1. DRILL: May 3, 2013 Quadratic Equation
What is wrong with the following “solution” of

2. CHAPTER 4 COORDINATE GEOMETRY Pages 197 - 232 Learning Outcomes
At the end of this chapter, you should be able to:
appreciate the importance of coordinate geometry
plot points in a coordinate system
determine the distance between two points
find the midpoint of a line segment
find the slope of a line
find the equation of a line
slope-intercept form
point-slope form
explain the meaning of slope and y-intercept in the real world context
enhance your critical thinking and self-management skills
Prior knowledge: Plane, Points, Lines, Linear equations, Solution of Linear Equation

3. CHAPTER 4: COORDINATE GEOMETRY
Why do we need to study Coordinate Geometry?
4.1 The Rectangular Coordinate System
Definition 4.1 Coordinate Geometry is a system where the position
of points on the plane is described using an ordered pair of numbers.
Let us recall that a plane is a flat surface that goes on forever in both directions..
If we were to place a point on the plane, coordinate geometry gives us a way to
describe exactly where it is by using two numbers.
What is the exact
location of “x”?
Consider this grid:
x is at (5, C)

4. The Rectangular Coordinate Plane or the Cartesian Planey 5 4 3 2 1
Quadrant II
(-,+)
Quadrant I
(+,+) x -1 -2 -3 -4 -5
Quadrant III
(-,-)
Quadrant I V
(+,-)
Figure 4.1: The Coordinate Plane

5. DRILL B( , ) A( , ) D( , ) E( , ) C( , )
The Rectangular Coordinate Plane or the Cartesian Plane y
Quadrant II
(-,+)
Quadrant I
(+,+) 5 4 3 2 1
DRILL
Plot the ff.
points,
label them.
F(-1, 5)
G(-3, - 4)
H(0, - 5.5)
J(5, - 1)
B( , )
A( , )
D( , )
E( , ) x -1 -2 -3 -4 -5
C( , )
Quadrant I V
(+,-)
Quadrant III
(-,-)
Figure 4.1: The Coordinate Plane

6. The Rectangular Coordinate Plane or the Cartesian Planey
Quadrant II
(-,+)
Quadrant I
(+,+) 5 4 3 2 1 x -1 -2 -3 -4 -5
Quadrant I V
(+,-)
Quadrant III
(-,-)
Figure 4.1: The Coordinate Plane

7. CHAPTER 4: COORDINATE GEOMETRY
Name: ___________________ Group: _____
DRILL
Point x y
Coordinates
Quadrant/Axis
CHAPTER 4: COORDINATE GEOMETRY

8. The X & Y-intercepts x y -1 -2 -3 -4 -5 5 4 3 2 1
y-intercept
(0, -1)
x-intercept
(0.5, 0)
line a
Self test:
Identify the x & y
Intercepts of
line b:
X-int: _______
Y-int: _______
line b
an x-intercept is a point on the graph where y is zero,
and a y-intercept is a point on the graph where x is zero.
Figure 4.3

9. In the same manner, the y-intercept is solved by setting x = 0
The X & Y-intercepts
Without the graph, we can also find the intercepts as long as the equation is given.
For example in 3x + 2y = 12; the x intercept is solved by setting y = 0
So, setting y = 0, we have: 3x + 2(0) = 12; Solving for x,
we have 3x = 12; therefore x = 4, the x-intercept.
In the same manner, the y-intercept is solved by setting x = 0
Therefore, setting x = 0, we have: 3(0) + 2y = 12; Solving for y,
we have 2y = 12; So, y = 6, the y-intercept.
Self Test: Find the x and y intercepts in the linear
equation 6x – 10y = 30
y-intercept: _________
x-intercept: _________
To enhance the concepts of intercepts, you can do Exercise 4.1
Board Work # 5 page 210 before proceeding to the next topic.

10. Distance and Midpoint Formulas Page 200y
Find the Distance & Midpoint
of points A & B. 5 4 3 2 1
The distance between
two points:
B(2, 3)
Midpoint
M(0, -1) : x : -1 -2 -3 -4 -5 .
The midpoint between
two points:
A(-2, -5) C
Figure 4.4: Distance & Midpoint Between Two Points

11. Distance and Midpoint Formulas Page 201
Self Test: Quadrilateral ABCD in Figure 4.5 is
drawn in a coordinate plane below.
Use the distance formula to show
that line segment
are equal in length. : :
Use the Midpoint formula to show
that the midpoints of the diagonals .
have the same coordinates.
Figure 4.5:
Challenge!
What are diagonals?
To enhance the concepts “distance bet. 2 points” and “midpoint”, you can do
Exercise 4.1 Board Work #4 & C applications on pages
before proceeding to the next topic.

12. each unit represents one mile.
The bus breaks down on your way to school. The conductor calls the garage for a tow truck.
There is a foot bridge halfway between the garage and the bus.
a) How far is the garage from the bus?
b) Give the exact location of the foot bridge. Draw the foot bridge on the map.
c) How far is the foot bridge from the school?
In the map
on the right,
each unit
represents
one mile.

14. CHAPTER 4: COORDINATE GEOMETRY
Slope of a Line: Page 202
Slope of a line is a number that measures its "steepness",
usually denoted by letter “m”.
What do you know about slope?
Consider the following definitions of slope.
On the coordinate plane, the slant of a line is called the slope.
It is the change in y for a unit change in x along the line.
Slope is the ratio of the change in the y-value over the change
in the x-value.
Slope of a line is the ratio of its rise to run.

15. CHAPTER 4: COORDINATE GEOMETRY
Slope of a Line: Page 202
m= “+”
m= “-”
“NO slope”
m = “0”

16. CHAPTER 4: COORDINATE GEOMETRY
Slope of a Line: Page 202

17. CHAPTER 4: COORDINATE GEOMETRY

18. CHAPTER 4: COORDINATE GEOMETRY
Slope of a Line: Page 202
Slope of a line is a number that measures its "steepness",
usually denoted by letter “m”.
Why do we use “m” for slope?
Slope" was once called the "modulus of slope",
the word "modulus" being used in its sense of
"number used to measure"
It is originated from the Arabic word
MOMAS means tangent.

19. CHAPTER 4: COORDINATE GEOMETRY
Slope of a Line: Page 202
Slope of a line is a number that measures its "steepness",
usually denoted by letter “m”.
…but…what is slope in the real world?
In the real world,
slope of a line tells us how something changes
over time.
If we find the slope we can find the rate of change over that period.
- Carpenters use the terms rise and run to
describe the steepness of a stairway or
a roofline.
- We can use rise and run to describe the
steepness of a hill.

20. Why do you think we are using “m” instead of “s” to represent slope?
CHAPTER 4: COORDINATE GEOMETRY
Challenge!
Why do you think we are using “m” instead of “s” to represent slope?
Step1: Choose two exact points
on the line & then connect
them by a straight line.
Step2: Draw a right triangle
using the two points you have
selected as the vertices of the
two acute angles.
Step3: To get the rise, count the
# of units of the vertical leg;
To get the run,, count the #
of units of the horizontal leg.
How do we find the slope?

21. CHAPTER 4: COORDINATE GEOMETRY5 4 3 2 1
B(2, 3)
Step1: Choose exact points
on the line & then connect
them by a straight line.
Step2: Draw a right triangle
using the two points you have
selected as the vertices of the
two acute angles.
Step3: To get the rise, count the
# of units of the vertical leg;
To get the run,, count the #
of units of the horizontal leg.
run = 2 units
rise = 4 units x -1 -2 -3 -4 -5
D(0,-1)
rise = 6 units
A(-2,-5)
run = 3 units
Figure 4.6

22. CHAPTER 4: COORDINATE GEOMETRY5 4 3 2 1
B(2, 3)
Use points A & B to
get the slope using
the ratio rise to the
run. x -1 -2 -3 -4 -5
What conclusion
can you draw
about the slope
of the line using
any 2 points on it?
D(0,-1)
A(-2,-5)
Figure 4.6

23. Using the formula, let us find the slope of the
CHAPTER 4: COORDINATE GEOMETRY
Using the formula, let us find the slope of the
Line in Fig. 4.6 using:
A] Points A(-2, -5) and B(2, 3).
B] Points A(-2, -5) and D(0, -1).

24. line a line b line c line d line “b” falls from left
to right; slope is negative.
m = _________
Direction of the Line and its Slope y
line a
line b 5 4 3 2 1
line “a” rises from left
to right; slope is positive
m = _________ x -1 -2 -3 -4 -5
line c
line “d” is vertical;
no run or run = 0
m = ________
line “c” is horizontal;
no rise or rise = 0
m = _________
line d
Figure 4.7

25. CHAPTER 4: COORDINATE GEOMETRY
INTERPRETING SLOPE (m) using lines in Fig. 4.7
Slope (m) is POSITIVE: This means, for every 5-unit increase in y,
x increases by 2 units. Line “a” rises from left to right.
Slope (m) is NEGATIVE: This means, for every 1-unit decrease in y,
x increases by 1unit. Line “b” falls from left to right.
Slope (m) is ZERO: This means, y does not change as x increases;
line c is horizontal; therefore the slope of horizontal line is “zero”. ; ;
Slope (m) is UNDEFINED or does not exist: This means,
as y increases, x does not change; thus, the two x coordinates
are the same, so the difference is zero. In short, vertical line
has NO defined slope.
Page 204

26. CHAPTER 4: COORDINATE GEOMETRY
Slopes of
parallel lines y
line “b” 5 4 3 2 1
Page 204
line “a”
What do you know
about the slopes
of parallel lines? x -1 -2 -3 -4 -5
Figure 4.8: Slopes of Parallel Lines

27. CHAPTER 4: COORDINATE GEOMETRY
Slopes of
Perpendicular lines x y -1 -2 -3 -4 -5 5 4 3 2 1
line “a”
line “b”
Figure 4.9: Perpendicular lines
Page 205
Please read the
word of caution
on page 205!
What do you know
about the slopes
of perpendicular
lines?
Who can interpret
the slope of:
- line a
- line b

28. CHAPTER 4: COORDINATE GEOMETRY
Slope &
steepness
of a line x y 5 4 3 2 1
Figure 4.10: 10 9 8 7 6
Page 206
Which between lines “a” and “b” is steeper? Justify.
Which between lines “c” and “d” is steeper? Justify.
Which lines have the same steepness? Justify.
Arrange the lines in order from the steepest to the least
Are lines “b” and “c” perpendicular to each other? Justify

29. CHAPTER 4: COORDINATE GEOMETRY
Slope &
steepness
of a line
Page 206
Complete the statements below:
The bigger the absolute value of the slope, the _______________
is the line.
___________________ line have the same slopes.
Two lines are perpendicular to each other if ___________________
______________________________________________________.
Horizontal lines have a slope equal to ____________.
___________________ lines have no slopes.
A line that rises from left to right has ______________ slope.
A line that ___________________________________
has a negative slope.

30. CHAPTER 4: COORDINATE GEOMETRY10 9 8 7 6
line a
line b
1. Find the slope of
line “a” only!
2. Approximate the
slopes of
- line b
- line c
- line d
- line e
3. Explain how did
you do it?
4. What concepts did
you use in this
exercise?
line c
line d 5 4 3 2 1
line e x
Figure 4.11:

31. CHAPTER 4: COORDINATE GEOMETRY10 9 8 7 6
Do the Self-Test on Pages ! B C 5 4 3 2 1 x D A
Do Page #2 (a to i)
Figure 4.11

32. CHAPTER 4: COORDINATE GEOMETRY
Page 207 y
Draw a line passing
through the point (-2, -1)
with
Graphing of Line
given its Slope &
a Point 5 4 3 2 1
Step 1:Plot (-2, -1)
Step 2: Decide on the direction.
Since m is +, the line goes up
to the right.
Step 3: Find another point
using m = 4/3; that is
4 units up from (-2, -1)
& 3 units to the right.
That point is ((1, 3). Or you
can go 4 units down from
(-2, -1) & 3 units left. That
point is (-5, -5).
Step 4: Connect the points by a
straight line.
(1, 3) x
Do the
self-test on
Page 208! -1 -2 -3 -4 -5
(-2, -1)
(-5, -5)
Figure 4.12:
Do the Mathematical Investigations on Page 210
before proceeding to the next lesson!

33. CHAPTER 4: COORDINATE GEOMETRY
REVIEW
Meaning of slope & y-intercept in the real world
Example: Taxi fare: y = 7.50x + 40
2. Finding the slope given a line:
Finding the slope given 2 points: (0, 6) & (8, -10)
3. Finding the slope & the y-intercept given the equation
y = mx + b
Example: y = 2x – 5; m = 2 & b = -5
2y = -6x m = - 3 & b = 5 4.

34. CHAPTER 4: COORDINATE GEOMETRY
Page 208 y 5 4 3 2 1
Draw a line passing
through the point (-1, 2)
with x -1 -2 -3 -4 -5
Draw a line with a slope
of 3 and
- a y-int of 2
- an x-int of 2
Go to Page 216, do Letter B (3-6)
In a graphing paper.
Figure 4.12:

35. Page 213 CHAPTER 4: COORDINATE GEOMETRY 4.2 Equation of a line
A straight line is defined by a linear equation whose general
form is Ax + By + C = 0, where A, B are not both 0.
Form
Equation
Remark
1. Slope Intercept
slope y-intercept
Use this form when you know the slope “m”and the y-intercept “b”.
y = mx + b where:
Example1.
Find the equation of a line with a slope of 3
and a y-intercept of 5.
Equation: y = 3x + 5
Your turn: Give the equation of the line with m = 6 & b = -1
Equation: y = 6x - 1

36. Page 213 CHAPTER 4: COORDINATE GEOMETRY 4.2 Equation of a line
A straight line is defined by a linear equation whose general
form is Ax + By + C = 0, where A, B are not both 0.
Form
Equation
Remark
2. Point Slope Form
Use this form when you know a point on the line and the slope (or can determine the slope).
Your turn: Give the equation of the line with m = 4 &
and passing through the point (0, 3).
y = 4x + 3
Example2.
Find the equation of the line with a slope of -3
and the line passes through the point (2, 4)
Equation: y = -3x + 10 10
What is the y-intercept “b”of this line? __________

37. Page 213 CHAPTER 4: COORDINATE GEOMETRY 4.2 Equation of a line
A straight line is defined by a linear equation whose general
form is Ax + By + C = 0, where A, B are not both 0.
Form
Equation
Remark
3. Horizontal Line
y = b
This equation also describes what is happening to the y-coordinates on the line. In this case, it is always “b”.
Example3.
Find the equation of the horizontal line
with a y intercept of -3.
y = - 3

38. Page 213 CHAPTER 4: COORDINATE GEOMETRY 4.2 Equation of a line
A straight line is defined by a linear equation whose general
form is Ax + By + C = 0, where A, B are not both 0.
Form
Equation
Remark
4. Vertical Line
x = k
This equation also describes what is happening to the x-coordinates on the line. In this case, it is always “k”.
Example3.
Find the equation of the line which is parallel to
y–axis and passing through (5, 0).
x = 5

39. Page 213 Do the Self-Test on page 214. Do Exercise 4.2 pages 215-216.
CHAPTER 4: COORDINATE GEOMETRY y
Page 213
4.2 Equation of a line
A straight line is defined by a linear equation whose general
form is Ax + By + C = 0, where A, B are not both 0.
Example 3:
Find the equation of the line whose slope is 4
and passing through the point (0, -3).
Example 4:
Find the equation of the line that passes
through the points (-3, 5) and (-5, -8).
First, find the slope:   
Example 5:
Find the slope and y-intercept for the
equation: 6x + 3y = 9.
Do the Self-Test on page 214.
Do Exercise 4.2 pages

40. CHAPTER 4: COORDINATE GEOMETRY
Page 221
CHAPTER 4: COORDINATE GEOMETRY y
4.3 Applications
Interpreting slope & y-intercept in the
real world context.
Example 1: Physical Fitness
The membership fee (f), in pesos, at a gym is based on
the number of people (p) that a new member recruits to join
the gym at the time of registration.
The membership fee formula is given by the equation:
f = – 300p , where:
f = membership fee, and
p = the number of recruits at the
time of registration
Relating the equation f = – 300p , to y = mx + b,
we have:
y = x
What is the slope of the membership formula?
What does the slope of this equation mean in the context of the problem?

41. CHAPTER 4: COORDINATE GEOMETRY
Page 221
CHAPTER 4: COORDINATE GEOMETRY y
4.3 Applications
Interpreting slope & y-intercept in the
real world context.
Example 1: Physical Fitness
f = – 300p y = x
What is the slope of the membership formula?
What does the slope of this equation mean in the context of the problem?
Answer: The slope (m) is – 300; Since the slope is negative, this means
that a membership fee is decreased by Php300 for every person that
a new member recruits at the time of registration.
What is the y-intercept of the membership formula?
What does the y-intercept of this equation mean in the context of the problem?
Answer: The y-intercept (b) is 5,000; This means the new member
will pay a membership fee of Php5,000 if he has no recruit at the
time of the registration (because x = 0); that is f = –(300)(0)+ 5,000;
f = 0 + 5,000; therefore f = Php5,000.

42. CHAPTER 4: COORDINATE GEOMETRY
Page 221
CHAPTER 4: COORDINATE GEOMETRY y
4.3 Applications
Interpreting slope & y-intercept in the
real world context.
Example 2: Food Handling
y = 200x
What is the slope of the bacteria formula?
What does the slope of this equation mean in the context of the problem?
Answer: The slope (m) is 200; since the slope is positive, this means that
the number of bacteria increases by 200 for every hour that the meat is in
the warm room.
What is the y-intercept of the bacteria formula?
What does the y-intercept of this equation mean in the context of the problem?
Answer:The y-intercept (b) 1,550; This means that, the meat has
already 1,550 bacteria when it was placed in the warm room;
that is x = 0, so y = 200(0) + 1,550; y = 1,550.

43. CHAPTER 4: COORDINATE GEOMETRY
Page 221
CHAPTER 4: COORDINATE GEOMETRY y
4.3 Applications
Interpreting slope & y-intercept in the
real world context.
Example 2: Food Handling
y = 200x
The Scientist-Chef left the piece of meat in the warm room
at 9:00 AM. What number of bacteria does the meat will have
at 1:30 p.m. of the same day? Justify your answer by showing
the solution.
Answer: The number of hours “x” that elapsed from 9:00AM to 1:30 PM
is 4.5.
So, since y = 200x + 1,550;
y = 200(4.5) + 1,550;
y = ,550;
y = 2,450, this is the number of bacteria which is in the meat at 1:30 P.M.

44. CHAPTER 4: COORDINATE GEOMETRY
Page 222
CHAPTER 4: COORDINATE GEOMETRY y
4.3 Applications
Interpreting slope & y-intercept in the
real world context.
Activity Worksheet 4.3b page 227
A tire company wants to determine how quickly the tread
on its tires wears down with average use.
- Let x represents the number of months the tire was used.
y represents the thickness of the tire thread, in millimeters.
An equation for a line that describes this relationship is
Explain the meaning of slope & y-intercept in the context
of the problem.

45. CHAPTER 4: COORDINATE GEOMETRY
Page 222
CHAPTER 4: COORDINATE GEOMETRY y
4.3 Applications
Interpreting slope & y-intercept in the
real world context.
Example 3: Demand & Price
BBZ company noticed that a linear relationship exits between the price of
The school bags and the number of bags sold. At Php1000 the company
sold 2,000 pieces. When the company raised the price to Php1,200 the
Company was able to sell only 1,500 pieces. Find an equation that
relates the price of school bags to the number of bags sold.
Show your solution
Let x = the number of bags sold
y = the price of bag

46. CHAPTER 4: COORDINATE GEOMETRY
Page 222
CHAPTER 4: COORDINATE GEOMETRY y
4.3 Applications
Interpreting slope & y-intercept in the
real world context.
Example 3: Demand & Price
1] Find an equation that relates the price of school bags to the number of bags sold.
Remember that if we have two points we can determine the
equation of a line. In this case, we will use “ point-slope form”:
First, solve for the slope using the two points: (2000, Php1000) & (1500, Php1200) ;
Use either point; say let
us use (2000, 1000);
we have:
Therefore, the equation that relates the price to the number of bags demanded is:

47. CHAPTER 4: COORDINATE GEOMETRY
Page 222
CHAPTER 4: COORDINATE GEOMETRY y
4.3 Applications
Interpreting slope & y-intercept in the
real world context.
Example 3: Demand & Price
Therefore, the equation that relates the price to the number of bags demanded is:
2] Explain the meaning of slope & y-intercept in the context of this problem.
The slope means “for every 2 pesos decrease in the price
of bag, an additional 5 pieces is sold”.
The y-intercept 1,800 means, “if the price is Php1,800, no bag is sold”.

48. CHAPTER 4: COORDINATE GEOMETRY
Page 222
CHAPTER 4: COORDINATE GEOMETRY y
4.3 Applications
Interpreting slope & y-intercept in the
real world context.
Example 3: Demand & Price
Therefore, the equation that relates the price to the number of bags demanded is:
3] What is the unit price if the number of bags
demanded/sold is 1000?
4] How many bags are demanded/sold if the
unit price is Php500?

49. CHAPTER 4: COORDINATE GEOMETRY
Page 222
CHAPTER 4: COORDINATE GEOMETRY y
4.3 Applications
Interpreting slope & y-intercept in the
real world context.
Example 3: Demand & Price
SMS department store sell 2000 bags when the unit price is Php450.
It was determined that it can sell 300 bags more with each Php100
reduction in the unit price.
a. Find the demand equation: y = mx+b
b. Explain the meaning of slope & y-intercept in the context
of this problem.
c. How many bags are demanded if the unit price is Php200?
d. What is the unit price if 3000 bags are demanded?