**5.1**

pg. 296

BELL RINGER

1.) What is the range of the function shown in the table?

A.) {1, 3, 5, 8}

B.) {4}

C.) {-2, 0, 2}

D.) {7}

2.) Which expression is equivalent to 2(3x - 4) - 8x + 3?

A.) 2x + 11

B.) -2x - 5

C.) 2x - 5

D.) -2x -11

x 1 3 5 8

f(x) 2 -2 0 2

VOCABULARY

A function whose graph forms a straight line

is called a

LINEAR FUNCTION

.

The equation for a linear function

is called a

LINEAR EQUATION

.

In a linear equation, the variables are never raised to any power other than the first, never multiplied by another variable, never in a denominator, and never inside another function (square root, absolute value, etc.)

Examples:

y = 2x -5

5x + 1 = y

2

20x + 25y = 15

y - 8 = 2(x -5)

IDENTIFYING BY GRAPH

Which graph(s), if any, represent linear functions:

*First, which graphs are functions? Then, which graphs are linear?

If a graph is both linear and a function, then it's a linear function.

Since these two are not functions, (they don't pass the vertical line test) they cannot be linear functions.

This graph is a function, but since it is not linear, (it's a curve not a line)

it cannot be a

linear function.

These three graphs are of linear functions. They pass the vertical line test and are functions AND they form a straight line.

IDENTIFYING BY EQUATION

Which equations, if any, represent linear functions:

3x + 2y = 10 3xy + x = 1

y = x + 3 x + y = -1

y = |x + 17| - 8 x + 6 = 12

y

3

3x + 2y = 10

y = x + 3

3xy + x = 1 x + y = -1

x + 6 = 12

y

y = |x + 17| - 8

These two are linear functions.

The variables are to the first power, they are not multiplied together, none are in a denominator, and there are no "other" functions that have the variable inside.

These NOT a linear function because variables are multiplied together.

This is NOT a linear function

because one variable is in absolute value.

This is NOT a linear function because one of the variables is in a denominator.

IDENTIFYING BY ORDERED PAIRS

Which of these is a linear function?

x y

2 4

5 3

8 2

11 1

x y

-10 10

-5 4

0 2

5 0

IDENTIFYING BY ORDERED PAIRS

First we need to identify any patterns.

x y

2 4

5 3

8 2

11 1

x y

-10 10

-5 4

0 2

5 0

+3

+3

+3

-1

-1

-1

+5

+5

+5

-6

-2

-2

IDENTIFYING BY ORDERED PAIRS

x y

2 4

5 3

8 2

11 1

+3

+3

+3

-1

-1

-1

Since a constant change of +3 in x corresponds to a constant change

of -1 in y, these ordered pairs satisfy a linear function.

IDENTIFYING BY ORDERED PAIRS

Since a constant change of +5 in x corresponds to different changes in y (-6 and -2), these points do NOT satisfy a

linear function.

x y

-10 10

-5 4

0 2

5 0

+5

+5

+5

-6

-2

-2

IDENTIFYING BY ORDERED PAIRS

One other way to tell whether or not these are functions is to graph each of them.

If the resulting graph is a line, then it is a linear function.

x y

2 4

5 3

8 2

11 1

x y

-10 10

-5 4

0 2

5 0

APPLICATION

Sue rents a manicure station in a salon and pays the salon owner $5.50 for each manicure she gives. The amount Sue pays each day is given by f(x) = 5.50x, where x is the number of manicures. Graph this function and give its domain and range.

This problem has 3 steps:

1.) Find a reasonable domain to use.

2.) Use the chosen domain to create a table of ordered pairs.

3.) Use the ordered pairs to graph the function.

APPLICATION

Sue rents a manicure station in a salon and pays the salon owner $5.50 for each manicure she gives. The amount Sue pays each day is given by f(x) = 5.50x, where x is the number of manicures. Graph this function and give its domain and range.

Step 1: Find a reasonable domain.

Since niether the amount she pays nor the number of manicures she gives can be negative, we should use all non-negative numbers.

Let's start with 0, since it is possible for her to give 0 manicures, and from there we can count up by one. Counting from 0 to 5 should give us enough points to recognize a pattern.

Let's also remember that since no negatives are involved, when we graph we will only need to graph Quadrant I of the coordinate plane.

APPLICATION

Sue rents a manicure station in a salon and pays the salon owner $5.50 for each manicure she gives. The amount Sue pays each day is given by f(x) = 5.50x, where x is the number of manicures. Graph this function and give its domain and range.

Step 2: Create a table

0 f(0) = 5.50(0) 0

1 f(1) = 5.50(1) 5.50

2 f(2) = 5.50(2) 11.00

3 f(3) = 5.50(3) 16.50

4 f(4) = 5.50(4) 22.00

5 f(5) = 5.50(5) 27.50

x f(x) = 5.50x f(x)

APPLICATION

Sue rents a manicure station in a salon and pays the salon owner $5.50 for each manicure she gives. The amount Sue pays each day is given by f(x) = 5.50x, where x is the number of manicures. Graph this function and give its domain and range.

Step 3: Graph

*Now we have a set of

ordered pairs for step 3:

(0, 0)

(1, 5.5)

(2, 11)

(3, 16.5)

(4, 22)

(5, 27.5)

(0, 0)

(1, 5.5)

(2, 11)

(3, 16.5)

(4, 22)

(5, 27.5)

Each point in this line is a solution in the equation.

The line shows that the trend continues.

APPLICATION

Sue rents a manicure station in a salon and pays the salon owner $5.50 for each manicure she gives. The amount Sue pays each day is given by f(x) = 5.50x, where x is the number of manicures. Graph this function and give its domain and range.

We can get the domain and range

from the table or the graph, but one thing to

remember is that they both continue on...

**16-20, 25-29**

ASSIGNMENT

Lesson 5.1

pg. 300-301

**x**

**y**

3

D: {0, 1, 2, 3, 4, 5, ...}

R: {0, 5.5, 11, 16.5, 22, 27.5, ...}

Think about what the domain

and range represent.

What does their

continuation mean?

Why is this important?