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