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1. What is a relation?
2. What is the domain?
3. What is the range?
4. What is a function?
5. What are the different ways you can determine if a
relation is a function?
6. What is functional notation?
7. How do you determine if a function is linear?
8. What is slope and how do you find it?
9. What are the different classifications of lines by their
slope?
Relation – a set of ordered pairs
Domain – set of first coordinates
(the x’s, input)
Range – set of second coordinates
(the y’s, output)
Function – each member of the domain is paired with
exactly one member of the range
You can determine if something is a function if…
• you are looking at a relation, then no x’s (domain) will repeat
• you are looking at a graph, then it will pass the vertical line test
• you are looking at a equation - you can graph it,
test points or look for things like y2’s
Functional Notatation – renames y using f(x) or other
variables in place of the y. (Usually used when
substituting values.)
* Every function is a relation, but every relation is not a function*
Consider the relation given by the ordered pair
(– 2, –3), (– 1, 1), (1, 3), (2, – 2), and (3, 1).
a.
Identify the domain and range.
D = –2, –1, 1, 2, and 3.
Is the relation a function?
R = –3, –2, 1, and 3.
Yes, x’s don’t repeat
b. Represent the relation using a graph and a
mapping diagram.
Mapping Diagram
Graph
Is the relation a function?
Yes, passes the
vertical line test
Is the relation a function?
Yes, each input has one output
Consider
the relation given by the ordered pairs
GUIDED
PRACTICE
(–4, 3), (–2, 1), (0, 3), (1, –2), and (–2, –4)
a.
Identify the domain and range.
D = –4, –2, 0, 1,
R = 3, 1, –2, –4
b.
Is the relation a function?
No, x’s repeat (-2)
Represent the relation using a table and a
mapping diagram.
Is the relation a function?
No, -2 x value has two
different y values
Is the relation a function?
No, -2 input has two outputs
Tell whether the relation is a function. Explain.
EXAMPLE
2
a.
The relation is a function because each input is mapped
onto exactly one output.
b.
The relation is not a function “1” has 2 outputs: – 1 and 2.
c. Tell whether the relation is a function. Explain.
Yes, x’s don’t repeat
How to graph an equation in two variables
EXAMPLE 4
Graph the equation y = – 2x – 1.
Make a t- chart or use the slope and intercept
xy
-2 3
-1 1
0 -1
1 -3
2 -5
y = – 2x – 1
1
Graph the equation y = 3x – 2.
Tell whether the functions are linear.
Then evaluate the functions when x = – 4.
a.
f (x) = – x2 – 2x + 7
The function f is not linear because it has an x2-term.
f (x) = – x2 – 2x + 7
f (– 4) = –(– 4)2 – 2(– 4) + 7
= –1
b.
g(x) = 5x + 8
The function g is linear because it has the form g(x) = mx + b.
g(x) = 5x + 8
g(–4) = 5(–4) + 8
= – 12
Tell whether the functions are linear.
Then evaluate the functions when x = –2.
c. f (x) = x – 1 – x3
The function f is not linear because it has an x3-term.
f (x) = x – 1 – x3
f (– 2) = – 2 – 1 – (– 2)3
=5
d. g (x) = –4 – 2x
The function f is linear because it has the form g(x) = mx + b.
g (x) = – 4 – 2x
g (– 2) = – 4 – 2(– 2)
=0
Slope (m) - rise = y2 – y1
x2 – x1
run
Positive Slope – rises,
(Right - hand line)
Negative Slope - falls
(Left – hand line)
Undefined Slope – vertical line,
5
0
Zero Slope – horizontal line,
0
5
What is the slope of the line passing through the following points?
a. (– 4, 9) and (– 8, 3)
3 – (9)
=
– 8 – (–4)
2
b. (0, 3), (4, 8)
8–3
4–0
3
5
=
4
What is the slope of the line passing through the following points?
c.
(– 5, 1), (5, – 4)
–4–1
5 – (–5)
d.
1
= – 2
(– 3/4, –2), (5/4, -3)
1
3 2
 3  ( 2 )


8
5 3
5
3
 ( )

4
4
4
4 4
1

2
Classify lines using slope
Without graphing, tell whether the line rises, falls, is horizontal, or is vertical.
a.
(– 5, 1), (3, 1)
0
1–1
m = 3– (–5) = 8 = 0
Because m = 0, the line is horizontal.
b.
(– 6, 0), (2, –4)
–4–0
–4
m = 2– (–6) = 8
1
=–
2
Because m is negative, the line falls.
Classify lines using slope
Without graphing, tell whether the line rises, falls, is horizontal, or is vertical.
c.
(–1, 3), (5, 8)
8–3
m = 5– (–1) =
5
6
Because m is positive, the line rises.
d.
(4, 6), (4, –1)
–1–6
m=
4–4 =
–7
0
Because m is undefined, the line is vertical.
HOMEWORK 2.1
p. 76 #3-12(EOP), 16, 17, 21-23,
25-38 (EOP)
HOMEWORK 2.2a
p. 86 #3-14(all)