Objective:
Point on a Graph
The coordinate
plane is a grid with an xaxis and a yaxis.
Every point on
a plane is an ordered pair, which consists of two numbers separated by a
comma and enclosed by parentheses (x, y). The first number is the x value
and the second number is the y value. Example: (4, 7) is the point on
the coordinate plane where x = 4 and y = 7.
The point where
xaxis and yaxis meet is called the origin and has the ordered pair (0, 0).
The x value in
the ordered pair is the value on the right (if positive) or to the left (if
negative) from the origin (0,0).
The y value in
the ordered pair is the value on the up (if positive) or to the down (if
negative) from the origin (0,0).
Exercise 1: Draw the point on the
graph.
Lines and Equations
Lines that
cross the xaxis and do not cross the yaxis have equations that contain
only the variable x and not the variable y. For example x = 3.
Lines that
cross the yaxis and do not cross the xaxis have equations that contain
only the variable y and not the variable x. For example y = 7.
Exercise 2: Draw the graph for given
equation.
Slope of a Line
When the two or
more points on a plane form a line, a special equation, called linear
equations, can be written to represent all of the points on the line.
Line that cross
both the xaxis and the yaxis have equations that contain both the
variables x and y. For example, y = x + 1
Every line has a slope, except vehicle line (have no slope).
Slope is found by comparing the positions of any two points on the line.
For example, for a given two points (x1, y1) & (x2, y2), the slope of the line
would be (y1y2) / (x1  x2). It is also described as (rise)/(run) or (the
change in y)/the change in x).
Ex: If two points on a line are given as (2,0) and (1, 02) then the slope of
the line is (y1y2)/(x1x2) =( 0(2))/(21) = 2/(3) = (2/3)
PARALLEL: If two lines are parallel then the slopes of both lines are
parallel.
PERPENDICULAR: If two lines are perpendicular then the slopes of
perpendicular lines will be negative reciprocals. For example, if the slope of
the line 1 is m1 and the slope of line 2 is m2, and the line l1 and l2 are
perpendicular then m1 =  (1/m2) or (m1)(m2) = 1.
Exercise 3: Draw the graph for given
inequality equation.
Graphing Lines
The equation for a line is y=mx +b where m is the slope of the line and b is
the yintercept (yvalue of the point where the line intersects yaxis).
Case I: The Horizontal lines have equations which simplify to the form y = b,
where b is the yintercept. The slope of these lines is zero.
Case II: The Vertical lines have no sope and the equation of the line takes
the form of x=c, where c is the xintercept.
Case III: Neither horizontal nor vertical line:
1. If the slope and yintercept value is given then substitute these numbers
in the slopeintercept form os a linear equation, y= mx+b, where m is the
slope and b is the yintercept.
2. If the slope (m) and a point (x1, y1) is given then solve for
equation y = mx +b
since m = (y y1)/(x  x1) => y  y1 = m (x  x1) => y  y1 = mx
 mx1 => y = mx  mx1 + y1 => y = mx + (y1  mx1)
3. Given 2 points (x1, y1) and (x2, y2) on the line, then solve for the
equation y = mx + b
Step 1: Find the slop, m =(y2y1) / (x2
y1) to form the equation y = mx + b
Step 2: Find the value of b, by
substituting the (x1, y1) and (x2, y2) values in the equation y = mx + b.
4. Given the equation of another line:
Option 1: To find the equation for the parallel
line, the slop of the equation will the same as the slope of the given line..
Option 2: To find the equation for the
perpendicular line, the slope of the line will be the negative reciprocal of
the given slope.
Exercise 4: Draw the graph for given
inequality equation.
Graphs of Inequalities
x > n means that any number x that is greater than n is number x is in the
solution set . For example, for a given equation x > 3 means that all real
numbers that are greater than 3 such as 4, 5, 6 are in the solution
set. However, numbers such as 1, 2 and 3 are not in the solution set.
Examples:
1.)
x > 2 means all real numbers greater than or
equal to –2.
2.)
x< 2 means all real numbers less than 2.
3.)
x< 2 means all real numbers less than or equal
to 2.
Example: Draw
the graph for equation y < 7
Exercise
Simple Inequalities
A simple inequality is solved basically
the same way as a firstdegree equation. The only difference is when
multiplying or dividing by a negative number, switch the inequality sign.
Examples:
7 > 10x + 13
=> 7 13 > 10x +13 13
=> 20 > 10x
=> 2 > x
=> 2 < x
Exercise
Compound Inequalities & Conjunction
Conjunction: “and” For a
conjunction to be true both parts of the inequality must be true. The graph
is the intersection of both solution sets.
Example: Solve x > 2 and x < 3
(x > 2)
(x < 3)
(2 < x < 3)
Solution: “The set of all real numbers x that x is greater than –2 and
less than 3”.
Exercise
Compound Inequalities & Disjunction
Disjunction:
“or” For a disjunction to be true at least one part of the inequality must be
true (both parts may be true). The graph is the union of both solution sets.
Example: Solve x > 2 or x < 3
(x > 2)
(x < 3)
(2 < x < 3)
Solution: “The set of all real numbers such that x is greater than –2 or
less than 3”.
Exercise
Review
If x>n, then a number (x) greater than number (n) will be with
in the solution set, and all the other value of x will not be in the
solution set.
Lessons
Learned
Quiz
None
