Download Reasoning Algebraically

Coordinate Geometry
and Functions
The principal goal of education is to
create individuals who are capable of
doing new things, not simply of
repeating what other generations have
done.
Jean Piaget
Swiss psychologist
Introduction to the Cartesian
Coordinate (x,y) Plane
Before the end of the European Renaissance,
math was cleanly divided into the two separate
subjects of geometry and algebra. You didn't
use algebraic equations in geometry, and you
didn't draw any pictures in algebra.
Then, around 1637, a French guy named René
Descartes came up with a way to put these two
subjects together
Seating Chart
The following seating chart has 5 rows of desks
with 4 desks in each row
Mary sits in the
third row at the
second desk
Row 1 Row 2
Row 3
Row 4
Row 5
Who is sitting in desk (4,2)? N
4
A
B
C
D
E
3
F
G
H
I
J
Row 4
Seat 2
2
K
L
M
N
O
1
P
Q
R
S
T
1
2
3
4
5
Mary sits in the third row at the second
desk (3,2) and George sits in the second
row at the third desk (2,3). Are these seats
the same?
No!! The seats (3,2) and (2,3) are called
ordered pairs because the order in which
the pair of numbers is written is important!!
Coordinate Plane
This was a handy way to reference
individuals in the room, telling you how far
over (row 2) and how far up (seat 4) you
needed to go.
Descartes did something similar for
mathematics
Coordinate Plane
Descartes'
breakthrough was in
taking a second
number line, standing
it up on its end, and
crossing the first
number line at zero:
Definition
x axis – a horizontal number line on a
coordinate grid.
0 1 2 3 4 5 6
x
Definition
y axis – a vertical number line on a
coordinate grid.
y
6
5
4
3
2
1
0
Coordinates – an ordered pair of
numbers that give the location of a point
on a grid. (3, 4)
6
5
(3,4)
4
3
2
1
00 1 2 3 4 5 6
Hint
The first number is always the x or first letter in
the alphabet. The second number is always the y
the second letter in the alphabet.
6
5
(3,4)
4
3
2
1
00 1 2 3 4 5 6
How to Plot Ordered Pairs
Step 1 – Always find the x value first, moving
horizontally either right (positive) or left (negative).
y
6
(2, 3)
5
4
3
2
1
00 1 2 3 4 5 6 x
How to Plot Ordered Pairs
Step 2 – Starting from your new position find the
y value by moving vertically either up (positive) or
down (negative).
y
6
(2,
3)
5
4
(2,3)
3
2
1
00 1 2 3 4 5 6 x
What is the ordered pair?
(3,5)
y
6
5
4
3
2
1
00 1 2 3 4 5 6
x
The x-axis and y-axis separate
the coordinate plane into four
regions, called quadrants.
II
(-, +)
I
(+, +)
III
(-, -)
IV
(+, -)
How to Plot in 4 Quadrants
Step 1 - Plot the x number first moving to the left
when the number is negative.
(-3, -2)
y
3
2
1
x
0
-1
-2
-3
-3 -2 -1 0 1 2 3
How to Plot in 4 Quadrants
Step 2 - Plot the y number moving from your new
position down 2 when the number is negative.
y
3
(-3, -2)
2
1
0
x
-1
-2
-3
-3 -2 -1 0 1 2 3
Write the ordered pairs that name
points A, B, C, and D.
5
A = (1, 3)
B = (3, -2)
C = (0, -4)
D = (-6, -1)
•A
5
-5
•D
•B
-5
•C
Linear Equation:
an equation whose graph forms a
line in an x,y coordinate plane
In linear equations, all variables are taken to
the first power.
x+y=1
3x – 4y = 2
Suppose you were asked to graph
the solution set of the linear
equation,
x+y=3
You could start by creating a table
of values.
Choose any number to use as x or y
x+y=3
x
y
0 3
3 0
-3 6
Substitute into
equation and solve
-3
0+y =3
3
y ==0
36
Plot the points and graph
x+y=3
x y
0 3
3 0
-3 6
(-3,6)
(0,3)
connect
the
points
with a
straight
line
(3,0)
Let’s graph another line:
3x – y = 2
x
y
-1 -5
0 -2
2 4
Substitute into
equation and solve
3(-1)––yy== 2
3(0)
3(2)
y=4
-5
-2
Plot the points and graph
3x – y = 2
(2,4)
x y
-1 -5
0 -2
2 4
connect
the
points
with a
straight
line
(0,-2)
(-1,-5)
Graphing Using Intercepts
The x-intercept of a line is the x-coordinate of the
point where the line intercepts the x-axis.
8
The line shown
intercepts the x-axis
at (2,0).
6
4
2
We say that the
x-intercept is 2.
-8 -6
-4
-2
-2
-4
-6
-8
2
4
6
8
Graphing Using Intercepts
The y-intercept of a line is the y-coordinate of the
point where the line intercepts the y-axis.
The line shown
intercepts the y-axis at
(0,-6).
8
6
4
We say that the
y-intercept is -6.
2
-8 -6
-4
-2
-2
-4
-6
-8
2
4
6
8
Graph using intercepts. 4x – 3y = 12
*To find the y-intercept, let x = 0.
4x – 3y = 12
4(0) – 3y = 12
0 – 3y = 12
-3y = 12
-3 -3
y = -4
x
0
y
-4
Graph using intercepts. 4x – 3y = 12
*To find the x-intercept, let y = 0.
4x – 3y = 12
4x – 3(0) = 12
4x - 0 = 12
4x = 12
4
4
x=3
x
0
3
y
-4
0
Graph using intercepts. 4x – 3y = 12
8
x
0
3
6
4
2
-8 -6
-4
-2
-2
-4
-6
-8
2
4
6
8
y
-4
0
Slope
Graphs of any line can easily be classified
by its y-intercept and by its steepness.
The steepness is often referred to as slope
Types of Slope
Zero
Negative
Positive
Undefined
or
No Slope
Slope
1
1
To walk up
the steps, the
man must go
up one step
and over one
step
1
To walk up
the steps, the
man must go
up two steps
and
over one
1
step
To walk up
the steps, the
man must go
up one step
and over two
steps
Slope
10
rise
Slope =
run
run
8
6
4
rise
2
-4
-2
-2
-4
-6
-8
-10
2
4
6
8
10
Graph
the
line slope
containing points (2,1) and (7,6)
Find
the
and find the slope.
rise 5
Slope =

1
run 5
10
8
run
6
rise
4
2
-4
-2
-2
-4
-6
-8
-10
2
4
6
8
10
Slope
rise change in y  coordinates
Slope =

change in x  coordinates
run
y2  y1
m
x2  x1
or
y1  y2
m
x1  x2
Find the slope of the line containing points (1,6)
and (5,4).
y2  y1
m
x2  x1
46
m
5 1
2
m
4
1
m
2
Slope of a Horizontal Line
10
What is the rise?
What is the run?
0
8
6
rise 0
Slope =

0
run
6
The slope of ANY
horizontal line is 0.
6
4
2
-4
-2
-2
-4
-6
-8
-10
2
4
6
8
10
Slope of a Vertical Line
10
What is the rise?
What is the run?
5
8
6
0
rise 5
Slope =


run
0
The slope of ANY
vertical line is
undefined.
4
2
-4
-2
-2
-4
-6
-8
-10
2
4
6
8
10
If given an equation of a line, there
is a nice way to find the slope and yintercept. Simply write the equation
in slope-intercept form, which is:
y = mx + b
slope
y-intercept
Find the slope and
y-intercept of the
following equations.
y = 3x + ½
slope= 3
y-intercept = ½
Find the slope and yintercept:
3x + 5y = 10
First, solve the equation for y.
5y = -3x + 10
y = -3/5 x + 2
m= -3/5
b=2
Find the equation of a line:
To find the equation, consider slope intercept
form: y = mx + b
•
•
•
•
•
•
Slope
If given, just substitute it into the equation
Otherwise calculate the slope
y-intercept
If given, just substitute it into the equation
Otherwise use a point and substitute into
equation for x and for y, then solve for b
Find the equation of a line
containing the following points:
(4,2) and (3,6)
m=?
b=?
Always find slope first!
Find the slope using the formula.
m=2–6
4–3
m = -4
y = -4x + b
(4, 2) and (3, 6)
m = -4
Find the y-intercept.
y = mx + b
2 = -4 ͯ 4 + b
2 = -16 + b
18 = b
Pick a point to plug
into the equation
(4,2)
→
x=4
y=2
Write equation:
y = -4x + 18
Relations

A relation is a mapping, or pairing, of
input values with output values.

The set of input values is called the
domain (x values)

The set of output values is called the
range (y-values)
Functions
A relation is a function provided there is
exactly one output for each input.
It is NOT a function if at least one input
has more than one output
In order for a relationship to be a function…
EVERY INPUT MUST
HAVE AN OUTPUT
TWO DIFFERENT
INPUTS CAN HAVE
THE SAME
OUTPUT
INPUT
(DOMAIN)
FUNCTION
MACHINE
ONE INPUT MUST
HAVE ONLY ONE
OUTPUT
OUTPUT (RANGE)
Which of the following relations are functions?
R= {(9,10, (-5, -2), (2, -1), (3, -9)}
S= {(6, a), (8, f), (6, b), (-2, p)}
T= {(z, 7), (y, -5), (r, 7) (z, 0), (k, 0)}
No two ordered pairs can have the
same first coordinate
(and different second coordinates).
Identify the Domain and Range. Then
tell if the relation is a function.
Input
Output
-3
3
1
1
3
-2
4
Domain = {-3, 1,3,4}
Range = {3,1,-2}
Function?
Yes: each input is
mapped onto
exactly one output
Identify the Domain and Range. Then
tell if the relation is a function.
Input
Output
-3
3
1
-2
4
1
4
Domain = {-3, 1,4}
Range = {3,-2,1,4}
Function?
No:
input 1 is
mapped
onto both
-2 & 1
The Vertical Line Test
If it is possible for a vertical
line to intersect a graph at
more than one point, then the
graph is NOT the graph of a
function.
Page 117
Which are functions?
Function
Not a
Function
Function
Function Notation
f (x)
“f of x”
Input = x
Output = f(x) = y
Before…
Now…
y = 6 – 3x
f(x) = 6 – 3x
x
y
x
f(x)
-2
12
-2
12
-1
9
-1
9
0
6
0
6
1
3
1
3
2
0
2
0
(x, y)
(x, f(x))
Evaluate the function:
f(x) = 2x2 – 3
Find f(0), f(3)
f(0) = 2(0)2 – 3
f(3) = 2(3)2 – 3
=0–3
= 18 – 3
=-3
= 15