Algebra 1Ch 4.8 – Functions and Relations

Do Now1. Find the slope between these two points:

(0,-4) and (-3,2)

2. Rewrite the equation in slope-intercept form, and then tell me the slope and y-intercept.

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6x − 3y = 30

3. Rewrite the equation in slope-intercept form, and then tell me the slope and y-intercept.

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−5x + y = −3

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m =y2 − y1

x2 − x1

=2 − (−4)

−3 − 0=

6

−3= −2

-6x -6x-3y = -6x + 30

y = 2x - 10

m = 2/1 b = (0,-10)

y = mx + b

+5x +5xy = 5x - 3

m = 5/1 b = (0,-3)

Do Now Cont’d

4. Rewrite the equation in slope-intercept form, and then graph it.

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3x − 2y +12 = 0

5. Decide whether the graphs of the two equations are parallel lines.

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y + 3x − 4 = 0

−2y + 6x = 5

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-3x -3x

-2y + 12 = -3x

-2y = -3x - 12

m = b = (0,6)-12 -12

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y =3

2x + 6

-2 -2 -2

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3

2

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y = −3x + 4

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y = 3x −5

2

NO THEY ARE NOT PARALLEL

DIFFERENT SLOPES!!

Functions A relationship where one thing depends upon

another is called a function. A function is a rule that establishes a

relationship between two quantities called the input and output.

In a function each input has exactly one output. More than one input can have the same output

You can’t have one input go to TWO different outputs!!!

Domain and Range

A relation is any set of ordered pairs.The set of all first components of the order pairs is called the domain of the function or relation, and the set of all second components is called the range of the function or relation.

xy

InputOutput

DomainRange

Ex 1: Determine whether each relation is a function.

a. {(4,5), (6,7), (8,8)}

b. {(5,6), (4,7), (6,6), (6,7)}

Yes!!!!

NO!!!!

Because the 6 outputs a 6 anda 7

Practice Exercises

Determine whether each relation is a

function. Give the domain and range

for each relation.

1. {( 7, 7), ( 5, 5), ( 3, 3), (0,0)}

2. {(4,1), (5,1), (4,2)}

Answers

1. Domain { 7, 5, 3,0}

Range { 7, 5, 3,0}

Given relation is a function.

2. Domain {4,5}

Range {1,2}

Given relation is not a function.

Function NotationThe special notation ( ), read " of "

or " at ," represents the value of the

function at the number .

The notation ( ) does not mean

" times ."

f x f x

f x

x

f x

f x

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f (x) = 2x − 3

y = 2x − 3

They mean the same thing

xy

InputOutput

DomainRange

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f (x)

Ex 3: Evaluating a Function

If f(x) = x2 – 10x – 3, evaluate each:

a. f(-1) b. f(2) c. f(-4)

(-1)2 – 10(-1) - 3 (2)2 – 10(2) - 3 (-4)2 – 10(-4) - 3

1 + 10 – 3

8

4 – 20 – 3

-19

16 + 40 – 3

53

Is this a function??? Look at the table to the

right…notice that each input has exactly one output…

Yes it is!!!

Input Output

5 3

6 4

7 5

8 6

Is this a function?? Look at the table to the

right…notice that the input of 9 has two different outputs (5 and 4 respectively)

Therefore, this set of data is not considered to be a function

Input Output

9 5

9 4

8 3

7 2

Is this a function??? Look at the table to the

right…notice that the input of 1 and 2 have the same output of 3

In this instance this is considered a function because each input has exactly one output…it’s ok to have different inputs with the same output

Input Output

1 3

2 3

3 4

4 4

Make a input-output table of the function y = 3x + 2, with the values 0, 1, 2, and 3

1. y = 3x + 2 Input Output

y = 3(0) + 2

y = 0 + 2

y = 2

2. y = 3x + 2

y = 3(1) + 2

y = 3 + 2

y = 5

3. y = 3x + 2

y = 3(2) + 2

y = 6 + 2

y = 8

4. y = 3x + 2

y = 3(3) + 2

y = 9 + 2

y = 11

0

1

2

3

2

5

8

11

Your Turn – Identifying a Function Does the table represent a function? Explain

Input Output

1 3

1 4

2 5

3 6

Input Output

1 1

2 3

3 6

4 10

Input Output

1 3

2 6

3 11

4 18

Input Output

5 9

4 8

3 9

2 7

1.

2.

3.

4.

YES

No

YES

YES

The Vertical Line TestThe vertical line test is used to determine if a graph is a function.

Created by:David W. Cummins

If a vertical line passes through a graph more than once, the

graph is not the graph of a

function.

Pass a pencil across the graph held

vertically to represent a vertical line.

The pencil crosses the graph more than once. This is not a

function because there are two y-values for the same x-value.So that means if we input the x-value of 2, it has two different outputs.

So this is a function!!!

So this is NOT a function!!!

So this is a function!!!

So this is NOT a function!!!

So this is a function!!!

So this is NOT a function!!!