Find the distance between the following two points. Then find the coordinates of the midpoint between them.

Find the distance between the following two points. Then find the coordinates of the midpoint between them.

)2,5( and )4,3(

Find the distance between the following two points. Then find the coordinates of the midpoint between them.

)2,5( and )4,3( d = 10

Midpoint is (1, -1)

A Review of Linear Functions

• Linear Function (aka Linear Equation): Establishes a consistent relation between 2 parameters (x and y). When these (x, y) pairs are plotted on a coordinate plane, they line up in a straight line.

• Example: means that for any value of x, the corresponding value of y can be found by multiplying x by , then adding 2 to that product.

232 xy

32

Let’s find some ordered pairs that follow this relation.

First, pick some values of x. How about the following:

34

3

0

3

6

7

9

x

x

x

x

x

x

x

232 xy

Now plug each value of x into the function to find

it's corresponding value of y

23

3

( 3) 2

2 2

4

( 3,4)

x

y

y

y

23

0

(0) 2

0 2

2

(0,2)

x

y

y

y

23

3

(3) 2

2 2

0

(3,0)

x

y

y

y

23

6

(6) 2

4 2

2

(6, 2)

x

y

y

y

23

143

83

83

7

(7) 2

2

(7, )

x

y

y

y

23

9

(9) 2

6 2

4

(9, 4)

x

y

y

y

34

323 4

12

52

3 54 2

( ) 2

2

( , )

x

y

y

y

Plot these points on a graph.What is special about this group of points?

83

3 54 2

( 3,4)

(0,2)

3,0

(6, 2)

(7, )

(9, 4)

( , )

Some Important Details

intercept (0,2)

intercept (3,0)

y

x

Slope – The rate of change in y for every unit change in x.

What does this mean?

An Explanation of Slope

83

23

23

In our example, when 6, 2 but when 7,

So when x is increased by 1, y decreases by

Therefore the slope is

x y x y

An Explanation of Slope

There are two other ways to find the slope

1. Take any pair of points from your equation.

Change in y Calculate the ratio

Change in x

Look at the points (6, 2) & (9, 4)

( 2 4) 2

(6 9) 3

Therefore the sl

23ope is

An Explanation of Slope

3

23

2

2. Remember that if an equation is written in the form.

The value of will ALWAYS correspond to the slope

2

Therefore the slope is

y mx b

m

y x

Independent and Dependent Variables

• We frequently refer to x as the independent variable

• We frequently refer to y as the dependent variable

• This is because YOU choose the value for x and use it to calculate the value of y

Find 3 ordered pairs that are solutions to each linear equation

3 4y x 43 1y x 4 5y x

Find the slope2 6y x

35 4y x

8 1y x

2y x

Find the slope2 6y x

35 4y x

8 1y x

2y x

2 8

3

51

Find the slope

(5, 2) & (4,6)P Q ( 2, 3) & ( 1,6)P Q

(2,0) & (3,4)P Q ( 9, 4) & (0, 5)P Q

Find the slope

(5, 2) & (4,6)P Q ( 2, 3) & ( 1,6)P Q

(2,0) & (3,4)P Q ( 9, 4) & (0, 5)P Q

4 9

4 1

9

Homework

Pg 409-410

#1-8 even,

#14 – 20 even, (do not graph)

#37, #38,

#42-46 even

Quiz

Wednesday

Warm Up Tues. Sep 14

Find 3 ordered pairs that are solutions to the following equation.

Find the slope. Find the x and y intercepts

32

5 xy

Warm Up Tues. Sep 14

32

5 xy

65

5slope (m)

2 intercept (0, 3)

intercept ( ,0)

y

x

Tue Sep 14 HW Solutions32

12 125 5

125

43

34

2. (0, 6);(8,0);(6, ); (4, 3)

4. (0, 4);(12,0);(3,3);( 6,6)

6. (0, ); (6,0);(1, 2);( 2, )

8. (0,12);( ,0);(3, 3);(2,2)

14. (5,0);(0,2)

16. (6,0);(0,2)

18. ( ,0);(0, 2)

20. ( ,0);(0,3)

Tue Sep 14 HW Solutions

98

6 337.

20 1032 8

38. 108 27

42. 1

44.

46. 0

m

m

m

m

m

Day2 Linear Function Applications

Ax By C Linear functions are frequently presented in Standard Form:

Therefore, we must know how to rewrite the equation in slope intercept form:

y mx b

632 yx

Convert from standard form to slope intercept form.

23

2 3 6

2 2

3 2 6

3 2 6

3 32

x y

x x

y x

y x

y x

Linear Function Applications

Sometimes it is useful to write out and solve a linear function that models a real world situation

Car Rental

A car rental company charges a flat fee of $35 plus $25 for each day that the car is rented. Write out a linear function to represent the total cost, y, to rent a car for x days

Car Rental

A car rental company charges a flat fee of $35 plus $25 for each day that the car is rented. Write out a linear function to represent the total cost, y, to rent a car for x days

25 35y x

Telephone Company

The telephone company charges a monthly service charge of $12.95. They also charge a usage charge of $0.07 per minute of calling. Write out a linear function to represent the total cost each month, f(x), to talk on the phone for x minutes a month.

( ) 0.07 12.95f x x

The telephone company charges a monthly service charge of $12.95. They also charge a usage charge of $0.07 per minute of calling. Write out a linear function to represent the total cost each month, f(x), to talk on the phone for x minutes a month.

Telephone Company

Pg 433-434 #55-6455. Suppose that a taxicab driver charges $1.50 per mile. Let x

represent the number of miles driven and f(x) represent the total charge.

f(x)=

f(0) =

f(1 )=

f(2) =

f(3) =

Pg 433-434 #55-6455. Suppose that a taxicab driver charges $1.50 per mile. Let x

represent the number of miles driven and f(x) represent the total charge.

f(x)=

f(0) = $0.00

f(1 )= $1.50

f(2) = $3.00

f(3) = $4.50

1.50x

56. Cost to Mail a Package Suppose that a package weighing x pounds costs f(x) dollars to mail to a given location, where

f(x) = 2.75x.(a) What is the value of f(3)?(b) Describe what 3 and the value f(3) mean in part (a), using the terminology independent variable and dependent variable.(c) How much would it cost to mail a 5-lb package? Write the answer using function notation.

56. Cost to Mail a Package Suppose that a package weighing x pounds costs f(x) dollars to mail to a given location, where

f(x) = 2.75x.(a)What is the value of f(3)? $8.25

(b) Describe what 3 and the value f(3) mean in part (a), using the terminology independent variable and dependent variable. 3 represents 3 lbs. f(3) represents the cost

(c) How much would it cost to mail a 5-lb package? Write the answer using function notation. f(5) = $13.75

57. Forensic Studies - Forensic scientists use the lengths of the tibia (t), the bone from the ankle to the knee, and the femur (r), the bone from the knee to the hip socket, to calculate the height of a person. A person’s height (h) is determined from the lengths of these bones using functions defined by the following formulas. All measurements are in centimeters.

tth

rrh

39.269.81)(

or 24.209.69)(

men,For

tth

rrh

53.257.72)(

or 32.241.61)(

For women,

(a) Find the height of a man with a femur measurement of 56 cm

(b) Find the height of a man with a tibia measurement of 40 cm

(c) Find the height of a woman with a femur measurement of 50 cm

(d) Find the height of a woman with a tibia measurement of 36 cm

57. Forensic Studies - Forensic scientists use the lengths of the tibia (t), the bone from the ankle to the knee, and the femur (r), the bone from the knee to the hip socket, to calculate the height of a person. A person’s height (h) is determined from the lengths of these bones using functions defined by the following formulas. All measurements are in centimeters.

tth

rrh

39.269.81)(

or 24.209.69)(

men,For

tth

rrh

53.257.72)(

or 32.241.61)(

For women,

(a) Find the height of a man with a femur measurement of 56 cm 194.53 cm

(b) Find the height of a man with a tibia measurement of 40 cm177.29 cm

(c) Find the height of a woman with a femur measurement of 50 cm194.53 cm

(d) Find the height of a woman with a tibia measurement of 36 cm163.65 cm

58. Pool Size for Sea Otters - Federal regulations set standards for the size of the quarters of marine mammals. A pool to house sea otters must have a volume of “the square of the sea otter’s average adult length (in meters) multiplied by 3.14 and by .91 meter” If x represents the sea otter’s average adult length and f(x) represents the volume of the corresponding pool size, this formula can be written as

2)14.3)(91.0()( xxf Find the volume of the pool for each of the following adult lengths (in meters). Round answers to the nearest hundredth.

(a) .8 (b) 1.0 (c) 1.2 (d) 1.5

58. Pool Size for Sea Otters - Federal regulations set standards for the size of the quarters of marine mammals. A pool to house sea otters must have a volume of “the square of the sea otter’s average adult length (in meters) multiplied by 3.14 and by .91 meter” If x represents the sea otter’s average adult length and f(x) represents the volume of the corresponding pool size, this formula can be written as

2)14.3)(91.0()( xxf Find the volume of the pool for each of the following adult

lengths (in meters). Round answers to the nearest hundredth.

(a) .8 (b) 1.0 (c) 1.2 (d) 1.51.83m3 2.86m3 4.11m3 6.43m3

59. Number of Post Offices The linear function

f(x) = —183x + 40,034

is a model for the number of U.S. post offices for the period 1990—1995, where x = 0 corresponds to 1990, x = 1 corresponds to 1991, and so on. Use this model to give the approximate number of post offices during the following years. (Source: U.S. Postal Service, Annual Report of the Postmaster General and Comprehensive Statement on Postal Operations.)

(a) 1991 (b) 1993 (c) 1995

59. Number of Post Offices The linear function

f(x) = —183x + 40,034

is a model for the number of U.S. post offices for the period 1990—1995, where x = 0 corresponds to 1990, x = 1 corresponds to 1991, and so on. Use this model to give the approximate number of post offices during the following years. (Source: U.S. Postal Service, Annual Report of the Postmaster General and Comprehensive Statement on Postal Operations.)

(a) 1991 (b) 1993 (c) 1995

39,851 39,485 39,119

is a model for U.S. defense budgets in millions of dollars from 1992 to 1996, where x = 0 corresponds to 1990, x = 2 corresponds to 1992, and so on. Use this model to approximate the defense budget for the following years:

(a) 1993 (b) 1995 (c) 1996

60. The linear function

U.S. Defense

( ) 6324 305,29

g t

4

Bud e

f x x

is a model for U.S. defense budgets in millions of dollars from 1992 to 1996, where x = 0 corresponds to 1990, x = 2 corresponds to 1992, and so on. Use this model to approximate the defense budget for the following years:

(a)1993 (b) 1995 (c) 1996

$286,322 million $286,322 million $286,322 million

60. The linear function

U.S. Defense

( ) 6324 305,29

g t

4

Bud e

f x x

61. Perian Herring stuffs envelopes for extra income during her spare time. Her initial cost to obtain the necessary information for the job was $200. Each envelope costs $0.02 and she gets paid $0.04 per envelope stuffed. Let x represent the number of envelopes stuffed? Write an equation to represent this scenario.

61. Perian Herring stuffs envelopes for extra income during her spare time. Her initial cost to obtain the necessary information for the job was $200. Each envelope costs $0.02 and she gets paid $0.04 per envelope stuffed. Let x represent the number of envelopes stuffed? Write an equation to represent this scenario.

( ) represents net income

( ) 0.04 (0.02 200)

I x

I x x x

62. Tony Motton runs a copying service from his home. He paid $3500 for the copier and a lifetime service contract. Each sheet of paper he uses costs $0.01 and he charges $0.05 per copy he makes. Write an equation to represent this scenario. Remember to define your variables.

62. Tony Motton runs a copying service from his home. He paid $3500 for the copier and a lifetime service contract. Each sheet of paper he uses costs $0.01 and he charges $0.05 per copy he makes. Write an equation to represent this scenario. Remember to define your variables.

represents the number of copies made

( ) represents net income

( ) 0.05 (0.01 3500)

x

I x

I x x x

63. Eugene Smith operates a delivery service in a southern city. His start-up costs amounted to $2300. He estimates that is costs him $3.00 per delivery and he charges $5.50 per delivery. Write an equation to represent this scenario. Remember to define your variables.

63. Eugene Smith operates a delivery service in a southern city. His start-up costs amounted to $2300. He estimates that is costs him $3.00 per delivery and he charges $5.50 per delivery. Write an equation to represent this scenario. Remember to define your variables.

represents the number of deliveries made

( ) represents net income

( ) 5.50 (3.00 2300)

x

I x

I x x x

64. Lisa Ventura bakes cakes and sells them at county fairs. Her initial cost for the Washington County Fair in 1996 was $40.00. She figures that each cake costs $2.50 to make, and she charges $6.50 per cake. Let x represent the number of cakes sold. (Assume that there were no cakes left over). Write an equation that represents this scenario.

represents the number of cakes sold

( ) represents net income

( ) 6.50 (2.50 40)

x

I x

I x x x

Pg 410 #47

a) +232 students/year

a) +232 students/year

b) Positive; Increase

a) +232 students/year

b) Positive; Increase

c) +232 students/year

d) -1.66 students/year

d) -1.66 students/year

e) Negative; Decreased

d) -1.66 students/year

e) Negative; Decreased

f) 1.66 students per year

Homework

• Page 411 #63 – 65. Hand In