Math 025 Section 7.1 Coordinate Systems

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Math 025 Section 7.1 Coordinate Systems
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y-axis
Quadrant I
Quadrant II
x-axis
Origin
Quadrant III
Quadrant IV
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Each point in the plane can be identified by an
ordered pair
(5, 7)
The ordered pair tells the location of the point
with reference to the origin
Example (5, 7)
The numbers in the ordered pair are called the
coordinates of the point
5 is the x-coordinate or abscissa
7 is the y-coordinate or ordinate
The graph of a point is a dot placed at the
location of the point
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Graph the following ordered pairs
A(-2, -3) B(3, -2) C(0,2) D(-3,0)
y
C
D
x
B
A
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Give the coordinates of A and B
y
Give the abscissa of C
B
Give the ordinate of D
C
A
D
x
Answers
Coordinates of A are (-4,2)
Coordinates of B are (4, 4)
Abscissa of C is -1
Ordinate of D is 1
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Objective To check solutions of an equation in
two variables.
Question Is (-3, 7) a solution of y -2x
1 ?
y -2x 1
Replace x with -3
7 -2(-3) 1
Replace y with 7
7 6 1
7 7
Both sides of the equation simplify to the same
thing.
Yes, (-3, 7) is a solution
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Objective To check solutions of an equation in
two variables.
Question Is (3, -2) a solution of 3x 4y
15 ?
3x 4y 15
Replace x with 3
3(3) 4(-2) 15
Replace y with -2
9 8 15
17 15
Both sides of the equation are not the same.
No, (3, -2) is not a solution
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Problem Graph the ordered-pair solutions of
2x 3y 6 when x -3, 0, 3 and 6
Solve 2x 3y 6 for y
- 3y -2x 6
y 2x 2 3
x
y
5
-3
2(-3) 2
-4
3
0
2(0) 2
-2
3
-5
3
2(3) 2
0
3
6
2(6) 2
2
3
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Problem Graph the ordered-pair solutions of y
2x 1 when x -2, 0, 1 and 3
y 2x - 1
y
x
3
-2
-5
0
-1
3
1
1
3
5
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Objective To determine if a set of ordered
pairs is a function
Definition of a relation
A relation is any set of ordered pairs.
Example (2, 3), (5, -4), (-7, 8), (-12, 8)
Definition of a function
A function is a relation in which no two ordered
pairs have the same first coordinate.
Example (2, 3), (5, -4), (-7, 8), (-12, 8)
is a function
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Objective To determine if a set of ordered
pairs is a function
State whether each of the following relations is
a function
No
(5, 3), (5, -4), (-7, 12), (-5, 12)
Yes
(3, 3), (6, -5), (-7, 12), (-5, 12), (8, 3)
(3, 3), (6, 3), (-7, 3), (-5, 3), (8, 3)
Yes
(2, 3), (2, -5), (2, 12), (2, 15), (2, 9)
No
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Does the equation express y as a function of x ?
y 0.5x 1 where x Î -4, 0, 2
x y
The relation for this domain is
-4 0 2
-1
(-4,-1), (0, 1), (2, 2)
1
2
Yes, the equation is a function
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Does the equation express y as a function of x ?
y x 2 where x Î -2, 0, 2
x y
When x 2, y 0 so y 0
0
-2
When x 0, y 2
so y 2 or y -2
2
0
When x 2, y 4
so y 4 or y - 4
-2
0
4
2
The relation for this domain is
-4
2
(-2, 0), (0, 2), (0, -2), (2, 4), (2, -4)
No, the equation is not a function
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Objective To evaluate a function that is
written in function notation.
When an equation such as y x2 3 defines y
as a function of x, the following function
notation is often used to emphasize that the
relation is a function
f(x) x2 3
f(x) is read the value of the function f at x
or f of x
The expression f(4) means the value of the
function when x 4
so f(4) (4)2 3
16 3
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This process is called evaluating the function
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Objective To evaluate a function that is
written in function notation.
Problem Given f(x) 5x 1 find
f(2)
Solution
f(x) 5x 1
f(2) 5(2) 1
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Problem Given Q(r) 4r2 r 3
find Q(3)
Solution
Q(r) 4r2 r 3
Q(3) 4(3)2 (3) 3
36 3 3
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