Advanced Mathematics Functions and Statistics...Unit 1, Activity 1, Graphically Speaking with...

Unit 1, Activity 1, Graphically Speaking

Blackline Masters, Advanced Math – Functions and Statistics Louisiana Comprehensive Curriculum, Revised 2008

Advanced Mathematics

Functions and Statistics

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Blackline Masters, Advanced Math – Functions and Statistics Page 1 Louisiana Comprehensive Curriculum, Revised 2008

a) Domain: _______________ b) Range: ________________ c) Max: ______________ d) Inc: __________________ e) Dec: __________________ f) Constant: ___________ g) f(-3) = ________________ h) f(x) = 0 ________________ i) f(x) > 0 ____________

a) Domain: _______________ b) Range: ________________ c) Min: ______________ d) Inc: __________________ e) Dec: __________________ f) Constant: ___________ g) f(0) = _________________ h) f(x) = 3 _________________ i) f(x) < 0 ____________

2.

1.



a) Domain: _______________ b) Range: ________________ c) Min: ______________ d) Inc: __________________ e) Dec: __________________ f) Constant: ___________ g) f(1) = ________________ h) f(x) = -1_________ ______ i) f(x) < 0 ____________

a) Domain: _______________ b) Range: ________________ c) Max: ______________ d) Inc: __________________ e) Dec: __________________ f) Constant: ___________ g) f(-1) = ________________ h) f(x) = -2 _______________ i) f(x) > 0 ____________

3.

4.

Unit 1, Activity 1, Graphically Speaking with Answers


a) Domain: __[-4, 5] b) Range: ____[-3, 4] ____ c) Max: _____4 _______ d) Inc: _____(-4, 1)________ e) Dec: ___(1, 2)___________ f) Constant: __(2, 5)_____ g) f(-3) = _____2__________ h) f(x) = 0 _x = -4 ; x = 1.5__ i) f(x) > 0 __(-4, 1.5)____

a) Domain: ____[-3, 6] _____ b) Range: ___[-1, 3] _______ c) Min: _____-1 ______ d) Inc: ______(-3,6) ______ e) Dec: _(-3, -1) ∪ (1, 3)_____ f) Constant: __(-1, 1) ___ g) f(0) = ______1 ________ h) f(x) = 3 __x = -3 ; x = 6___ i) f(x) < 0 __ (2, 5) ____

1.

2.

Unit 1, Activity 1, Graphically Speaking with Answers


3.

a) Domain: ____(-5, ∞) _____ b) Range: _ _[-4, 3)______ c) Min: _____-4 ______ d) Inc: _______(0, 4) ____ e) Dec: _____(-5, 0) ______ f) Constant: ___(4, ∞)___ g) f(1) = ____-3 _______ h) f(x) = -1__x = -2 ; x ≥ 4___ i) f(x) < 0 ___(-4, ∞) ___

a) Domain: ____(-∞, ∞) ____ b) Range: ____(-2, ∞)_______ c) Max: ___none_______ d) Inc: __(-2, -1) ∪ (2, ∞) __ e) Dec: _____(-1, 2)_________ f) Constant: __(-∞, -2)___ g) f(-1) = ___ 2___________ h) f(x) = -2 _____x = 2_______ i) f(x) > 0 (-∞, 0) ∪ (4,∞)

4.

Unit 1, Activity 2, Family of Functions


Function

Unit 1, Activity 2, Family of Functions with Answers


Function Graph Domain Range Extrema Increasing/Dec. Constant f(x) = k

(-∞, ∞)

[k]

None

None

Linear f(x) = x

(-∞, ∞)

(-∞, ∞)

None

Inc. (-∞, ∞)

Quadratic f(x) = x2

(-∞, ∞)

[0, ∞)

(0, 0)

Inc. (0, ∞)

Dec. (-∞, 0)

Cubic f(x) = x3

(-∞, ∞)

(-∞, ∞)

None

(-∞, ∞)

Square Root f(x) = x

[0, ∞)

[0, ∞)

None

Inc.[0, ∞)

Unit 1, Activity 2, Family of Functions with Answers


Function Graph Domain Range Extrema Increasing/Dec. Cube Root f(x) = 3 x

(-∞, ∞)

(-∞, ∞)

None

Inc. (-∞, ∞)

Exponential f(x) = ex

(-∞, ∞)

(0, ∞)

None

Inc. (-∞, ∞)

Logarithmic f(x) = ln x

(0, ∞)

(-∞, ∞)

None

Inc. (0, ∞)

Absolute Value f(x) = ⏐x⏐

(-∞, ∞)

[0, ∞)

Min (0, 0)

Inc. (0, ∞)

Dec. (-∞, 0)

Greatest Integer f(x) = [ x ]

(-∞, ∞)

Integers

None

None

Unit 1, Activity 3, Translations, Dilations, and Reflections


Graph Type of

Function Description of

Change Equation

Unit 1, Activity 3, Translations, Dilations, and Reflections


Equation Description of Change Graph – Parent Graph - Final

f(x) = 2

1+x

+ 3

f(x) = x4− - 1

f(x) = ½ (x – 1)2 - 2

f(x) = -3 ln (2x)

Unit 1, Activity 3, Translations, Dilations, and Reflections with Answers


Graph Type of Function

Description of Change

Equation

Linear

Up 2

Vertical stretch of

factor 3

Reflect over x-axis

f(x) = -3x + 2

Cubic

Right 1

Vertical stretch of

factor 2

Reflect over x-axis

f(x) = -2(x - 1)3

Cube root

Vertical

compression of factor 2

Horizontal stretch

of factor 2

f(x) = 21

3

21 x

Exponential

Right 1

Down 2

Reflect over y-axis

f(x) = e1-x - 2

Unit 1, Activity 3, Translations, Dilations, and Reflections with Answers


Equation Descripti

on of Change

Graph – Parent Graph - Final

f(x) = 2

1+x

+ 3

Left 2

Up 3

f(x) = x4− - 1

Reflect over y-

axis

Horizontal compressi

on of factor 4

Down 1

f(x) = ½ (x – 1)2 - 2

Vertical

compression of

factor 2

Right 1

Down 2

f(x) = -3 ln ( ½ x)

Reflect over x-

axis

Vertical stretch of factor 3

Horizontal stretch of factor 2

Unit 1, Activity 4, In Pieces


Tax Model #1 Citizens earning $5000 and up to $80,000 will pay a personal income tax of 10%. Citizens earning $80,000 and up to $200,000 will pay 20%. Citizens making $200,000 and above will pay an income tax of 25%. 1. Write a function to model this tax structure. 2. Draw the graph of the tax model. 3. Is this particular tax structure fair? Why or why not?

Unit 1, Activity 4, In Pieces with Answers


Tax Model #1 Citizens earning $5000 and up to 80,000 will pay a personal income tax of 10%. Citizens earning $80,000 and up to $200,000 will pay 20%. Citizens making $200,000 and above will pay an income tax of 25%. 1. Write a function to model this tax structure.

f(x) = {

2. Draw the graph of the tax model.

3. Is this particular tax structure fair? Why or why not? Answers will vary, but students should identify this tax structure as being progressive since the tax increases as the income increases.

.15x ; 5,000 ≤ x < 80,000

.20x ; 80,000 ≤ x < 200,000

.25x ; 200,000 ≤ x

Unit 1, Activity 7, Inverse Functions


Split-Page Notetaking Topic: Inverse Functions Date: ___________

Verbal Representation Example #1 amount you pay for gas number of gallons purchased Function The total cost of the gas is dependent on the number of gallons purchased. Ordered Pairs (number of gallons, total cost) Inverse Function The number of gallons that can be purchased depends on the amount of money you have. Inverse Ordered Pairs (total cost, number of gallons) Example #2 Number of hours worked amount of paycheck Function The amount of your paycheck is dependent on the number of hours you worked. Ordered Pairs (number of hours, amount of paycheck) Inverse Function The number of hours you need to work depends on the amount of money you need to earn. Inverse Ordered Pairs (amount of paycheck, number of hours)

Unit 2, Activity 1, Solving Right Triangles


Solve each triangle. 1. A A ____________ a ___________ 4 m B ____________ b ___________ C ____________ c ___________ C 10 m B 2. A 9 ft C A ____________ a ___________ B ____________ b ___________ 15 ft C ____________ c ___________ B 3. A A ____________ a ___________ B ____________ b ___________ 14 km C ____________ c ___________ C B

45°

Unit 2, Activity 1, Solving Right Triangles


B 4. A ____________ a ___________ B ____________ b ___________ C ____________ c ___________ A 11 m C 5. C B A ____________ a ___________ B ____________ b ___________ 24 yd C ____________ c ___________ A 6. A A ____________ a ___________ B ____________ b ___________ 35 m C ____________ c ___________ B C

38°

60°

53°

Unit 2, Activity 1, Solving Right Triangles with Answers


Solve each triangle. 1. A A ≈ _68°11′55″__ a = __10 m____ 4 m B ≈ _21°48′5″___ b = ___4 m____

C = ____90°____ c = m292 C 10 m B 102 + 42 =c2 tan A = 10/4 c2 = 116 A = tan -1 (10/4) A ≈ 68.2° 2. A 9 ft C A ≈ _53° 7′48″__ a = _12 ft_____ B ≈ _36°52′12″__ b = __ 9 ft_____ 15 ft C = ___90°______ c = __15 ft____ B a2 = 152 - 92 OR 3 - 4 - 5 cos A = 9/15 a2 = 144 9-12-15 A = cos -1 (9/15) A ≈ 53.1°

3. A A = ____45°_____ a = km27

14 km B = ____45°_____ b = km27 C = ____90°_____ c = __14 km___ C B

a = 2

14

45°

Unit 2, Activity 1, Solving Right Triangles with Answers


B 4. A = ___38°_____ a ≈ __8.6 m____ B = ____52°_____ b = __11 m____ C = ____90°_____ c ≈ __14.0 m___ A 11 m C tan 38° = a/11 cos 38° = 11/c a = 11 tan 38° c = 11/cos 38°

5. C B A = ___60°______ a = yd312 B = ___30°______ b = __12_yd___ 24 yd C = __90°_______ c = __24 yd ___ A short leg • 2 = 24 long leg = 12 • 3 6. A A = ___27°_____ a ≈ __21.1 m___ B = ___53°______ b ≈ __28.0 m___ 35 m C = ___90°______ c = ___35 m___ B C cos 53° = a/35 sin 53° = b/35 a = 35 cos 53° b = 35 sin 53°

38°

60°

53°

Unit 2, Activity 2, Applications of Right Triangles


Problem Solution Check 1. Height of an object

2. Angle of elevation or depression

3. Vector components (hor. & vert. OR dir. & mag.)

Unit 2, Activity 3, Discovering the Law of Sines


ABC is an oblique triangle. C A B

1. Draw an altitude from vertex C. 2. Label the altitude x. 3. Use right triangle trigonometry to complete the ratios below.

sin A = sin B =

4. Solve each of the above equations for x.

5. Set the above equations equal to each other to form a new equation. Why is this possible?

6. Regroup the variables so that each capital letter is on the same side of the equal sign as its lower case counterpart.




7. Draw an altitude from vertex B. 8. Label the altitude x. 9. Use right triangle trigonometry to complete the ratios below.

sin A = sin C =







13. Draw an altitude from vertex A. 14. Label the altitude x. 15. Use right triangle trigonometry to complete the ratios below.

sin B = sin C =




19. Use the results from 1-18 to write the Law of Sines.

Unit 2, Activity 3, Discovering the Law of Sines with Answers



1. Draw an altitude from vertex C. 2. Label the altitude x. 3. Use right triangle trigonometry to complete the ratios below.

sin A = bx sin B =

ax

4. Solve each of the above equations for x. x = b sin A x = a sin B


b sin A = a sin B The transitive property makes this possible.


b

Ba

A sinsin= OR

Bb

Aa

sinsin=

x




7. Draw an altitude from vertex B. 8. Label the altitude x. 9. Use right triangle trigonometry to complete the ratios below.

sin A = cx sin C =

ax

10. Solve each of the above equations for x. x = c sin A x = a sin C


c sin A = a sin C The transitive property makes this possible.


c

Ca

A sinsin= OR

Cc

Aa

sinsin=

x



ABC is an oblique triangle. C A B 13. Draw an altitude from vertex A.

14. Label the altitude x.

15. Use right triangle trigonometry to complete the ratios below.

sin B = cx sin C =

bx

16. Solve each of the above equations for x. x = c sin B x = b sin C 17. Set the above equations equal to each other to form a new equation. Why is this

possible? c sin B = b sin C The transitive property makes this possible.


c

Cb

B sinsin= OR

Cc

Bb

sinsin=

19. Use the results from 1-18 to write the Law of Sines.

c

Cb

Ba

A sinsinsin== OR

Cc

Bb

Aa

sinsinsin==

x

Unit 2, Activity 3, Law of Sines: Split-Page Notetaking


Split-Page Notetaking

Topic: Law of Sines Date: _____________________ AAS Example

Unique triangle B = 180°- 88°- 43° = 49°

°

=° 88sin

1149sinb b ≈ 8.3 m

B C

°

=° 88sin

1143sinc c ≈ 7.5 m

*To make the calculations easier, put the unknown value in the numerator. SSA – Obtuse Angle

Unique triangle 34106sin

17sin °

=A A ≈ 28.7° ≈ 28°43′36″

B C ≈ 180° - 28.7° - 106° ≈ 45.3° ≈ 45°16′24″

A C °

=° 106sin

343.45sin

c c ≈ 25.1 km

*Since a triangle can have only one obtuse angle, a unique triangle exists. SSA – Obtuse Angle No triangle The Law of Sines is not needed; however, it will reveal no triangle. A triangle can have only one A B obtuse angle. In this case, it is C. Thus, c must be the longest side. Since b > c, no triangle with these measurements exists.

C 25117sin

28sin °

=A A ≈ 86.3° ≈ 86°18′37″

A + C > 180°

88° 43° 11 m

A

106° 17 km 34 km

25 m 28 m 117°

Unit 2, Activity 4, Discovering the Law of Cosines


b a A c B 1. Draw an altitude from vertex C and label it h. 2. The altitude divides c into two different pieces. Label one piece x. How can you label the

other piece in terms of x and c? 3. Using the Pythagorean Theorem, write two different equations for each right triangle. 4. Solve each equation for h2. 5. Set the two equations equal to each other to form a new equation. Why can this be done? 6. Which variable in the equation is not a side of ΔABC? 7. Solve the equation for b2 and expand (c – x)2. What happens to the x2? 8. Since x is not a side of ΔABC, it needs to be eliminated. What do you suggest?

C

Unit 2, Activity 4, Discovering the Law of Cosines


9. Write an equation relating x and cos B and then solve for x. 10. Replace x in the equation from #7 with its equivalent expression found above. 11. This part of the Law of Cosines finds the length of side b. Based on the work for #1-10,

write equations to find the lengths of sides a and c. 12. Rewrite each of the three equations to find angles A, B, and C.

Unit 2, Activity 4, Discovering the Law of Cosines with Answers


b h a A c - x x B 1. Draw an altitude from vertex C and label it h. 2. The altitude divides c into two different pieces. Label one piece x. How can you label the

other piece in terms of x and c? 3. Using the Pythagorean Theorem, write two different equations for each right triangle. (c – x)2 + h2 = b2 x2 + h2 = a2 4. Solve each equation for h2. h2 = b2 – (c – x)2 h2 = a2 – x2 5. Set the two equations equal to each other to form a new equation. Why can this be done? b2 – (c – x)2 = a2 – x2 The transitive property makes this possible. 6. Which variable in the equation is not a side of ΔABC? x 7. Solve the equation for b2 and expand (c – x)2. What happens to the x2? b2 = a2 – x2 + (c – x)2 The x2s cancel each other out. b2 = a2 + c2 – 2cx 8. Since x is not a side of ΔABC, it needs to be eliminated. What do you suggest? Answers will vary

C

Unit 2, Activity 4, Discovering the Law of Cosines with Answers


9. Write an equation relating x and cos B and then solve for x.

cos B = ax x = a cos B

10. Replace x in the equation from #7 with its equivalent expression found above. b2 = a2 + c2 – 2c(a cos B) b2 = a2 + c2 – 2ac cos B 11. This part of the Law of Cosines finds the length of side b. Based on the work for #1-10,

write equations to find the lengths of sides a and c. It is not necessary to rework the steps #1-10. Students should be able to use patterns to generate the other two equations. a2 = b2 + c2 – 2bc cos A c2 = a2 + b2 – 2ab cos C 12. Rewrite each of the three equations to find angles A, B, and C. 2bc cos A = b2 + c2 – a2 2ac cos B = a2 + c2 – b2

cos A = bc

acb2

222 −+ cos B = ac

bca2

222 −+

A = cos -1 ⎟⎟⎠

⎞⎜⎜⎝

⎛ −+bc

acb2

222

B = cos -1 ⎟⎟⎠

⎞⎜⎜⎝

⎛ −+ac

bca2

222

2ab cos C = a2 + b2 – c2

cos C = ab

cba2

222 −+

C = cos -1 ⎟⎟⎠

⎞⎜⎜⎝

⎛ −+ab

cba2

222

Unit 2, Activity 5, Applications of Oblique Triangles


1. To find the width of a lake, a surveyor stands 136 m from one end of the lake and 162 m from the other end at an angle of 78°. What is the width of the lake?

2. A surveying crew needs to find the distance between two points, A and B, but a boulder between the two points makes a direct measurement impossible. Thus, the crew moves to a point C that is at an angle of 110° to points A and B. The distance between C and B is 422 ft and the angle from A is 30°. What is the distance between points A and B? 3. Two coast guard stations are 150 miles apart. A ship at sea sends out a distress call that is

received by both stations. The angle from one station to the ship is 55°. The angle from the other station to the ship is 36°. How far is the ship from the closest station?

4. A Major League baseball diamond is a square with sides measuring 90 ft each. The pitching rubber is 60.5 ft from home plate on a line joining home plate and second base. How far is it from the pitching rubber to first base?

Unit 2, Activity 5, Applications of Oblique Triangles


5. The area of an oblique triangle can be found using the formula A = ½ ab sin C. To use this formula, two sides and the included angle must be known. Find the appropriate side in order to determine the area of the triangle below.

17 in. 6. Heron’s formula, A = ( )( )( )csbsass −−− , is used to find the area of an oblique triangle

when all three sides are known. The variable s represents the semi-perimeter (half the perimeter). Find the third side and then use Heron’s formula to find the area of the triangle.

22 m 16 m 7. A boat is traveling 8 knots at a bearing of 100°. After two hours, the boats turns and travels at a bearing of 55° for three hours at 10 knots. Find the magnitude and the direction of the displacement vector. 8. A plane is flying due East at 300 mph. A tailwind is blowing 25° west of North at 15 mph.

What is the actual direction and velocity of the plane?

53° 30°

43°

Unit 2, Activity 5, Applications of Oblique Triangles with Answers


1. To find the width of a lake, a surveyor stands 136 m from one end of the lake and 162 m from the other of the lake at an angle of 78°. What is the width of the lake?

a2 = 1622 + 1362 – 2(162)(236) cos 78° 162 m 136 m a ≈ 188.6 m a

2. A surveying crew needs to find the distance between two points, A and B, but a boulder between the two points makes a direct measurement impossible. Thus, the crew moves to a point C that is at an angle of 110° to points A and B. The distance between C and B is 422 ft and the angle from A is 30°. What is the distance between points A and B?

°

=° 30sin

422110sinc

422 ft c ≈ 793.1 ft A c B 3. Two coast guard stations are 150 miles apart. A ship at sea sends out a distress call that is

received by both stations. The angle from one station to the ship is 55°. The angle from the other station to the ship is 36°. How far is the ship from the closest station?

°

=° 89sin

15036sina

a a ≈ 88.2 mi 150 mi 4. A Major League baseball diamond is a square with sides measuring 90 ft each. The pitching rubber is 60.5 ft from home plate on a line joining home plate and second base. How far is it from the pitching rubber to first base?

a2 = 60.52 + 902 – 2(60.5)(90) cos 45° a a ≈ 63.7 ft 60.5 ft 90 ft

78°

110°

30°

C

55° 36°

ship

45°

Unit 2, Activity 5, Applications of Oblique Triangles with Answers


5. The area of an oblique triangle can be found using the formula A = ½ ab sin C. To use this formula, two sides and the included angle must be known. Find the appropriate side in order to determine the area of the triangle below.

17 in. °

=° 53sin

1797sina A ≈ ½ (17)(21.1)sin 30°

A ≈ 89.7 in2 a ≈ 21.1 in. a

6. Heron’s formula, A = ( )( )( )csbsass −−− , is used to find the area of an oblique triangle when all three sides are known. The variable s represents the semi-perimeter (half the perimeter). Find the third side and then use Heron’s formula to find the area of the triangle.

22 m A 16

43sin22

sin °=

C C ≈ 180° - 69.7° ≈ 110.3°

°

=° 43sin

167.26sin

b b ≈ 10.5 m

C Area ≈ ( )( )( )5.1025.242225.241625.2425.24 −−− Area ≈ 78.7 m2

7. A boat is traveling 8 knots at a bearing of 100°. After two hours, the boats turns and travels at a bearing of 55° for three hours at 10 knots. Find the magnitude and the direction of the displacement vector. Extended angle = 45° N N B = 180°- 45° = 135° b C b2 = 162+302-2(16)(30)cos135° A b ≈ 42.8 n. mi 16 n. mi 55° 30 n. mi

B 8.42

135sin30

sin °=

∠BAC ∠BAC ≈ 29.7°

Direction: bearing of 100°- 29.7° ≈ 70.3° 8. A plane is flying due East at 300 mph. A tailwind is blowing 25° west of North at 15 mph.

What is the actual direction and velocity of the plane? Extended angle = 90° N C N B = 180°- 90°- 25° = 65° b b2 = 152 + 3002 – 2(15)(300)cos 65° b ≈ 294.0 mph

A 300 mph B 294

65sin15

sin °=

∠BAC ∠BAC≈2.7°

Direction: bearing of 90°-2.7°≈ 87.3° (east of North)

53° 30°

43° B 16 m

100°

15mph 25°

Unit 3, Activity 1, Know Thyself


Rate your understanding of each mathematical term with a “+” if you understand the term well, a “√” if you have a limited understanding of the term, or a “-” if you have no understanding of the term at all. You should continually revise your entries as you progress through unit 2. Since this is a self-awareness activity, you will not share your entries with the rest of the class. So, be honest with yourself!

Term(s) + √ - Definition Example Power Function

Polynomial Function

Domain

Range

Zero

Zero Multiplicity

End Behavior

Extrema

Increasing Intervals

Decreasing Intervals

Symmetry

Even Function

Odd Function

Unit 3, Activity 2, Power Functions – Positive Integer Exponents


Fill in the following word grid for pxy = . Start the first row with p = 1. Fill in the first column with important function properties and components.

f(x) = xp

Unit 3, Activity 2, Power Functions – Positive Integer Exponents with Answers


Fill in the following word grid for pxy = . Start with p = 1.

f(x) = xp

f(x) = x f(x) = x2 f(x) = x3 f(x) = x4

Graph

Domain

(-∞, ∞)

(-∞, ∞)

(-∞, ∞)

(-∞, ∞)

Range

(-∞, ∞)

[0, ∞)

(-∞, ∞)

[0, ∞)

Behavior as ∞→x

y → ∞

y → ∞

y → ∞

y → ∞

Behavior

as −∞→x

y → -∞

y → ∞

y → -∞

y → ∞

Extrema

None

(0, 0)

None

(0, 0)

Symmetry

Origin

y-axis

Origin

y-axis

Unit 3, Activity 4, Polynomial Functions & Their Graphs


Use technology to complete the chart. Function

f(x) = (x+1)2 f(x) = -x(x-2)(x+3) f(x) = -x2(x-5)2 f(x) = x3(x2-4)

Sketch

Parent

Zeros

Root Characteristics

End Behavior x →∞

End Behavior x →-∞

Relative and Absolute Extrema



Unit 3, Activity 4, Polynomial Functions & Their Graphs with Answers



f(x) = (x+1)2 f(x) = -x(x-2)(x+3) f(x) = -x2(x-5)2 f(x) = x3(x2-4)

Sketch

Parent

f(x) = x2 f(x) = x3 f(x) = x4 f(x) = x5

Zeros

-1 0, 2, -3 0, 5 0, -2, 2

Root Characteristics

Double root and tangent to x-axis at x = -1

Crosses at 0 Crosses at 2 Crosses at -3

Double root and tangent to x-axis at x = 0 and at x = 5

Crosses at 0 Crosses at -2 Crosses at 2

End Behavior x →∞

y →∞

y →-∞

y →-∞

y →∞

End Behavior x →-∞

y →∞

y →∞

y →-∞

y →-∞

Relative and Absolute Extrema

Ab. Min. (-1, 0)

Rel. Min. ≈(-1.786, -2.209)

Rel. Max. ≈(1.120, 4.061)

Ab. Max. (0, 0) & (5, 0)

Rel. Min. (2.5, -39.0625)

Rel. Max. ≈(-1.549, 5.949)

Rel. Min. ≈(1.549, -5.949)


(-1, ∞)

≈(-2.209, 1.120)

(-∞,0) ∪ (2.5,5)

≈(-∞, -1.549) ∪ ≈(1.549, ∞)


(-∞, -1)

≈(-∞, -2.209) ∪ ≈(1.120, ∞)

(0, 2.5) ∪ (5, ∞)

≈(-1.549, 1.549)

Unit 3, Activity 5, Polynomial Functions & Their Linear Factors


1. Graph the lines y = x + 2 and y = x – 4 on the coordinate axes below. y x 2. Sketch the quadratic function formed by multiplying the linear expressions in #1 on the same coordinate axes. 3. What do you notice about the x- and y-intercepts of the parabola? 4. Use the graphs above to complete the sign chart below. Quadratic function: f(x) = (x + 2)(x – 4) Linear factor: x - 4 Linear factor: x + 2 -2 4 5. Use the sign chart above to answer the questions below. a) Is the y value of the quadratic function positive or negative when x = 0? ______________ b) Is the y value of the quadratic function positive or negative when x = -7? _____________ c) For what values of x is (x + 2)(x – 4) > 0? ______________

Unit 3, Activity 5, Polynomial Functions & Their Linear Factors


6. Graph the lines y = x + 1, y = 2 – x, and y = x – 5 on the coordinate axes below y x 7. Sketch the cubic function formed by multiplying the linear expressions in #1 on the same coordinate axes. 8. What do you notice about the x- and y-intercepts of the cubic function? 9. Use the graphs above to complete the sign chart below. Cubic function: f(x) = (x + 1)(2 – x)(x - 5) Linear factor: x - 5 Linear factor: 2 - x Linear factor: x + 1 -1 2 5 10. Use the sign chart above to answer the questions below. a) Is the y value of the quadratic function positive or negative when x = 13? ______________ b) Is the y value of the quadratic function positive or negative when x = -2? _____________ c) For what values of x is (x + 1)(2 – x)(x + 5) ≤ 0? ______________

Unit 3, Activity 5, Polynomial Functions & Their Linear Factors with Answers


1. Graph the lines y = x + 2 and y = x – 4 on the coordinate axes below.

2. Sketch the quadratic function formed by multiplying the linear expressions in #1 on the same coordinate axes. 3. What do you notice about the x- and y-intercepts of the parabola? The x-intercepts of the parabola are the same x-intercepts of the lines. The y-intercept of the parabola is the product of the y-intercepts of the lines. 4. Use the graphs above to complete the sign chart below. + 0 - 0 + Quadratic function: f(x) = (x + 2)(x – 4) - - 0 + Linear factor: x - 4 - 0 + + Linear factor: x + 2 -2 4 5. Use the sign chart above to answer the questions below. a) Is the y value of the quadratic function positive or negative when x = 0? ___negative__ b) Is the y value of the quadratic function positive or negative when x = -7? ___positive___ c) For what values of x is (x + 2)(x – 4) > 0? _ (-∞, -2) ∪ (4, ∞)___

Unit 3, Activity 5, Polynomial Functions & Their Linear Factors with Answers


6. Graph the lines y = x + 1, y = 2 – x, and y = x – 5 on the coordinate axes below

7. Sketch the cubic function formed by multiplying the linear expressions in #1 on the same coordinate axes. 8. What do you notice about the x- and y-intercepts of the cubic function? The x-intercepts of the cubic function are the same as the x-intercepts of the lines. The y-intercept of the cubic function is the same as the product of the y-intercepts of the lines. 9. Use the graphs above to complete the sign chart below. + 0 - 0 + 0 - _ Cubic function: f(x) = (x + 1)(2 – x)(x - 5) - - - 0 + Linear factor: x - 5 + + 0 - - Linear factor: 2 - x - 0 + + + Linear factor: x + 1 -1 2 5 10. Use the sign chart above to answer the questions below. a) Is the y value of the quadratic function positive or negative when x = 13? negative___ b) Is the y value of the quadratic function positive or negative when x = -2? positive___ c) For what values of x is (x + 1)(2 – x)(x - 5) ≤ 0? ___[-1, 2] ∪ [5, ∞)___

Unit 3, Activity 7, Applications of Polynomial Functions I


Situation: A farmer has 800 m to enclose a rectangular pen for his goats. If he uses a stream as one side of the pen, what dimensions will maximize the area of the pen? What is the maximum area of the pen? Diagram/Picture Algebraic Model Graphical Model Limitations of the models Solution

Unit 3, Activity 7, Applications of Polynomial Functions I with Answers


Situation: A farmer has 800 m to enclose a rectangular pen for his goats. If he uses a stream as one side of the pen, what dimensions will maximize the area of the pen? What is the maximum area of the pen? Diagram/Picture 800 – 2x x x stream Algebraic Model Graphical Model

Limitations of the models Side x of the rectangle can only be so large. The domain restriction is 0 < x < 400 m. The model also assumes that the stream is as long as the side 800 – 2x. Solution The dimensions that will maximize the area are 200 m by 400 m. The maximum area of the rectangular pen is 80,000 m2.

A = x(800 – 2x) or A = 800x – 2x2

Unit 3, Activity 7, Applications of Polynomial Functions II


Situation: A box, without a top, is to be made from a 20 in by 24 in piece of cardboard by cutting equal size squares from each corner and then folding up the sides. What size square should be cut out from each corner in order to maximize the volume? What are the dimensions of the box? What is the maximum volume? Diagram/Picture Algebraic Model Graphical Model Limitations of the models Solution

Unit 3, Activity 7, Applications of Polynomial Functions II with Answers


Situation: A box, without a top, is to be made from a 20 in by 24 in piece of cardboard by cutting equal size squares from each corner and then folding up the sides. What size square should be cut out from each corner in order to maximize the volume? What are the dimensions of the box? What is the maximum volume? Diagram/Picture

24 – 2x

20 – 2x

Algebraic Model Graphical Model

Limitations of the models The size of the square can only be so big. Domain restrictions are 0 < x < 10 in. Solution Square Size: ≈ 3.6 in Dimensions: ≈ 3.6 in by 16.8 in by 12.8 in Maximum Volume: ≈ 774.1 in3

x x

x x x x x x

V = x(24 – 2x)(20 – 2x)

Unit 4, Activity 1, Power Functions – Negative Integer Exponents


Fill in the following modified word grid for pxy = . Start with p = -1 and continue to p = -4.

f(x) = xp

Unit 4, Activity 1, Power Functions – Negative Integer Exponents with Answers


Fill in the following word grid for pxy = . Start with p = -1 and continue to p = -4.

f(x) = xp

f(x) = x -1 f(x) = x -2 f(x) = x -3 f(x) = x -4

Graph

Domain

(-∞, 0) ∪ (0, ∞)

(-∞, 0) ∪ (0, ∞)

(-∞, 0) ∪ (0, ∞)

(-∞, 0) ∪ (0, ∞)

Range

(-∞, 0) ∪ (0, ∞)

(0, ∞)

(-∞, 0) ∪ (0, ∞)

(0, ∞)

Vertical

Asymptote

x = 0

x = 0

x = 0

x = 0

Horizontal Asymptote

y = 0

y = 0

y = 0

y = 0

Behavior as ∞→x

y → 0

y → 0

y → 0

y → 0

Behavior

as −∞→x

y → 0

y → 0

y → 0

y → 0

Extrema

None

None

None

None

Symmetry

Origin

y-axis

Origin

y-axis

Unit 4, Activity 6, Applications of Rational Functions


1. The gravitational acceleration (in m/s2) of an object r meters above the earth’s surface

is g(r) = ( )26

14

10378.610987.3

r+•

• .

Question Answer Check



2. The concentration (in micrograms) of a certain drug in a patient’s bloodstream t hours after

injection is C(t) = 11

302 +t

t .




3. The daily cost (in thousands of dollars) of manufacturing x sports cars is

C(x) = 0.6x3 – 2.4x2 + 43.2


Unit 4, Activity 6, Applications of Rational Functions with Answers


1. The gravitational acceleration (in m/s2) of an object r meters above the earth’s surface

is g(r) = ( )26

14

10378.610987.3

r+•

• .

Question Solution Check What is the gravitational acceleration 1 million meters above the earth’s surface?

≈ 7.32 m/s2

What is the gravitational acceleration at the surface of the earth?

9.8 m/s2

What are the asymptotes of this function?

There is no vertical asymptote since the denominator cannot equal zero. The horizontal asymptote is y = 0 because the larger degree is in the denominator.

Use the graph of the function to determine if it is possible to escape the pull of gravity.

Since the horizontal asymptote for the function is y = 0, the gravitational acceleration for extremely large values of r will approach but never equal zero. Thus, it impossible to ever fully escape the pull of gravity.



2. The concentration (in micrograms) of a certain drug in a patient’s bloodstream t hours after

injection is C(t) = 11

302 +t

t .

Question Solution Check What is the concentration of the drug 10 hours after injection?

≈ 2.7 micrograms

What happens to the concentration of the drug as the time after injection increases?

The concentration decreases as the time increases. In fact, since the horizontal asymptote is y = 0, the concentration will approach 0 as time continues to pass.

Use the graph of the function to determine when the concentration of the drug is highest.

≈ 3.32 hours

What is the highest possible concentration?

≈ 4.52 micrograms



3. The daily cost (in thousands of dollars) of manufacturing x sports cars is

C(x) = 0.6x3 – 2.4x2 + 43.2

Question Solution Check Write the average cost function.

xxxxC 2.434.23.0)(

23 +−=

What is the average cost of manufacturing 5 sports cars per day?

$4,140

What are the asymptotes for the average cost function?

The vertical asymptote is x = 0.

There is no horizontal or oblique asymptotes since the degree of the

numerator is 2 larger than the degree of the denominator.

Use the graph of the average cost function to find the minimum average cost of manufacturing a widget.

6 sports cars per day

What is the minimum average cost per day?

$3,600

Unit 5, Activity 1, Power Functions – Fractional Exponents


Fill in the following word grid for pxy = . Start with p = 1/2 and continue to p = 1/5.

f(x) = xp

Graph

Domain

Range

Behavior as ∞→x

Behavior

as −∞→x

Extrema

Symmetry

Unit 5, Activity 1, Power Functions – Fractional Exponents with Answers


Fill in the following word grid for pxy = . Start with p = 1/2 and continue to p = 1/5.

f(x) = xp

f(x) = x1/ 2 f(x) = x1/ 3 f(x) = x1/ 4 f(x) = x1/ 5

Graph

Domain

[0, ∞)

(-∞, ∞)

[0, ∞)

(-∞, ∞)

Range

[0, ∞)

(-∞, ∞)

[0, ∞)

(-∞, ∞)

Behavior as ∞→x

y → ∞

y → ∞

y → ∞

y → ∞

Behavior

as −∞→x

Does Not Exist

y → -∞

Does Not Exist

y → -∞

Extrema

Min (0, 0)

None

Min (0, 0)

None

Symmetry

None

Origin

None

Origin

Unit 5, Activity 4, Solving Radical Equations


Use a modified version of the story chain to solve each equation.

Equation Step Partner Check

2 1−x = x

x – 2 = x212 −

Unit 5, Activity 4, Solving Radical Equations



13 +x + 3 = x

32 +x - 1−x =1

Unit 5, Activity 4, Solving Radical Equations with Answers


Use a modified version of the story chain to solve each equation.

Equation Steps May Vary Partner Check

2 1−x = x

4(x - 1) = x2

4x – 4 = x2

x2 – 4x + 4 = 0

(x – 2)(x – 2) = 0

x = 2

*There is no extraneous root.

x – 2 = x212 −

x2 – 4x + 4 = 12 – 2x

x2 – 2x – 8 = 0

(x – 4)(x + 2) = 0

x = 4

*x=-2 is an extraneous root

Unit 5, Activity 4, Solving Radical Equations with Answers



13 +x + 3 = x

13 +x = x - 3

3x + 1 = x2 – 6x + 9

x2 – 9x + 8 = 0

(x – 1)(x – 8) = 0

x = 8

*x = 1 is an extraneous root

32 +x - 1+x =1

32 +x = 1 + 1+x

2x + 3 = 1 + 2 1+x + x + 1

x + 1 = 2 1+x

x2 + 2x + 1 = 2x + 2

x2 - 1 = 0

(x + 1)(x – 1) = 0

x =-1 *x = 1 is an extraneous root

Unit 5, Activity 6, Pendulum Experiment


Setup

1. Attach string to fishing weight. The lengths should vary from group to group. 2. Place the motion detector facing the path of the pendulum. Make sure that the motion

detector is at least 18 inches from the pendulum. It may help to set the motion detector on a small stack of books.

3. Plug the motion detector into the Sonic port on the CBL or EA 100. 4. Connect the graphing calculator to the CBL or EA 100. 5. Run the Physics program on the graphing calculator. 6. In the home menu, choose set up probes. 7. Enter 1 for the number of probes. 8. Choose motion. 9. Choose collect data. 10. Choose time graph. 11. Enter 150 measurements at 0.05 second apart.

Procedure 1. Gently swing the pendulum in the direction of the motion detector. 2. Press Enter on the calculator to begin taking measurements. 3. Continue until you see 3-6 periods on the graph. 4. If you do not get a satisfactory graph, repeat the process until you do.

Data

1. Measure from the top of the string to the middle of the fishing weight to find the length of the pendulum in inches.

2. Find the period of the pendulum by dividing the total time (in seconds) by the number of periods.

3. Record your results on the board.

Unit 6, Activity 1, Graphs of Exponential Functions


My Opinion

Statements Calculator Findings Lessons Learned

1. Exponential functions of the form f(x) = bx are always increasing.

2. Exponential functions of the form f(x) = bx have domains of (-∞, ∞).

3. Exponential functions of the form f(x) = bx have ranges of (-∞, ∞).

4. Exponential functions of the form f(x) = bx exhibit asymptotic behavior.

5. Exponential functions of the form f(x) = bx

have y-intercepts of 1.

6. Exponential functions of the form f(x) = bx are always concave down.

Unit 6, Activity 1, Graphs of Exponential Functions With Answers


My Opinion

Statements Calculator Findings Lessons Learned

1. Exponential functions of the form f(x) = bx are always increasing.

False: Exp. functions will decrease when: 1) 0<b<1 or 2) b>1 with a negative exponent.

2. Exponential functions of the form f(x) = bx have domains of (-∞, ∞).

True

3. Exponential functions of the form f(x) = bx have ranges of (-∞, ∞).

False: Exp. functions of the form f(x) = bx will have (0, ∞) as their ranges.

4. Exponential functions of the form f(x) = bx exhibit asymptotic behavior.

True: Exp. functions of the form f(x) = bx will be asymptotic to the x-axis.

5. Exponential functions of the form f(x) = bx have y-intercepts of 1.

True

6. Exponential functions of the form f(x) = bx are always concave down.

False: Exp. Functions of the form f(x) = bx are always concave up.

Unit 6, Activity 2, Graphs of Logarithmic Functions


Fill in the following modified word grid. For this grid, b > 1. f(x) = logb x

Unit 6, Activity 2, Graphs of Logarithmic Functions With Answers


Fill in the following modified word grid. For this grid, b > 1. f(x) = logb x

f(x) = log2 x f(x) = log3 x f(x) = log x f(x) = ln x

Exp. Form

2y = x

3y = x

10y = x

ey = x

Graph

Asymptote

x = 0

x = 0

x = 0

x = 0

Domain

(0, ∞)

(0, ∞)

(0, ∞)

(0, ∞)

Range

(-∞, ∞)

(-∞, ∞)

(-∞, ∞)

(-∞, ∞)

Increasing

(-∞, ∞)

(-∞, ∞)

(-∞, ∞)

(-∞, ∞)

Decreasing

Never

Never

Never

Never

Concave Up

Never

Never

Never

Never

Concave

Down

(-∞, ∞)

(-∞, ∞)

(-∞, ∞)

(-∞, ∞)

x-Intercept

(1, 0)

(1, 0)

(1, 0)

(1, 0)

Unit 6, Activity 3, Translations, Dilations, and Reflections of Exponential Functions



f(x) = 2x-1 - 4 f(x) = -3(1/3)x f(x) = ½ (4)-x + 1 f(x) = 5e1/3 x

Parent

Translations, Dilations, & Reflections

Sketch

Domain

Range

Asymptote



Concavity

Unit 6, Activity 3, Translations, Dilations, and Reflections of Exponential Functions With Answers



f(x) = 2x-1 - 4 f(x) = -3(1/3)x f(x) = ½ (4)-x + 1 f(x) = 5e1/3 x

Parent

f(x) = 2x f(x) = (1/3)x f(x) = 4x f(x) = ex


Right 1 Down 4

Reflect over x-axis; Vertical

stretch of factor 3

Vertical compression of factor 2; Reflect over y-axis; Up 1

Vertical stretch of factor 5;

Horizontal stretch of factor 3

Sketch Domain

(-∞, ∞) (-∞, ∞) (-∞, ∞) (-∞, ∞)

Range

(-4, ∞) (-∞, 0) (1, ∞) (0, ∞)

Asymptote

y = -4 y = 0 y = 1 y = 0


(-∞, ∞) None None (-∞, ∞)


None (-∞, ∞) (-∞, ∞) None

Concavity

Concave Up (-∞, ∞)

Concave Down (-∞, ∞)



Unit 6, Activity 3, Translations, Dilations, and Reflections of Log Functions



f(x)=log2 (x+1)-1 f(x) = -2log1/3 x f(x) = log (-x) + 3 f(x) = ln (2x)

Parent


Sketch

Domain

Range

Asymptote



Concavity

Unit 6, Activity 3, Translations, Dilations, and Reflections of Log Functions With Answers



f(x)=log2 (x+1)-1 f(x) = -2log1/3 x f(x) = log (-x) + 3 f(x) = ln (2x-4)

Parent

f(x) = log2 x f(x) = log1/3 x f(x) = log x f(x) = ln x


Left 1 Down 1

Reflect over x-axis; Vertical

stretch of factor 2

Reflect over y-axis; Up 3

Horizontal compression of

factor 2; Right 2

Sketch Domain

(-1, ∞) (0, ∞) (-∞, 0) (0, ∞)

Range

(-∞, ∞) (-∞, ∞) (-∞, ∞) (-∞, ∞)

Asymptote

x = -1 x = 0 x = 0 x = 2


(-∞, ∞) (-∞, ∞) None (-∞, ∞)


None None (-∞, ∞) None

Concavity

Concave Down (-1, ∞)

Concave Up (0, ∞)

Concave Down (-∞, 0)

Concave Down (0, ∞)

Unit 6, Activity 5, Solving Logarithmic Equations


1. log 3 (x2 - 6x) = 3

Steps – Incorrect Order Steps – Correct Order 1. (x – 9)(x + 3) = 0 2. x2 – 6x – 27 = 0 3. x = 9 4. 33 = x2 – 6x 2. log 4 (x2 + 6x) = 2 3. log 7 (2x - 9) = - 1 4. log 2 (x – 8) + log 2 (x – 1) = 3

Steps – Incorrect Order Steps – Correct Order 1. x2 – 9x = 0 2. log 2 (x2 – 9x + 8) = 3 3. x = 9 4. 23 = x2 – 9x + 8 5. x(x – 9) = 0 5. log 4 (x – 4) – log 4 (9x + 6) = -2 6. log 5 (2x + 7) - log 5 (x – 1) = log 5 3

Unit 6, Activity 5, Solving Logarithmic Equations


7. 2 ln x – 3 ln 2 = ln 18

Steps – Incorrect Order Steps – Correct Order

1. ln8

2x = ln 18

2. x = 12

3. 8

2x = 18

4. ln x2 – ln 23 = ln 18

5. x2 = 144

8. 3 log x + log 2 – log 5 = log 50 9. ½ log 3 x + 2 log 3 3 = 4 10. log 2 4 – 1/3 log 2 x = -4 11. 2/3 log 9 x – 3/2 log 9 4 = log 9 18

Unit 6, Activity 5, Solving Logarithmic Equations with Answers


1. log 3 (x2 - 6x) = 3

Steps – Incorrect Order Steps – Correct Order 1. (x – 9)(x + 3) = 0 33 = x2 – 6x 2. x2 – 6x – 27 = 0 x2 – 6x – 27 = 0 3. x = 9 (x – 9)(x + 3) = 0 4. 33 = x2 – 6x x = 9 2. log 4 (x2 + 6x) = 2 3. log 7 (2x - 9) = - 1 42 = x2 + 6x 7-1 = 2x - 9 16 = x2 + 6x 1/7 = 2x - 9 x2 + 6x – 16 = 0 64/7 = 2x (x + 8)(x – 2) = 0 x = 32/7 x = -8 x = 2 The only solution is x = 2 because -8 is not in the domain of the log function. 4. log 2 (x – 8) + log 2 (x – 1) = 3

Steps – Incorrect Order Steps – Correct Order 1. x2 – 9x = 0 log 2 (x2 – 9x + 8) = 3 2. log 2 (x2 – 9x + 8) = 3 23 = x2 – 9x + 8 3. x = 9 x2 – 9x = 0 4. 23 = x2 – 9x + 8 x(x – 9) = 0 5. x(x – 9) = 0 x = 9 5. log 4 (x – 4) – log 4 (9x + 6) = -2 6. log 5 (2x + 7) - log 5 (x – 1) = log 5 3

log 4 694+−

xx = -2 log 5 1

72−+

xx = log 5 3

4-2 = 694+−

xx

172

−+

xx = 3

694

161

+−

=xx 2x + 7 = 3x - 3

9x + 6 = 16x – 64 10 = x 70 = 7x 10 = x

Unit 6, Activity 5, Solving Logarithmic Equations with Answers


7. 2 ln x – 3 ln 2 = ln 18

Steps – Incorrect Order Steps – Correct Order

1. ln8

2x = ln 18 ln x2 – ln 23 = ln 18

2. x = 12 ln

8

2x = ln 18

3. 8

2x = 18 8

2x = 18

4. ln x2 – ln 23 = ln 18 x2 = 144

5. x2 = 144 x = 12

8. 3 log x + log 2 – log 5 = log 50 9. ½ log 3 x + 2 log 3 3 = 4

log 5

2 3x = log 50 log 3 9 x = 4

5

2 3x = 50 34 = 9 x

2x3 = 250 81 = 9 x x3 = 125 9 = x x = 5 81 = x 10. log 2 4 – 1/3 log 2 x = -1 11. 2/3 log 9 x – 3/2 log 9 4 = log 9 18

log 2 3

4x

= -1 log 9 8

32

x = log 9 18

2-1 = 3

4x

8

32

x = 18

21 =

3

4x

x2/3 = 144

3 x = 8 x = 1443/2 x = 512 x = 1728

Unit 6, Activity 6, Exponential Growth & Decay


Create an exponential growth or decay story chain modeled after one of the examples covered in class.

STORY LINES AUTHOR

Create three questions based on the story chain. 1. 2. 3.

Unit 6, Activity 6, Money Investments


Create a money investment story chain modeled after one of the examples covered in class.

STORY LINES AUTHOR

Create three questions based on the story chain. 1. 2. 3.

Unit 6, Activity 7, Loudness of Sound


SQPL Statement: Some sounds can barely be heard; while others can be painful.

Your Questions Answers

Classmates’ Questions Answers

1. How many times more intense is a sound of 80 dB than one of 50 dB? 2. How many times more intense is a sound of 115 dB than one of 70 dB? 3. Find the loudness, in decibels, of a washing machine that operates at an intensity of 10-5 watt per square meter.

Unit 6, Activity 7, Loudness of Sound With Answers


NOTE: Answers will vary. Some important questions are listed below.

SQPL Statement: Some sounds can barely be heard; while others can be painful. Your Questions Answers

Classmates’ Questions Answers Is the SQPL statement true?

yes

How is sound measured?

Decibels: watts per square meter

What sounds are barely audible?

Whisper: 10 decibels Light Rain: 20 decibels

What sounds are painful?

Jet taking off from 100 ft away: 140 decibels Shotgun Blast: 140 decibels

How do you use the Decibel Scale?

The scale starts at 0 and counts by 10 up to 140. To compare sounds, find the difference between their decibels and calculate 10 to that difference.

Is there a decibel formula?

Yes, L(x) =10 log ⎟⎠⎞

⎜⎝⎛

−1210x ; where x is the

intensity of sound in watts per square meter 1. How many times more intense is a sound of 80 dB than one of 50 dB?

80 – 50 = 30 (which is 3 steps on the decibel scale) ∴ 103 = 1000 times more intense 2. How many times more intense is a sound of 115 dB than one of 70 dB?

115 – 70 = 45 (which is 4½ steps on the decibel scale) ∴ 104.5 ≈ 31,622.8 times more intense 3. Find the loudness, in decibels, of a washing machine that operates at an intensity of 10-5 watt per square meter.

L(10-5) = 10 log ⎟⎟⎠

⎞⎜⎜⎝

⎛−

−

12

5

1010 = 10 log 107 = 70 dB

Unit 6, Activity 7, Magnitude of Earthquakes


SQPL Statement: The earthquake with the largest magnitude occurred in the Indian Ocean. Your Questions Answers

Classmates’ Questions Answers

1. The 1906 earthquake in San Francisco had a magnitude of 6.9. The 1985 earthquake in

New Mexico had a magnitude of 8.1. Compare the intensities of the two earthquakes. 2. Find the magnitude of an earthquake whose seismographic reading is 10 mm at a distance of 100 km from the epicenter.

Unit 6, Activity 7, Magnitude of Earthquakes with Answers


NOTE: Answers will vary. Some important questions are listed below.

1. The 1906 earthquake in San Francisco had a magnitude of 6.9. The 1985 earthquake in

New Mexico had a magnitude of 8.1. Compare the intensities of the two earthquakes.

8.1 – 6.9 = 1.2 (which is 1.2 steps on the Richter Scale) ∴ 101.2≈ 15.85 times more intense 2. Find the magnitude of an earthquake whose seismographic reading is 10 mm at a distance

of 100 km from the epicenter.

M(10) = log ⎟⎠⎞

⎜⎝⎛

−31010 = log (104) = 4

SQPL Statement: The earthquake with the largest magnitude occurred in the Indian Ocean. Your Questions Answers

Classmates’ Questions Answers Is the SQPL statement true?

No, the earthquake with the largest magnitude occurred in Chile. It measured a 9.5 on the Richter Scale.

How is magnitude measured?

The logarithmic ratio of the seismographic reading of the earthquake that occurred to the zero-level earthquake whose seismographic reading is 10-3 at a distance of 100km from the epicenter.

How do you use the Richter Scale?

The Richter Scale is used to compare the magnitudes of earthquakes. Since it is logarithmic in nature, each whole number increase in Richter value represents a ten-fold increase in magnitude.

What was the worst earthquake in US history?

San Francisco, April 18, 1906 Magnitude = 7.9

Is there a formula for determining the magnitude of an earthquake?

Yes, M(x) = log ⎟⎟⎠

⎞⎜⎜⎝

⎛− 310

x; where x is the

seismographic reading in millimeters 100 km from the epicenter

Unit 6, Activity 8, Linearizing Exponential Data


Year 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 Population (millions)

92.0 105.7 122.8 131.7 150.7 179.3

203.3 226.5 246.8 281.4

1. Enter the year 1910 as 1, 1920 as 3, and so on. 2. Which of the following models best fits the data? Justify your answer! Linear Power Exponential 3. Write the equation of the model of best fit. 4. Linearize the data. Show your work!

Unit 6, Activity 8, Linearizing Exponential Data With Answers


Year 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 Population (millions)

92.0 105.7 122.8 131.7 150.7 179.3

203.3 226.5 246.8 281.4

1. Enter the year 1910 as 1, 1920 as 3, and so on. 2. Which of the following models best fits the data? Justify your answer! Linear: r ≈ .9909 Power: r ≈ .9559 Exponential: r ≈ .9980 *Since this model had the largest correlation coefficient, it is the best fit for this data. 3. Write the equation of the model of best fit. y ≈ 82.423(1.1325)x 4. Linearize the data. Show your work! m = log (1.1325) ≈ .0540 b = log (82.423) ≈ 1.9160 y ≈ 0.540x + 1.9160

Unit 7, Activity 1, Vocabulary Cards


STATISTICS

Def: the study of collecting, organizing, and interpreting data Ex: Statistics are used to determine car insurance rates.

INDIVIDUAL

Def: a person or object in the study Ex: If a study is about teachers, each teacher interviewed or observed is called an individual.

VARIABLE

Def: the characteristic of the individual to be observed or measured

Ex: test scores

QUANTITATIVE

VARIABLE

Def: variable that quantifies (assigns a numerical value) Ex: a person’s weight

QUALITATIVE

VARIABLE

Def: variable that categorizes or describes Ex: gender

POPULATION

Def: every individual of interest Ex: all living presidents – not just a few of them

Unit 7, Activity 1, Vocabulary Cards


SAMPLE

Def: a subset of the population (some of the individuals of interest)

Ex: some living presidents

NOMINAL DATA

Def: data consisting of only names or qualities – no numerical values

Ex: colors

ORDINAL DATA

Def: data that has an order but differences between data values are meaningless

Ex: student high school rankings 1st, 9th , 28th , etc.

INTERVAL DATA

Def: data that has an order, meaningful differences, but may or may not have a starting point which makes ratios meaningless

Ex: temperature readings

RATIO DATA

Def: data with the same characteristics as interval data but with a starting point which makes ratios meaningful

Ex: measures of height

DESCRIPTIVE STATISTICS

Def: the practice of collecting, organizing, and summarizing information from samples or populations Ex: graphs, measures of center and spread

INFERENTIAL STATISTICS

Def: the practice of interpreting sample values gained from descriptive techniques and drawing conclusions about the population Ex: polling 100 voters and using the results to predict a winner

Unit 7, Activity 2, Collecting and Organizing Univariate Data


1. Collect data on the number of siblings for each student in the class. Identify the data set as a sample or a population. 2. Organize the data using a box-whisker plot. 3. Organize the data using the display of your choice. 4. Organize the data using another display of your choice.

Unit 7, Activity 2, Collecting and Organizing Univariate Data with Answers


1. Collect data on the number of siblings for each student in the class. Identify the data set as a sample or a population. Copy the data from one of the students so that you can create the same graphs as the students. The data set is from a population since the number of siblings was collected from each student in the class 2. Organize the data using a box-whisker plot. The box-whisker plot cannot be provided since it will depend on the data collected in class. 3. Organize the data using the display of your choice. Displays will vary. 4. Organize the data using another display of your choice. Displays will vary.

Unit 7, Activity 2, Data Displays: Advantages and Disadvantages


Complete the modified word grid below.

Type of Graph Advantages Disadvantages

Line Plot

Bar Graph

Circle Graph

Stem-Leaf Plot

Box Plot

Unit 7, Activity 2, Data Displays: Advantages and Disadvantages with Answers


Complete the modified word grid below.

Type of Graph Advantages Disadvantages

Line Plot

Individual data is not lost

Easy to create

Shows range, minimum, maximum,

gaps, clusters, & outliers

Can be cumbersome if there are a large number of data values

Needs a small range of data

Bar Graph

Easy to create

Easy to read

Makes comparisons easy

Only used for discrete data

Individual data is lost

Circle Graph

Easy to read

Shows percentages

Only used for discrete data


Good for only about 3-7 categories

Total is often missing

Stem-Leaf Plot

Easy to create

Stores a lot of data in a

smaller space

Shows range, minimum, maximum, gaps, clusters, & outliers

Can be cumbersome if there are a

large number of data values

Can be difficult to read

Not visually appealing

Box Plot

Identifies outliers

Makes comparisons easy

Shows 5-point summary

(minimum, maximum, 1st Quartile, Median, & 3rd Quartile)


Can be confusing to read

Not visually appealing

Unit 7, Activity 3, Frequency Tables and Histograms


The average lengths of the North American geese and ducks are given below.

Name of Bird Average Length

Name of Bird Average Length

Fulvous Whistling Duck 50 cm Black-bellied Whistling Duck 53 cm White-fronted Goose 72 cm Snow Goose 74 cm Ross’ Goose 61 cm Brant 66 cm Canada Goose (small) 61 cm Canada Goose (large) 101 cm Wood duck 69 cm Green-winged Teal 35 cm American Black Duck 52 cm Mottled Duck 53 cm Mallard 59 cm Northern Pintail (male) 69 cm Northern Pintail (female) 55 cm Blue-winged Teal 39 cm Cinnamon Teal 40 cm Northern Shoveler 47 cm Gadwall 50 cm Eurasian Wigeon 49 cm American Wigeon 52 cm Canvasback 55 cm Redhead 51 cm Ring-necked Duck 41 cm Tufted Duck 43 cm Greater Scaup 45 cm Lesser Scaup 42 cm Common Eider 64 cm King Eider 55 cm Harlequin Duck 44 cm Oldsquaw (male) 52 cm Oldsquaw (female) 41 cm Black Scoter 48 cm Surf Scoter 48 cm White-winged Scoter 55 cm Common Goldeneye 46 cm Barrow’s Goldeneye 47 cm Bufflehead 35 cm Hooded Merganser 44 cm Common Merganser 63 cm Red-breasted Merganser 57 cm Ruddy Duck 39 cm Masked Duck 33 cm

Class Lower Limit Upper Limit Number of birds

or Frequency Relative Frequency =

; 43f nn

=

Cumulative Relative Frequency

≤ x < ≤ x < ≤ x < ≤ x < ≤ x < ≤ x < ≤ x < ≤ x <

≤ x < ≤ x ≤

Unit 7, Activity 3, Frequency Tables and Histograms with Answers


Class Lower Limit Upper Limit Number of birds


; 43f nn

=


33 ≤ x < 40 5 .12 .12 40 ≤ x < 47 9 .21 .21 + .12 = .33 47 ≤ x < 54 13 .30 .30 + .33 = .63 54 ≤ x < 61 6 .14 .14 + .63 = .77 61 ≤ x < 68 5 .12 .77 + .12 = .89 68 ≤ x < 75 4 .09 .09 + .89 = .98 75 ≤ x < 82 0 0 .98 82 ≤ x < 89 0 0 .98 89 ≤ x < 96 0 0 .98 96 ≤ x ≤ 103 1 .02 .02 + .98 = 1.00

Unit 7, Activity 3, Frequency Tables and Histograms with Answers


Unit 7, Activity 3, Math Test Grades


Math Test Grades (0-100 pts)

Complete the table.

Class Lower Limit Upper Limit Number of scores


; 43f nn

=


Draw a relative frequency histogram on the back of this BLM.

Student Test Grade Student Test Grade Alvin 83 Kay 42 Amy 59 Keller 93 Brett 90 Kim 84 Cedric 88 Lamar 77 Charles 66 Lance 63 Connie 52 Lee 78 Debra 79 Leon 91 Dexter 36 Mai 95 Diane 77 Mason 76 Dion 85 Nicole 84 Edrick 83 Ouida 80 Evan 91 Pablo 77 Fredrick 99 Penny 80 Grace 80 Patrice 86 Gregory 85 Patrick 88 Hakim 88 Pedro 92 Helen 69 Stephanie 55 Janice 71 Trevor 66 Jay 76 Tyler 78 Jose 99 Xavier 81

Unit 7, Activity 3, Math Test Grades with Answers


Class Lower Limit Upper Limit Number of scores


; 43f nn

=


36 ≤ x < 44 2 .05 .05 44 ≤ x < 52 0 0 .05+0=.05 52 ≤ x < 60 3 .075 .075+.05=.125 60 ≤ x < 68 3 .075 .075+.125=.20 68 ≤ x < 76 2 .05 .05+.20=.25 76 ≤ x < 84 14 .35 .35+.25=.60 84 ≤ x < 92 11 .275 .275+.60=.875

92 ≤ x < 100 5 .125 .125+.875=1.00 Relative Frequency Histogram .35 .30 .25 .20 .15 .10 .05

36≤x<44

44≤x<52

52≤x<60

60≤x<68

68≤x<76

76≤x<84

84≤x<92

92≤x<100

Unit 7, Activity 4, Tropical Cyclones


Year Last Named Tropical Cyclone

Number of Hurricanes

Total Number of Tropical Cyclones

Date of First Tropical Cyclone

Date of Last Tropical Cyclone

1991

Grace 3 June 29 October 28

1992

Frances 4 August 16 October 22

1993

Harvey 4 June 18 September 18

1994

Gordon 3 June 30 November 8

1995

Tanya 11 June 3 October 27

1996

Marco 9 June 17 November 18

1997

Grace 3 June 30 October 16

1998

Nicole 10 July 27 November 24

1999

Lenny 8 June 11 November 13

2000

Nadine 8 August 4 October 19

2001

Olga 9 June 5 November 24

2002

Lili 4 July 14 October 14

2003

Peter 7 April 21 December 9

Unit 7, Activity 6, Distribution Shapes


Complete the chart by matching the name, definition, and example of data from the next page with its appropriate shape.

Example Shape of Histogram

Name and Definition Example of Data

A.

B.

C.

D.

E.

Unit 7, Activity 6, Distribution Shapes


Names & Definitions 1. Symmetrical, normal or triangular – both sides of the distribution are identical. Also called a bell-shaped distribution. 2. Left skewed or negatively skewed – the tail is to the left 3. Bi-modal – the two classes with the highest frequencies are separated by at least one class 4. Right skewed or positively skewed – the tail is to the right. 5. Uniform or rectangular – the bars are all the same height Examples of Data I. Heights of a group of people containing both males and females II. Heights of a group of males III. Grades on a test where most students perform poorly IV. Ages of people getting their first driver’s license V. Rolls of a regular die

Unit 7, Activity 6, Distribution Shapes with Answers


Example Shape of Histogram

Name and Definition Example of Data

A.

3. Bi-modal – the two classes with the highest frequencies are separated by at least one class

I. Heights of a group of people containing both males and females

B.

2. Left skewed or negatively skewed – the tail is to the left

IV. Grades on a test in which most students do fairly well

C.

5. Right skewed or positively skewed – the tail is to the right

III. Ages of people getting their first driver’s

license

D.

4. Uniform or rectangular – the bars are all the same height

V. Rolls of a regular die

E.

1. Symmetrical, normal or triangular – both sides of the distribution are identical. Also called a bell-shaped distribution.

II. Heights of a group of males

Unit 7, Activity 7, Normal Distribution


Describe why each distribution is not normal.

3.

4.

5. Draw and label a normal distribution for exam grades (0-100 pts) if the mean is 78 and the

standard deviation is 5.

1.

2.

Unit 7, Activity 7, Normal Distribution


6. Determine the number of standard deviations either above or below the mean for an exam

score of 68.

7. What is the probability that a student scored between 88 and 93 pts?

8. What is the probability that a student scored at least a 73? 9. If 160 students took the exam, how many got a C using the grading scale 78-88 pts. 10. What is the probability that a student scored a 90?

Unit 7, Activity 7, Normal Distribution with Answers


Describe why each distribution is not normal.

3.

4.

5. Draw and label a normal distribution for exam grades (0-100 pts) if the mean is 78 and the

standard deviation is 5.

2.

1.

63 68 73 78 83 88 93

The curve crosses the horizontal axis. The curve is not symmetrical about the mean. The curve has two peaks and is not bell-shaped. Thus, the highest point does not lie directly above the mean. The end behavior of the curve does not follow the horizontal axis.

Unit 7, Activity 7, Normal Distribution with Answers


6. Determine the number of standard deviations either above or below the mean for an exam

score of 68. 68 is two standard deviations below the mean

7. What is the probability that a student scored between 88 and 93 pts? 2.35%

8. What is the probability that a student scored at least a 73? .34 + .34 + .135 + .0235 +.0015 = .84 or 84% OR 1 - .135 - .0235 - .0015 = .84 or 84% 9. If 160 students took the exam, how many got a C using the grading scale 78-88 pts. 160 • .475 = 76 students 10. What is the probability that a student scored a 90?

z = 5

7895 − = 2.4 Reading the Z-table, the probability is 49.2%.

Unit 8, Activity 1, Bivariate Vocabulary Cards


SCATTERPLOT

Def: a graphical display of the pairs of values of two variables

Height •

Ex: • • • age

CORRELATION

Def: a relationship between two variables Ex: number of calories eaten and a person’s weight

CORRELATION COEFFICIENT

Def: a number (r) from -1 to 1 that measures the linear relationship between two variables

Ex: the number of movie tickets sold and the total cost is a perfect Linear relationship; thus, the correlation coefficient would be 1

COEFFICIENT OF DETERMINATION

Def: a number that measures the proportion of variance in the response variable explain-ed by the regression line and explanatory variable (0 ≤r2 ≤ 1)

Ex: an r2 value of .70 indicates that 70% of the variance in the response variable can be accounted for by the explanatory variable

RESIDUAL

Def: the difference between the observed value and the value suggested by the regression line Ex: y - y

REGRESSION LINE

Def: line that describes how the response variable changes as the explanatory variable changes

Height • • • Ex: •

age

∧

Unit 8, Activity 1, Bivariate Vocabulary Cards


LEAST SQUARES LINE

Def: line that makes the sum of squares of the vertical distances of the data points from the line as small as possible

Ex: y = a + bx

EXPLANATORY VARIABLE

Def: the independent variable which is used as a predictor of the response variable Ex: number of calories eaten

RESPONSE VARIABLE

Def: the dependent or predicted variable

Ex: a person’s weight

EXTRAPOLATION

Def: to infer or estimate by extending or projecting known information Ex: known independent variable data ranges from 0-50 and a prediction is made for an independent value of 60

INTERPOLATION

Def: inferring or estimating a value that lies between known values Ex: known independent variable data ranges from 0-50 and a prediction is made for an independent value of 40

CAUSATION

Def: the relationship between a cause and its effect which can only be determined by conducting an experiment

Ex: experimental studies have shown that smoking causes lung cancer

∧

Unit 8, Activity 2, Scatterplots and Correlations


Label each scatterplot as a perfect positive correlation, perfect negative correlation, strong positive correlation, strong negative correlation, weak positive correlation, weak negative correlation, or no correlation.

1. 2. 3.

4. 5.

6. 7.

Unit 8, Activity 2, Scatterplots and Correlations with Answers


Label each scatterplot as a perfect positive correlation, perfect negative correlation, strong positive correlation, strong negative correlation, weak positive correlation, weak negative correlation, or no correlation.

perfect positive strong positive weak negative correlation correlation correlation

strong negative weak positive correlation correlation

perfect negative no correlation correlation

1. 2. 3.

4. 5.

6. 7.

Unit 8, Activity 2, Regression Line and Correlation


Step One

Go to the following web site: http://illuminations.nctm.org/LessonDetail.aspx?ID=L456 . Step Two

Use the interactive math applet below to help you answer these questions:

1. Compare the r-values for the following three situations.

a) Create a scatterplot that you think shows a strong positive linear association between the two variables. What is the r-value? Draw the regression line.

b) Create a scatterplot that you think shows a strong negative linear association between the two variables. What is the r-value? Draw the regression line. c) Create a scatterplot that you think shows no linear association between the two variables. What is the r-value?

2. For each r-value below, create a scatterplot that has that exact r-value.

a) r = 1 b) r = -1 c) r = 0

3. Plot several points that exhibit a strong positive linear trend, and then plot one outlier.

a) Overall, is this scatterplot roughly linear? b) Is the r-value close to 1?

4. In the lower left corner of the coordinate plane, plot 10 points that exhibit no trend (this is sometimes called a "cloud" of points). Then plot one point in the upper right corner.

a) Overall, is this scatterplot linear? b) Is the r-value close to 1?

5. a) Does a high r-value necessarily mean that the data are definitely linear?

b) Does an r-value close to zero always mean that the data are not linear?

http://illuminations.nctm.org/LessonDetail.aspx?ID=L456

Unit 8, Activity 2, Regression Line and Correlation with Answers


Step One

Go to the following web site: http://illuminations.nctm.org/LessonDetail.aspx?ID=L456 . Step Two

Use the interactive math applet below to help you answer these questions: 1. Compare the r-values for the following three situations.

a) Create a scatterplot that you think shows a strong positive linear association between the two variables. What is the r-value? Draw the regression line. r values will vary, but should be close to 1

b) Create a scatterplot that you think shows a strong negative linear association between the two variables. What is the r-value? Draw the regression line. r values will vary, but should be close to -1 c) Create a scatterplot that you think shows no linear association between the two variables. What is the r-value? r values will vary, but should be close to 0

2. For each r-value below, create a scatterplot that has that exact r-value.

a) r = 1 b) r = -1 c) r =0 points make a straight points make a straight points randomly scattered line with a positive slope line with a negative slope with no linear pattern

3. Plot several points that exhibit a strong positive linear trend, and then plot one outlier.

a) Overall, is this scatterplot roughly linear? b) Is the r-value close to 1? The farther the outlier is from the The farther the outlier is from the rest of the data, the less linear the rest of the data, the farther the relationship. r-value is from 1.

4. In the lower left corner of the coordinate plane, plot 10 points that exhibit no trend (this is sometimes called a "cloud" of points). Then plot one point in the upper right corner.

a) Overall, is this scatterplot linear? b) Is the r-value close to 1? no yes

5. a) Does a high r-value necessarily mean that the data are definitely linear?

no

b) Does an r-value close to zero always mean that the data are not linear?

no - The moral is that the correlation coefficient, r, is a valuable tool for studying the linear association between two variables, but it does not fully explain the association (in fact, no statistic does).

http://illuminations.nctm.org/LessonDetail.aspx?ID=L456

Unit 8, Activity 2, RAFT Writing


Student example of RAFT writing in math.

R – A whole number between 1 and 9

A – A whole number equal to 10 minus the number used (from R)

F – A letter

T – Why it is important to be a positive role model for the fractions less than one.

Dear Number 7,

It has come to my attention that you are not taking seriously your responsibilities as a role

model for the fractions. With this letter I would like to try to convince you of the importance of

being a positive role model for the little guys. Some day, with the proper combinations, they, too,

will be whole numbers. It is extremely important for them to understand how to properly carry

out the duties of a whole number. For them to learn this, it is imperative for them to have good

positive role models to emulate. Without that, our entire numbering system could be in ruins.

They must know how to respond if ever asked to become a member of a floating point gang.

Since they are not yet whole, it is our duty to numberkind to make sure they are brought up

properly to the left of the decimal.

Thank you in advance for your support,

The number 3

Unit 8, Activity 3, Least Squares Line


1. Which line seems to best fit the data?

2. Complete the chart.

Possible line 1 Possible line 2 Possible line 3

City Cost of Living Index

Average Annual Pay

x y x2 xy San Francisco,

CA 169.8 56,602

Washington, D.C. 138.8 48,430

Houston, TX 91.6 42,712

Atlanta, GA 97.6 41,123

Huntsville, AL 91.8 38,571

Saint Louis, MO 101.3 36,712

Brazoria, TX 90.5 36,253

Memphis, TN 90.7 35,922

SUM



3. Calculate the least squares line y = a + bx using the formulas below.

b = x

xy

SSSS

; SSxy = ( )( )

∑ ∑∑−n

yxxy SSx =

( )nx

x2

2 ∑∑ −

a = xby − ( y and x are the means for each respective variable) 4. Compare the least squares line from number 3 with the least squares line generated by the

graphing calculator.

∧



5. Using the calculator’s least squares, state the real-life meaning of the slope and y-intercept. 6. Use the calculator’s least squares line to find the average annual salary for a city with a cost of living index of 100. 7. Use the calculator’s least squares line to find the average annual salary for a city with a cost of living index of 80. 8. State limitations of the linear model.

Unit 8, Activity 3, Least Squares Line with Answers


1. Which line seems to fit the data the best?

Line #1 appears to have the smallest vertical distances between the data points and the line of best fit. Therefore, its sum of squares will be smaller than that of the other two lines. 2. Complete the chart.

Possible line 1 Possible line 2 Possible line 3

City Cost of Living Index

Average Annual Pay

x y x2 xy San Francisco,

CA 169.8 56,602 28832.04 9611019.6

Washington, D.C. 138.8 48,430 19265.44 6722084

Houston, TX 91.6 42,712 8390.56 3912419.2

Atlanta, GA 97.6 41,123 9525.76 4013604.8

Huntsville, AL 91.8 38,571 8427.24 3540817.8

Saint Louis, MO 101.3 36,712 10261.69 3718925.6

Brazoria, TX 90.5 36,253 8190.25 3280896.5

Memphis, TN 90.7 35,922 8226.49 3258125.4

SUM 872.1 336,325 101119.47 38057892.9



3. Calculate the least squares line y = a + bx using the formulas below.

SSxy = ( )( )

∑ ∑∑−n

yxxy = 38,057,892.9- ( )( )

8325,3361.872 = 1,394,263.838

SSx = ( )

nx

x2

2 ∑∑ − = 101,119.47- ( )8

1.872 2

≈ 6,049.66875

b = x

xy

SSSS

= 66875.049,6

938.263,394,1 ≈ 230.5

a = xby − = ⎟⎠

⎞⎜⎝

⎛−8

1.8725.230

8325,336

≈ 16,913.2

y ≈ 16913.2 + 230.5x

4. Compare the least squares line from number 3 with the least squares line generated by the graphing calculator. Calculator’s least squares line: y ≈ 16,916.6 + 230.5x The slopes are identical, but the y-intercepts vary by 3.4.

∧

∧

∧



5. Using the calculator’s least squares, state the real-life meaning of the slope and y-intercept.

Slope: For each increase of 1 in the cost of living index, the average annual salary will increase by $230.50

y-intercept: For a cost of living index of 0, the average annual salary would be $16,916.6 Note: The y-intercept is meaningless for this particular data set since the cost of living index will never equal zero. 6. Use the calculator’s least squares line to find the average annual salary for a city with a cost of living index of 100. y ≈ 16,916.6 + 230.5(100) ≈ $39,966.6 7. Use the calculator’s least squares line to find the average annual salary for a city with a cost of living index of 80. y ≈ 16,916.6+ 230.5(80) ≈ 35,356.60 8. State limitations of the linear model.

Answers will vary. One possible limitation is that there are many factors that affect the average annual salary for a particular city (population, industries, unemploy- ment index, etc.).

∧

∧

Unit 8, Activity 3, Hospitals


1. Is a linear model appropriate for this data set? Justify your answer. 2. Calculate the least squares line.

State Population (in millions) Number of hospitals Alabama 4.501 106 Alaska 0.649 19

Mississippi 2.881 91 Ohio 11.436 168

Oklahoma 3.512 105 Louisiana 4.496 128

Utah 2.351 42 California 35.484 464

Texas 22.119 383 Maine 1.306 37

Unit 8, Activity 3, Hospitals


3. Give the real-life meaning of the slope and y-intercept. 4. Use your regression line to predict the number of hospitals for a city with a population of 6

million people. Is this an example of interpolation or extrapolation? 5. State limitations of the model.

Unit 8, Activity 3, Hospitals with Answers


1. Is a linear model appropriate for this data set? Justify your answer. The scatterplot reveals a positive linear relationship since the number of hospitals continues to increase by about the same amount as the population increases. 2. Calculate the least squares line.

SSxy = ( )( )

∑ ∑∑−n

yxxy =28,700.321 - ( )( )

101543735.88 = 15,008.4205

SSx = ( )

nx

x2

2 ∑∑ − =1,947.90787 - ( )10735.88 2

= 1,160.517848

b = x

xy

SSSS

=517848.160,1

4205.008,15 ≈ 12.93252019

a = xby − =154.3 – 12.93252019(8.8735) ≈ 39.54328209

y ≈ 39.54328209 + 12.93252019x

State Population (in millions) Number of hospitals Alabama 4.501 106 Alaska 0.649 19

Mississippi 2.881 91 Ohio 11.436 168

Oklahoma 3.512 105 Louisiana 4.496 128

Utah 2.351 42 California 35.484 464

Texas 22.119 383 Maine 1.306 37

∧

Unit 8, Activity 3, Hospitals with Answers


3. Give the real-life meaning of the slope and y-intercept. Slope: For every increase of 1 million people, there are approximately 11 more hospitals. y-intercept: For a population of 0, there are about 48 hospitals. This does not make sense for this data set. 4. Use your regression line to predict the number of hospitals for a city with a population of 6

million people. Is this an example of interpolation or extrapolation?

y ≈ 39.54328209 + 12.93252019(6) ≈ 117 hospitals This is an example of interpolation since 6 million people lies within the given range of the independent variable. 5. State limitations of the model. Limitations will vary. Ex. Linear extrapolation can be misleading because there is a chance that the linear tendency might level off for larger data values. Also, the size of the hospitals is not known. A few large hospitals could service the same number of people as a large number of small hospitals.

∧

Unit 8, Activity 4, Correlation Coefficient and Coefficient of Determination


Year Number of students in the United States who took the AP Statistics Exam

2000 34118 2001 41609 2002 49824 2003 58230 2004 65878

1. Calculate the correlation coefficient using the appropriate formulas. 2. Calculate the coefficient of determination and interpret its meaning.

Unit 8, Activity 4, Correlation Coefficient and Coefficient of Determination with Answers


Year Number of students in the United States who took the AP Statistics Exam

2000 34118 2001 41609 2002 49824 2003 58230 2004 65878

1. Calculate the correlation coefficient using the appropriate formulas.

xy

x y

xy

2 22

x

2 22 10

y

SSr =

SS SS

( x )( y ) 10010×249659SS = xy - = 499897459 - = 80141n 5

( x ) 10010SS = x - = 20040030 - = 10n 5

( y) 249659SS = y - = 1.310842157×10 - = 642498308.8n 5

80141r = = .999810×642498308.8

∑ ∑∑∑∑∑∑

2. Calculate the coefficient of determination and interpret its meaning. r2 = .9996 The coefficient of determination states that approximately 99.96% of the variance in the number of U.S. high school students taking the AP statistics exam can be accounted for by the year. Thus, the number of years can accurately be used to explain the number of students taking the exam.

Unit 8, Activity 5, Residual Plots


1. Which residual plot states that the linear regression model is a good fit? Explain your answer.

Height (cm) 95 106 120 133 150 162 166 174 Arm Span (cm) 95.5 100 107 122.5 128 136 137.5 145.5

2. Draw the scatterplot using the graphing calculator. What type of model appears to be a good

fit for the data? Explain your answer. 3. Write the linear regression equation and interpret the meaning of the slope and y-intercept. 4. Record and interpret the correlation coefficient and coefficient of determination.

a) b) c)

Unit 8, Activity 5, Residual Plots


5. Draw and label the residual plot. 6. What does the residual plot tell you about the data?

Unit 8, Activity 5, Residual Plots with Answers


1. Which residual plot states that the linear regression model is a good fit? Explain your answer.

Residual plot b demonstrates that a linear regression model is a good fit because the points are randomly dispersed.

Height (cm) 95 106 120 133 150 162 166 174 Arm Span (cm) 95.5 100 107 122.5 128 136 137.5 145.5

2. Draw the scatterplot using the graphing calculator. What type of model appears to be a good

fit for the data? Explain your answer. A linear model appears to be a good fit since the data points are increasing at about the same rate. 3. Write the linear regression equation and interpret the meaning of the slope and y-intercept.

y ≈ 34.2542971 + 0.63107199x

m ≈ 0.631 which means that for every cm of height, the arm span increases by about 0.631 cm

b ≈ 34.25 ; which means that for a height of 0 cm, the arm span would be about 34.25 cm (The interpretation for b does not make sense for this data set.) 4. Record and interpret the correlation coefficient and coefficient of determination. r ≈ 0.99259244 ; which indicates a strong positive linear relationship r2 ≈ 0.98523975 ; which indicates that about 98.52% of the variation in arm span can be accounted for by the explanatory variable height

a) b) c)

Unit 8, Activity 5, Residual Plots with Answers


5. Draw and label the residual plot. Residuals 5 4 3 2 1 0 -1 -2 -3 -4 -5 Height (cm) 6. What does the residual plot tell you about the data? Since the points are randomly dispersed, a linear model is appropriate for this data set.

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Unit 8, Activity 6, Achieving Linearity


Display the following information on a poster board. If necessary, use both sides of the poster board. 1. Collect and record data on two variables that you think are linearly related. Be sure to include

units and at least eight values for each variable. 2. Draw and label a scatterplot of the raw data. 3. Does a linear model appear to be a good fit for the raw data? Explain your answer. 4. Find the linear regression equation using the graphing calculator. 5. Interpret the real-life meaning of the slope and y-intercept. 6. Record and interpret the correlation coefficient and coefficient of determination. 7. Draw the residual plot for the raw data. 8. What does the residual plot say about the fit of the linear model? 9. Does the graphical analysis of fit agree or disagree with the numerical analysis of fit?

Explain your answer.

10. If your graphical and numerical analysis of fit does not agree, find a better model for the raw data. Explain why you chose that particular model.

11. Describe the data transformation used to linearize the data. 12. Write the linear regression for the transformed data. 13. Draw the residual plot for the transformed data. 14. What does the residual plot for the transformed data tell you about the raw data?

Advanced Mathematics Functions and Statistics...Unit 1, Activity 1, Graphically Speaking with...

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