1. POD
1. What is the longest distance between any
two points in a crate with the following
dimension:
Length 10 feet
Width 9 feet
Height 4 feet
2. Functions Unit 4
Part 1
CC8.F.1 Understand that a function is a rule that assigns
to each input exactly one output. The graph of a function
is the set of ordered pairs consisting of an input and the
corresponding output.
3. What is a Relation?
A rule that gives an output number for every
valid input number
A set of ordered pairs for which all x and y
values are related in the same way.
No special rules need apply.
The following are examples of relations:
{(1,2), (1, 4), (1, 5), (1, 6), (1, -3)}
{(1,2), (2, 4), (3, 5), (2, 6), (1, -3)}
4. What is a Function?
A rule of matching elements of two sets of
numbers in which an input value from the
first set has only one output value in the
second set.
Every value of x has a unique value of y.
function: {(1,2), (2, 4), (3, 5), (4, 6), (5,-3)}
5. What Will You Get?
If you combine cake mix, eggs and milk and put it in the oven, what will come out?
Cake
mix
6. What Will You Get
If you combine the ingredients again and put it in the oven, what will come out?
Cake
mix
7. Domain
• In a function, the possible values
for x in the given situation.
• It is the set of values of the
independent variable of a given
function.
function: {(1,2), (2, 4), (3, 5), (4, 6), (5,-3)}
Domain: {1, 2, 3, 4, 5}
8. Range
• In a function, the possible values
for y in the given situation.
• It is the set of values of the
dependent variable of a given
function.
function: {(1,2), (2, 4), (3, 5), (4, 6), (5,-3)}
Range: {2, 4, 5, 6, -3}
9. Relations and Functions
• Relations and functions can also be
represented as relationships between two
sets of elements
Input Output Input Output
x-values y-values x-values y-values
Domain Range Domain Range
1 1 2
2
3 3 4
4
5 5 6
6
7 7 8
Relation/Function Relation/Not a Function
10. Relations and Functions
• Now you try. Determine whether each set
is a relation, a function, or both.
3
5 Amy Bob
6
10 Liz Joe
9
15 Sara Dan
21
a b
2 2
e c
4 4
i d
6 6
o f
8 8
11. Relations and Functions
• We will look at functions in four different
ways
1. Numerically; tables and ordered pairs
2. Graphically
3. Verbally
4. Algebraically
12. Functions-- Numerically
• For each x value, you can have one, and
only one, y value
• Check each table for repeating x’s
x y x y
4 2 4 4
2 2 2 2
0 0 0 0
-2 -2 2 -2
-4 -4 4 -2
13. Functions-- Numerically
• For each x value, you can have one, and
only one, y value
• Check each set of ordered pairs for
repeating x’s
• {(4,4), (2,2), (0,0), (-2,-2), (-4,-4)}
• {(4,4), (2,2), (0,0), (2,-2), (4,-4)}
14. Functions-- Graphically
• For each x value, you can have one, and
only one, y value
• Check that each x-coordinate is related to
only one y-coordinate
15. Functions--Graphically
• For each x value, you can have one, and
only one, y value
• Check that each x-coordinate is related to
only one y-coordinate
16. Functions--Verbally
• It is a surprising biological fact that most crickets chirp at
a rate that increases as the temperature increases. For the
snowy tree cricket (Oecanthus fultoni), the relationship
between temperature and chirp rate is so reliable that this
type of cricket is called the thermometer cricket. We can
estimate the temperature (in degrees Fahrenheit) by
counting the number of times a snowy tree cricket chirps
in 15 seconds and adding 40. For instance, if we count 20
chirps in 15 seconds, then a good estimate of the
temperature is
20 + 40 = 60 F◦
18. Functions--Algebraically
• For each x value, you can have one, and only
one, y value
• Check that each x-value in your domain relates
to only one y-value in your range.
x y=x+1 y
-2 -2 + 1 -1
y=x+1 -1 -1 + 1 0
0 0+1 1
1 1+1 2
2 2+1 3
3 3+1 4
Discuss ordered pairs, x values and y values.x is the input or independent variabley is the output or dependent variablePoint out the relationship between the x and y values in each ordered pair.
Again point out the relationship in each ordered pair, but this time show that each x value has a unique y value.
If you mix the ingredients and put it in the oven, only a cake can come out.
If you mix the ingredients again can you get a turkey out? Can you get a basketball out? No, you will always get a cake out.In a function, if you put the same thing in, you will always get the same thing out—for every input (x) value there is only one output (y) value.
Point out the inputs (x values) in the set.
Point out the Outputs (y values) in the set.
Show different representation of a relation/function.
Point out that relations/functions can be sets of any kind of elements , not just numbers
Point out that in the absolute value function when x is 1, y is 1 and nothing else; and when x is -1, y is 1 and nothing else.Emphasize that each x (input) value relates to only one y (output) value.
Point out x coordinates that relate to more than one y coordinate. Emphasize that if any x coordinate has more than one y coordinate the relation is NOT a function.
Point out that you wouldn’t say “I have 5 gallons of paint, how much wall can I paint?” Instead, you would say “the walls in my room have 400 square feet, how much paint will I need?”