Class Notes for Lecture Day 14
The Great Algebra Triad:
Equations, Tables, and Graphs
Descartes was a French philosopher,
mathematician, physicist, and writer who
spent most of his adult life in the Dutch
Republic. He has been dubbed the
"Father of Modern Philosophy", and
much of subsequent Western philosophy
is a response to his writings, which
continue to be studied closely to this day.
In particular, his Meditations on First
Philosophy continues to be a standard
text at most university philosophy
departments. Descartes' influence in
mathematics is also apparent; the
Cartesian coordinate system—allowing
geometric shapes to be expressed in
algebraic equations—was named after
him. He is credited as the father of
analytical geometry. Descartes was also
one of the key figures in the Scientific
Revolution.
Cited from Wikipedia.org
René Descartes
(31 March 1596 – 11 February 1650)
Horizontal Number Line
Vertical Number
Line
“Cogito ergo sum”
“I think, therefore I am”
The Cartesian Plane
y
5
4
3
2
1
-5 -4 -3 -2 -1-1
-2
-3
-4
-5
1 2 3 4 5 x
Coordinate Axes
x-axis
y-axis
Quadrants
Cartesian Coordinate System
Ordered Pair
x-coordinate (abscissa)
y-coordinate (ordinate)
origin
“Pretty Points”
Linear Equation in Two Variables
Solutions
Table of Values, T-table, x-y Table
Linear Graph
Intercepts
x-intercept
y-intercept
Slope
The Great Algebra Triad
Linear
Equation
Equation to Table:
Slope-intercept Form:
y = mx + b
Decide on values of x to use and list
them under “x” on the table (be sure
to include zero). Substitute each
value into the equation to calculate
the corresponding value of y and list
them under “y” on the table
Equation to Graph:
Using the slope-intercept form y
= mx + b, plot the y-intercept (b),
then find more points using the
slope (m), “Rise over run”
Table to Equation:
Choose any two ordered pairs to
work with. Use the equations
m=(y1-y2)/(x1-x2) and y=mx+b to
find values of m and b, and write
these two values as y = mx + b
Table of
Values
List of Specific Solutions
Table to graph:
Plot the x and y values across from each
other as (x, y) points on the graph. Use a
straight edge to draw a straight line
connecting the points.
Graph to Equation:
Identify the y-intercept (b),
and several “Pretty Points” to
calculate slope (m), and write
these two values as y=mx + b
Graph
Pictorial Representation
Graph to Table:
Identify as many “Pretty Points” as
needed and organize their x and y
coordinates into an x-y T-table
Equation:
Graph:
y = 3x + 4
y
10
Table of Values:
x
y
5
-10
-5
0
5
10
x
5
10
x
-5
-10
Equation:
Graph:
y = -½x – 3
y
10
Table of Values:
x
y
5
-10
-5
0
-5
-10
Equation:
Graph:
y
10
Table of Values:
x
y
-3
7
-2
5
-1
3
0
1
1
-1
2
-3
3
-5
Equation:
5
-10
-5
0
5
10
x
5
10
x
-5
-10
Graph:
y
10
Table of Values:
x
y
-9
-9
-6
-5
-3
-1
0
3
3
7
6
11
9
15
5
-10
-5
0
-5
-10
Equation:
Graph:
y
10
Table of Values:
x
y
5
-10
-5
0
5
10
x
5
10
x
-5
-10
Equation:
Graph:
y
10
Table of Values:
x
y
5
-10
-5
0
-5
-10