The Coordinate System and Graphs - §3.1 - 3.2
Fall 2013 - Math 1010
(Math 1010)
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Roadmap
I
Plotting ordered pairs.
I
The distance formula.
I
The midpoint formula.
I
Graphs of equations.
I
Intercepts.
I
Verifying solutions to equations.
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§3.1 - Plotting Points
3
A•
2
B•
-3
-2
1
-1
C
0•
0
-1
1
2
D•
3
E•
F•
-2
-3
Point B is the ordered pair (-2,1). What are the coordinates of each point?
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§3.1 - Plotting Points
Previous slide:
Point
A
B
C
D
E
F
(Math 1010)
Coordintes
(3, 2)
(−3, 1)
(0, 0)
(3, 0)
(2.6, −1)
(−1, −2)
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§3.1 - Plotting Points
To plot a point, the ordered pair, (x, y ) describes the coordiante by moving
x units along the horizontal axis, and y units along the vertical axis.
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§3.1 - Plotting Points
To plot a point, the ordered pair, (x, y ) describes the coordiante by moving
x units along the horizontal axis, and y units along the vertical axis.
Tips:
I
Plot with a solid dot.
I
Give the point a clean label.
(Math 1010)
M 1010 §3.1 - 3.2
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§3.1 - Plotting Points
To plot a point, the ordered pair, (x, y ) describes the coordiante by moving
x units along the horizontal axis, and y units along the vertical axis.
Tips:
I
Plot with a solid dot.
I
Give the point a clean label.
An ordered pair can lie in one of four quadrants, or on the border.
(Math 1010)
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§3.1 - Plotting Points
3
•
2
•
-3
-2
1
-1
0•
0
-1
1
•
3
2
•
• -2
-3
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§3.1 - Plotting Points
3
•
2
•
-3
-2
1
-1
0•
0
1
-1
•
3
2
•
• -2
-3
Any problems with this plot?
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§3.1 - Plotting Points
Quadrant II
3
Quadrant I
A•
2
B•
-3
-2
1
-1
C
0•
0
-1
1
D•
3
2
E•
F•
-2
Quadrant III
-3
Quadrant IV
Labels return. Which quadrant is each point within? (Quadrants are
labelled for convenience.)
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§3.1 - Distance Formula
Distance between two reals is the absolute value of the distance.
distance = |b − a|
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§3.1 - Distance Formula
Distance between two reals is the absolute value of the distance.
distance = |b − a|
From the Pythagorean Theorem,
”The sum of the squares of the legs of a right triangle equals the square of
the hypotenuse.”
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§3.1 - Distance Formula
Distance between two reals is the absolute value of the distance.
distance = |b − a|
From the Pythagorean Theorem,
”The sum of the squares of the legs of a right triangle equals the square of
the hypotenuse.”
The distance between two points in the Cartesian coordinate frame,
(x1 , y1 ) and (x2 , y2 ) is
q
distance = (x2 − x1 )2 + (y2 − y1 )2
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§3.1 - Distance Formula
5
•(2,4)
4
3
(-1,2)•
2
1
0
-2
-1
0
1
2
3
4
-1
Distance? This is the hypotenuse of some right triangle.
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§3.1 - Distance Formula
5
•(2,4)
4
3
(-1,2)•
2
2
3
1
0
-2
-1
0
1
2
3
4
-1
√
+ p= 13. The distance is 13.√
Also, (2 − (−1))2 + (4 − 2)2 = 13.
32
22
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§3.1 - Midpoint
Two points make a line segment. The middle of that segment is the
midpoint.
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§3.1 - Midpoint
Two points make a line segment. The middle of that segment is the
midpoint.
Example: The points (−5, −3) and (9, 3) can be connected by a line
segment. The midpoint is
4 0
(−5) + 9 (−3) + 3
+
=
+
= (2, 0)
2
2
2 2
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§3.1 - Midpoint
Two points make a line segment. The middle of that segment is the
midpoint.
Example: The points (−5, −3) and (9, 3) can be connected by a line
segment. The midpoint is
4 0
(−5) + 9 (−3) + 3
+
=
+
= (2, 0)
2
2
2 2
The Midpoint Formula between (x1 , y1 ) and (x2 , y2 ) is
x1 + x2 y1 + y2
Midpoint =
,
2
2
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§3.2 - Graphs of Functions
How would you graph all points that are the same distance from (0, 0) and
(3, −2)?
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§3.2 - Graphs: Point-Plotting Method of Sketching A
Graph
1. (Ideal) Rewrite the equation by isolating one of the variables.
2. Make a table of values showing several solution points.
3. Plot these ordered pairs on a system with a reasonable scale.
4. Smoothly connect the points. (Sharp turns are graphed from
experience.)
Example: Plot 4x − y = −1.
(Math 1010)
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§3.2 - Graphs: Point-Plotting Method of Sketching A
Graph
1. (Ideal) Rewrite the equation by isolating one of the variables.
2. Make a table of values showing several solution points.
3. Plot these ordered pairs on a system with a reasonable scale.
4. Smoothly connect the points. (Sharp turns are graphed from
experience.)
Example: Plot 4x − y = −1.
(Math 1010)
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§3.2 - Absolute Value Equations
Absolute value equations are sharp at one or more places.
Example: Use the point-plotting method on y = |x|. Choose values for x
from {−3, −2, . . . , 2, 3}.
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§3.2 - Absolute Value Equations
Absolute value equations are sharp at one or more places.
Example: Use the point-plotting method on y = |x|. Choose values for x
from {−3, −2, . . . , 2, 3}.
x
y
Solution point
(Math 1010)
-3
3
-2
2
-1
1
0
0
1
1
2
2
3
3
(-3,3)
(-2,2)
(-1,1)
(0,0)
(1,1)
(2,2)
(3,3)
M 1010 §3.1 - 3.2
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§3.2 - Absolute Value Equations
Absolute value equations are sharp at one or more places.
Example: Use the point-plotting method on y = |x|. Choose values for x
from {−3, −2, . . . , 2, 3}.
x
y
Solution point
-3
3
-2
2
-1
1
0
0
1
1
2
2
3
3
(-3,3)
(-2,2)
(-1,1)
(0,0)
(1,1)
(2,2)
(3,3)
For the plot, do any patterns emerge? What point is the sharp corner?
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§3.2 - Intercepts
Intercepts are points on a graph that also intersects an axis. An
x−intercept has the corrdinate (a, 0), and a y −intercept has the
coordinate (0, b). That is, either the y or the x variable is zero, and the
other is a value that solves the original equation.
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§3.2 - Intercepts
Find the intercepts of y = −4x − 8.
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§3.2 - Intercepts
Find the intercepts of y = −4x − 8.
x: Let y = 0. Then,
0 = −4x − 8
8 = −4x
x = −2
This is (−2, 0).
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§3.2 - Intercepts
Find the intercepts of y = −4x − 8.
x: Let y = 0. Then,
0 = −4x − 8
8 = −4x
x = −2
This is (−2, 0).
y : Let x = 0. Then,
y =0−8
y = −8
This is (0, −8).
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Assignment
Assignment:
For Wednesday:
1. Exercises from §3.1, 3.2 due Wednesday, September 18.
2. Pre-Exam #1: September 18. Chapters 1 & 2
3. Read section 3.3.
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