the equation at all!


EXAMPLE: y = 4 ends up as a horizontal line at y = 4


Still don't get this???


Pick of few points with the y value of 4:
(0, 4) (2, 4) (-3, 4)


Graph those and join them in a line.
What do you get???


A horizontal line!

Math 6H ( Periods 3, 6, & 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!!

Algebra Period 4

GRAPHING ORDERED PAIRS:
7-1

Review of x y Coordinate Plane Graphing from Pre-Algebra

Cartesian plane: Named after French mathematician Descartes.


plane: a two dimensional (across and up/down) flat surface that extends infinitely in all directions.


quadrant: 2 perpendicular lines called axes split the plane into 4 regions....

quad means 4
quadrant

names: begin in the top right (where you normally write your name!) and go counterclockwise in a big "C" (remember it for "C"oordinate)


They are named I, II, III, IV in Roman Numerals



coordinate - A coordinate is the position of a point in the Cartesian plane


coordinate = "co" means goes along with (COefficient, COworker, CO-president, CO-champions)



"ordinate" means in order

So coordinate means numbers that go along with each other in a certain order


The numbers are the x and y values and the order is that the x always comes first



Also called an ordered pair (x y "ordered" and they are a "pair" of numbers)


Ordered pairs are recognized by the use of ( x , y ) format



origin = (0, 0) the center of the graph (its beginning or origin)


When you count the coordinate' s position, you count from the origin.



x comes before y in the alphabet so the order is (x, y) 
always go right or left first, then up or down



the x axis is the horizontal axis (goes across)


Remember that because the number line also is horizontal and you learn that first


(the pattern to remember is x is always first and the number line is before going up and down)



NOW LET'S GET TO WHAT YOU ACTUALLY DO!!!


1) Count your x value:


positive x, count right from origin (positive numbers are to the right of zero on number line)


negative x, count left from origin


2) Count your y value:


positive y value, count up from where your x value was (up is the positive direction)


negative y value, count down from where your x value was (down is the negative direction)



EXAMPLE:
(3, 5) Count 3 to the right from the origin, then 5 up


(3, -5) Still count 2 to the right, but now count 5 down


(-3, 5) Count 3 to the left from the origin, then count 5 up


(-3, -5) Again count 3 to the left, but now count 5 down



BUT WHAT HAPPENS WHEN
 ONE OF THE VALUES IS ZERO?


If the y value is zero it means that you move right or left, but don't go up or down:


SO YOUR POINT WILL BE ON THE x AXIS........

x axis is where y = 0 
Example: (3, 0) is a point on the x axis, 3 places to the RIGHT


Example: (-3, 0) is a point on the x axis, 3 places to the LEFT



If the x value is zero it means that you don't move right or left, you just go up or down.


SO YOUR POINT WILL BE ON THE y AXIS...........

y axis is where x = 0 


Example: (0, 3) is a point on the y axis, 3 places UP


Example: (0, -3) is a point on the y axis, 3 places DOWN