exploring quadratic …math.buffalostate.edu/~it/projects/markle.pdfsolve systems of equations ......

EXPLORING QUADRATIC

FUNCTIONS/INEQUALITIES

Course/ Grade Level: Math B/ Grade 11

Time SPAN: 42 minute class periods Tools: ti 83 Plus calculator Overhead projector with TI 83 Plus adapter Graph paper Course III Amsco 3rd edition textbook Worksheets

By: Michele Markle

DESCRIPTION OF RESOURCES USED

Amsco, Course III Integrated Mathematics, Third Edition, Edward P. Keenan and Ann Xavier Gantert, Chapter 13, pages 659-677, copyright 2000. Examgen computer program I2T2 Worksheet, page 27, Warm Up 27. Old New York State Math B Exam questions Quadratic Functions/Inequalities COURSE III –

MATH B Lesson Objectives Assignments

Identify key info. Represented by min, max, zeros, 1 connect relation of x & y Worksheet

Determine window on calculator Find solutions, min, max using calculator for

2 quadratic functions Worksheet Even numbers

3 Sketch Graphs Worksheet Solve Systems of Equations

4 Review Discriminant Finish Review Sum & Product of Roots

5 Solve Quadratic Inequalities Worksheet

6 Applications Worksheet

7 UNIT TEST

STANDARDS USED

New York State Learning Standards: Standard 1: Analysis, Inquiry, and Design Standard 2: Information Systems Standard 3: Mathematics Standard 7: Interdisciplinary Problem Solving NCTM Learning Standards: Numbers and Operations Algebra Problem Solving Communication Connections Representation

DESCRIPTION OF MATERIALS AND EQUIPMENT NEEDED

Day One:

Each student needs their TI-83 Plus calculators Overhead and TI-83 Plus overhead unit Quadratic Function Worksheets Review Worksheet for group work Homework Assignment

Day Two: Each student needs their TI-83 Plus calculators Overhead and TI-83 Plus overhead unit Warm up worksheet on previous days work Previous night’s homework assignment Worksheet – Finding the Roots of an Equation Worksheet – Finding the Min. or Max Review Worksheet for group work

Day Three: Each student needs their TI-83 Plus calculators Overhead and TI-83 Plus overhead unit Warm up worksheet on previous days work Previous night’s homework assignment Graph Paper Amsco Course III Textbook Worksheet - Solving Systems of Equations Review Worksheet for group work

Day Four: Each student needs their TI-83 Plus calculators Overhead and TI-83 Plus overhead unit Warm up worksheet on previous days work Previous night’s homework assignment Amsco Course III Textbook Review Worksheet for group work

Day Five: Each student needs their TI-83 Plus calculators Overhead and TI-83 Plus overhead unit Warm up worksheet on previous days work Previous night’s homework assignment Amsco Course III Textbook Worksheet – Quadratic Inequalities Review Worksheet for group work

Day Six: Each student needs their TI-83 Plus calculators Overhead and TI-83 Plus overhead unit Warm up worksheet on previous days work Previous night’s homework assignment Applications Worksheet

OVERVIEW OF ENTIRE UNIT

Day One: Transform an application to calculator language to be modeled (graphed) Discuss independent and dependent variables Discuss maximum and minimum Demonstrate finding minimum and maximums on calculator Find maximum and minimum of real life applications

Assignment of finding dependent/independent variables, max and min, zero, and window settings Group work on review topics

Day Two: Use calculator to find critical values in a quadratic function Calculator instruction on finding min, max, roots solutions of quadratic functions Describe relationships of quadratic function graphs Guided Practice on finding roots Assignment on finding roots of an equation and finding min, and max Group work on review topics Day Three: Sketching graphs on paper and calculator and solving system of Equations Discussion on what information is needed to sketch quadratic functions Guided practice on sketching graphs on paper and calculator Discussion on solving system of equations graphically Guided practice on sketching systems of equations on graph paper and calculator Assignment on graphing quadratic functions and system of equations Group work on review topics Day Four: Review discriminant and nature of roots and sum and product of roots Discussion on quadratic formula and discriminant Guided practice on quadratic formula and discriminant Description of nature of roots Looking at ways to find sum and product of roots Assignment is to finish review topic worksheet Day Five: Solving an inequality and graphing it on a number line Instruction on solving quadratic inequalities – by hand and by calculator Guided practice on solving quadratic inequalities Assignment on graphing inequalities Group work on review topics Day Six: Applying quadratic inequalities to real life situations Group work on applications regarding quadratic functions No assignment Test Prep due on test day

Lesson Plan – Day One

Quadratic Functions/Inequalities Class: Course III – Math B

Objectives: Identify key information represented by min., max, and zeros Connect relation of x and y Determine windows on calculator

Materials: Each student needs their TI-83 Plus calculators, Overhead and TI-83 Plus overhead unit, Quadratic Function Worksheets, Review Worksheet for group work, Homework Assignment

Anticipatory Set: Hand out and explain test prep which is due the day of the test.

Procedures: 20 minutes – Calculator Instruction (see worksheet) Explain Independent/Dependent Variables Explain Maximum and Minimum 10 Minutes- Guided Practice – Using Calculator 10 Minutes- Group Work to hand in (See review sheet)

Good Questions to Check for Understanding: How can we estimate the max and min? Can we use the axis of symmetry to help us? Closure: Ask if there are any questions, hand out homework and explain it.

Homework: Quadratic Functions worksheet

Adjustments for next time:

Quadratic Functions

A function is a relation between two variables

X- Min: The lowest point for _———————————— variable

X-Max: The highest point for —————————————variable Y Min: The lowest point for ______________ variable

Y Max: The highest point for _____________ variable

THESE NEED TO BE SET SO THE WHOLE MODEL (graph) CAN BE SEEN (PERTINENT).

KEY Quadratic Functions

A fonction is a relation between two variables

X- Min: The lowest point for _—Independent—— variable

X-Max: The highest point for —Independent———variable Y Min: The lowest point for __Dependent___ variable

Y Max: The highest point for ___Dependent_ variable

THESE NEED TO BE SET SO THE WHOLE MODEL (graph) CAN BE SEEN (PERTINENT).

negative

Positive

Quadratic Functions

For the following situations name the dependent and independent variables. Determine whether the model (graph) of the situation will have a maximum or a minimum. Determine what the zero between the first and fourth quadrants represents. Determine what window settings your calculator will need to see the whole graph.

The height in feet of a rock thrown upward at an initial speed of 64 ft/sec from a cliff 50 ft. above the ocean beach is given by the function s(t) = -16t2 + 64t + 50, where t is the time in seconds. Find the maximum height above the beach that the rock will attain.

Dependent variable: ________________

Independent Variable:

Max or Min: ______ Zero: _____

Window Settings: x min: ___ x max: __ y min: ___ y max:___

The height in feet of a ball thrown upward at an initial speed of 80 ft/sec from a platform 50 ft. high is given by the function s(t) = - 16t2 +80t + 50., where t is the time in seconds. Find the maximum height above the ground that the ball will attain.


Independent Variable:

Max or Min: ______ Zero: _____

Window Settings: x min: ___ x max: __ y min: ___ y max:____

A manufacturer of a camera lenses estimated that the average monthly cost (C) of a lens if given by the function C(x) = 0.1 x2 - 20x + 2000, where x is the number of lenses produced each month. Find the number of lenses the company should product in order to minimize the average cost.


Independent Variable: _______________

Max or Min: _______ Zero:_______

Window Settings: x min:______ x max:______y min:______ y max:_____

A pool is treated with a chemical to reduce the amount of algae. The amount of algae in the pool t days after the treatment can be approximated by the function A(t) = 40t2 – 400t + 500. How many days after treatment will the pool have the least amount of algae?



Max or Min: _______ Zero:_______

Window Settings: x min:______ x max:______y min:______ y max:_____

Go into y = and put the above equation in.

Go to windows and decide what the x min, x max, y min, y max should be.

How do you decide this? Well look at the last number – this is going to be your y-intercept so try going above that number for the y-max and leave the others.

Now go to graph to see if you can see the whole picture.

You can’t so decide where you need to go. You need to go further up on the y axis to see the turning point so make the y max larger. (hint: you could have found the axis of symmetry and gone to the table to find the y value and used that as your y max.( a little above it).

Key to the rest of the problems: Quadratic Functions


The height in feet of a rock thrown upward at an initial speed of 64 ft/sec from a cliff 50 ft. above the ocean beach is given by the function s(t) = -16t2 + 64t + 50, where t is the time in seconds. Find the maximum height above the beach that the rock will attain. 114 ft

Dependent variable: ___height s(t)_____________

Independent Variable: Speed (t)

Max or Min: _Max_____ Zero:Where the rock hits the ocean

Window Settings: x min: -10_ x max: 10__ y min: -10___ y max:__150_

The height in feet of a ball thrown upward at an initial speed of 80 ft/sec from a platform 50 ft. high is given by the function s(t) = - 16t2 +80t + 50., where t is the time in seconds. Find the maximum height above the ground that the ball will attain. 150 ft

Dependent variable: ____height s(t)____________

Independent Variable: Speed (t)

Max or Min: _Max_____ Zero: Where ball hits the platform_____

Window Settings: x min: -10___ x max: 10__ y min: -10___ y max:_250___

A manufacturer of a camera lenses estimated that the average monthly cost (C) of a lens if given by the function C(x) = 0.1 x2 - 20x + 2000, where x is the number of lenses produced each month. Find the number of lenses the company should product in order to minimize the average cost. 100 lenses

Dependent variable: ___cost c(x)_____________

Independent Variable: ___# produced per month (x)_

Max or Min: __Min_____ Zero:__none_____

Window Settings: x min:_-10_____ x max:_200_____y min:_-1000_____ y max:_2210____

A pool is treated with a chemical to reduce the amount of algae. The amount of algae in the pool t days after the treatment can be approximated by the function A(t) = 40t2 – 400t + 500. How many days after treatment will the pool have the least amount of algae?

Dependent variable: ___amount of algae A(t)___

Independent Variable: ___Days (t)_________

Max or Min: __Min_____ Zero:__where there is no algae____

Window Settings: x min:_-10_____ x max:__200 y min:__-10____ y max:_2000____ REVIEW GROUP WORK ANSWER KEY: 1) Find the inverse for: {(8,5), (6,8), (4,11), (2,14)} {(5,8), (8,6), (11,4), (14,2)}

2) Find the inverse for: y = 4x + 7 y = x ! 7

4

3) f(x) = x2 – 2x – 3; g(x) = x-1 what is (g(f(x)) x2 –2x – 4

4) g(x) = -2x3 – 5x2 + 4x –3 domain: -3 ≤ x ≤ 5 what is the largest value for the range? -108

5) g(x) = 3-2x does (1,-2) appear on its graph? NO

REVIEW GROUP WORK

1) Find the inverse for: {(8,5), (6,8), (4,11), (2,14)} 2) Find the inverse for: y = 4x + 7 3) f(x) = x2 – 2x – 3; g(x) = x-1 what is (g(f(x))

4) g(x) = -2x3 – 5x2 + 4x –3 domain: -3 ≤ x ≤ 5 what is the largest value for the range?

5) g(x) = 3-2x does (1,-2) appear on its graph?

HOMEWORK- DAY ONE

Quadratic Functions


The suspension cable that supports a small foot bridge hands in the shape of a parabola. The height in feet (h) of the cable above the bridge is given by the function h(x) = 0.025x2 - 0.8x + 25, where x is the distance in feet from one end of the bridge. What is the minimum height of the cable above the bridge?


Independent Variable: Max or Min: ________

Zero: ___________

Window Settings: x min: ___ x max: __ y min: ___ y max:

The net annual income of a family physician can be modeled by l(x) = - 290(x - 48)2 + 148,000, where x is the age of the physician and 27 < x < 70. Find the age at which the physician's income will be a maximum. What is the physician's maximum income?


Independent Variable: Max or Min: ________

Zero: ___________


Karen is throwing an orange to her brother Jim, who is standing on the balcony of their home. The height, h (in feet), of the orange above the ground t seconds after it is thrown is given by h(t) = - 16t2 + 32t + 4. If Jim's outstretched arms are 18 ft. above the ground, will the orange ever be high enough so that he can catch it?


Independent Variable:________________

Max or Min: ________________

Zero:____________________

Window settings: x min__________ x max____________ y min_____________ y max_____

Some football fields are built in a parabolic mound shape so that water will drain off the field. A model for the shape of the field is given by h(x) = -0.00023475x2 + 0.0375x, where h is the height of the field in feet at a distance of x feet from the sideline. What is the maximum height? Round to the nearest tenth.



Max or Min: ______

Zero: ______________

Window Settings: x min: ___ x max: __ y min: ___ y max: ______

The Buningham Fountain in Chicago shoots water from a nozzle at the base of the fountain. The height, h (in feet), of the water above the ground t seconds after it leaves the nozzle is given by h(t) = - 16t2 + 90t •»• 15. What is '"N the maximum height of the water spout to the nearest tenth of a foot?

Dependent variable: _____________________

Independent Variable: _____

Max or Min: __________

Zero: _________


On wet concrete, the stopping distance, s (in feet), of a car traveling v miles per hour is given by s(v) = 0.055v2 + 1.1v. At what speed could a car be traveling and still stop at a stop sign 44 ft away?

Dependent variable: _________________________


Max or Min: __________

Zero: _________

Window Settings: x min: _____ x max: __ y min: __ y max;______

HOMEWORK- DAY ONE

KEY

Quadratic Functions


The suspension cable that supports a small foot bridge hands in the shape of a parabola. The height in feet (h) of the cable above the bridge is given by the function h(x) = 0.025x2 - 0.8x + 25, where x is the distance in feet from one end of the bridge. What is the minimum height of the cable above the bridge?18.6

Dependent variable: _height(h)_______________

Independent Variable: Distance (x)

Max or Min: __Min

Zero: __None – cable on the ground

Window Settings: x min: -10___ x max: _30_ y min: -10___ y max:25

The net annual income of a family physician can be modeled by I(x) = - 290(x - 48)2 + 148,000, where x is the age of the physician and 27 < x < 70. Find the age at which the physician's income will be a maximum. What is the physician's maximum income? 48 years old

Dependent variable: ____net annual income__

Independent Variable: age of the physician

Max or Min: _max_______

Zero: __income is at 0_________

Window Settings: x min: _27__ x max: 70__ y min: _-10__ y max:150000

Karen is throwing an orange to her brother Jim, who is standing on the balcony of their home. The height, h (in feet), of the orange above the ground t seconds after it is thrown is given by

h(t) = - 16t2 + 32t + 4. If Jim's outstretched arms are 18 ft. above the ground, will the orange ever be high enough so that he can catch it? Yes because the orange will reach a max of 20 ft.

Dependent variable: ___height h(t)_____________

Independent Variable:____seconds (t)____________

Max or Min: ___max_____________

Zero: where the orange hits the ground

Window settings: x min___-10_______ x max__10___ y min___-10___ y max__30___

Some football fields are built in a parabolic mound shape so that water will drain off the field. A

model for the shape of the field is given by h(x) = 0.00023475x2 + 0.0375x, where h is the height of the field in feet at a distance of x feet from the sideline. What is the maximum height? Round to the nearest tenth. 1.5 ft

Dependent variable: __Height h(x)______________

Independent Variable: __feet from the side line (x)___

Max or Min: __max____

Zero: ___where the field is at water level_

Window Settings: x min: -80___ x max: 200__ y min: -10___ y max: _10_____

The Buningham Fountain in Chicago shoots water from a nozzle at the base of the fountain. The height, h (in feet), of the water above the ground t seconds after it leaves the nozzle is given by h(t) = - 16t2 + 90t + 15. What is the maximum height of the water spout to the nearest tenth of a foot? 141.56 ft

Dependent variable: _____height h(t)________________

Independent Variable: ___seconds (t)__

Max or Min: __Max________

Zero: __Water hits ground_______

Window Settings: x min: _-10__ x max: 10__ y min: -10___ y max:145

On wet concrete, the stopping distance, s (in feet), of a car traveling v miles per hour is given by s(v) = 0.055v2 + 1.1v. At what speed could a car be traveling and still stop at a stop sign 44 ft away? 154.88 mph

Dependent variable: _____Stopping distance S(v)____________________

Independent Variable: __MPH (v)_____________

Max or Min: __min________

Zero: ___Stopping distance is at 0______

Window Settings: x min: _-30____ x max: _10_ y min: _-10_ y max;__10____

Lesson Plan – Day Two Quadratic Functions/Inequalities

Class: Course III – Math B

Warm – up

Objectives: Use calculator to find critical values in a quadratic function

Materials: Each student needs their TI-83 Plus calculators, Overhead and TI-83 Plus overhead unit, Warm up worksheet on previous days work, Previous night’s homework assignment, Worksheet – Finding the Roots of an Equation, Worksheet – Finding the Min. or Max, Review Worksheet for group work

Anticipatory Set: 3 minute warm-up and go over previous night’s homework

Procedures: 10 minutes – Go over Homework and complete warm-up

10 minutes – Calculator Instruction – (give steps) Finding Min, Max, Roots, solutions of quadratics 10 minutes – Guided Practice using calculator to find the roots

10 minutes - Review Group work and hand out homework

Good Questions to Check for Understanding: What does zero mean? What does root mean? Where do we find roots? Describe relationships of quadratic function graphs.

Closure: Ask if there are any questions, hand out homework and explain it.

Homework: Finding min and max and roots worksheets


Course III – Math B

Name ________________________ Date__________________________________ A pool is treated with a chemical to reduce the amount of algae. The amount of algae in the pool t days after the treatment can be approximated by the function A(t) = 40t2 – 400 t + 500. How many days after treatment will the pool have the least amount of algae? Dependent variable: _________________________ Independent variable:________________________ Max or Min: _________________________ Zero: ________________________ Window Settings: x min:_________x max: ___ y min: ________ y max: ____________

Warm – up


Name _KEY_______________________ Date__________________________________ A pool is treated with a chemical to reduce the amount of algae. The amount of algae in the pool t days after the treatment can be approximated by the function A(t) = 40t2 – 400 t + 500. How many days after treatment will the pool have the least amount of algae? 5 days Dependent variable: ____amount of algae A(t)_____________________ Independent variable:______days (t)__________________ Max or Min: ____Min._____________________ Zero: ____Where there is no algae____________________ Window Settings: x min:___-10____ x max: 10 y min: __-500__ y max: ____10____

Notes: The place where the graph of a quadratic equation intersects the x-axis is called its solutions or roots. When a graph doesn’t cross the x-axis, it is said to have complex or imaginary roots. Practice solving equations using factoring and quadratic formula first:

3x2 +7x +2 = 0 3x2 = 2(x+2) (3x+1) (x + 2) = 0 3x2 = 2x + 4 3x+1 = 0 x+2 = 0 3x2 - 2x – 4=0 3x = -1 x = -2 need to use quadratic formula:

x = -1/3 xb b ac

a=! ± !

24

2

{-1/3, -2} therefore, the graph x =2 4 4 3 4

6

1

3

52

6

± ! != ±

( )( )

intersects the x-axis at –1/3 and –2. x= 1.54, -.869 therefore, the graph intersects the x-axis at 1.54 and -.869

Finding solutions (roots) using the calculator: 1.Press Y= and CLEAR to clear the Y1 prompt. Press ∇ and CLEAR to clear additional prompts. 2. Set the equation to zero. Enter your equation for y1. For example, enter 3x2 –2 = 0 for y1 by pressing 3 X,T,θ ,n x2 -2 ENTER. Press ZOOM 4 to view the equation in the decimal window. 3. Find where the equation is equal to zero by pressing 2nd CALC 2 (zero), set the lower bound for the calculation by moving the cursor to the left (above in this case) of the left zero by pressing β and then press ENTER. Set the upper bound for the calculation by Pressing α to move the cursor to the right (below in this case) of the zero and press ENTER ENTER. An accurate approximate for the left zero will appear at the bottom of the screen. Repeat

for the right zero. What does the zeros mean in a quadratic function? Ex) The height of an object is given by the function: h(t) = -16x2 + 4x + 52. Find the zero and tell me what it

means? X = 1.9 and at 1.9 sec. After release the object has 0 height or hits the ground (or water)

Describe relationships of a quadratic graph. H(t) H (height is a function of the time passed)

o Every point on the graph is (x,y) x = time, y = height

(t) When will an equation have a minimum value (from yesterday)? When the a-term is positive. When will an equation have a maximum value (from yesterday)?When the a-term is negative. The max. of a quadratic function that is a relationship between height & time is the time from the start(x=0 is the starting time) when the object reaches its max. height. Will this type of function have a negative or positive a – term? Negative Using the calculator to find min or max. 1. Enter the function and graph it as taught from yesterday. Use the example above.

2. Press 2nd calc.

3. Select Maximum for our example 4. Follow prompts on screen. GO to the left of the curve, then right and then a guess (similar to finding zeros)

Have the students try a few of finding roots and min and max. Same sheet as their homework. Use same sheet that says finding roots for finding min and max .If time permits do a couple of group work problems as well.

Answer Key to Finding the roots and min and max.

3. Roots: None 4. Roots: {-4,1} Min. (1,2) Min.(1-1.5,-6.3) 5. Roots: None 6. Roots: {-9,-2.4} Max: (1,-1) Max: (-.8,5.1) 7. Roots: {-2,0} 8.Roots: {-1.5,0) Min: (-1,-2) Max: (-.8,1.1) 9. Roots: None 10. Roots: None Max: (1,-3) Max: (.2,-5.9) 11. Roots {-2.9,1.4} 12. Roots: {-2,1} Min: (-.8,-9.1) Max: (-.5,2.3) 13. Roots: {-1.5,.5} 14.Roots: {.7,4.3} Min: (-.5,-2.75) Min: (2.5,-3.25) 15. Roots: None 16. Roots: {-2.7,.2} Min: (.7,-.7) Max: (-1.25,4.125) 17. Roots: {-.7} Min: (-.8,-.1) 23. 18.6 Feet 26 1.5 feet

Key to group review 1a. {(9,2), (5,4), (8,13), (-10,-1)}

2a. y = 5 30

2

x +

3a. 0 4a. –5x2 +82 5a. x >2 or x < -2 5b. all real numbers except 2

Lesson Plan – Day Three Quadratic Functions/Inequalities



Objectives: Sketch given graphs on paper given information from the calculator Solve systems of equations with quadratics

Materials: Each student needs their TI-83 Plus calculators, Overhead and TI-83 Plus overhead unit, Warm up worksheet on previous days work, Previous night’s homework assignment, Graph Paper, Amsco Course III Textbook, Worksheet - Solving Systems of Equations ,Review Worksheet for group work

Anticipatory Set: Go over previous day’s homework and complete warm-up

Procedures: 10 min – go over homework and complete warm-up 5 min – Instruction in graphing neat graphs 5 min – Guided Practice drawing graphs 5 min – Solving Systems of Equations 10 min – Guided Practice solving Systems of Equations p.670-671 #2,10,14,15,16 5 min – Group work Good Questions to Check for Understanding: Can anyone describe the procedure of graphing and solving systems of equations? How do you know your solutions are correct?


Homework: Graphing and Solving Systems of Equations Worksheet


Warm-Up

Find the roots of each equation by using the quadratic formula and check. 1. 2x2 +5 = 11x

2. x (x+4) =1 Answer key to warm- up

1. {5, .5}

2. x or=! ± !

±5 2 5

2

5

25 or { .24,-4.24}

Go through p. 670 showing how to graph and then how to solve the systems of equations. 2. What does it mean if the two graphs of a system do not intersect? No solutions

10. y = -x2 +4 and y = x + 2 [ students already know how to graph the line so just review this] Step 1: Using calculator graph the first equation

Step 2: We will now graph it on graph paper [you need at least 7 points so go to 2nd graph to see the table.

Find where the turning point is [ # where 3 above it and 3 below it are identical]. These are the points you will graph on your paper. Plot (-3,-5), (-2,6), (-1,3),(0,4) and so forth Step 3: ALWAYS LABEL YOUR GRAPH WITH THE ORIGINAL EQUATION. Step4: Now do you recall how to graph a line? What is your y-intercept? 2 What is your slope?1 Start at (0,2) and go up 1 to right 1 and draw a line and label it. (You can also put it in your calculator and go to the table at plot points that way)

Step 5: We need the solutions. So press 2nd Trace 5: Intersect

Place cursor on first curve, ENTER, place cursor on 2nd curve (2nd equation), ENTER, then place cursor where you think they intersect (guess)

You now have one solution (-2,0)Now do the same thing on the other side of your graph.

Second solution is (1,3) Step 6: Always substitute the numbers back in and check your solutions. You can also check the table if both

the y values are the same that is your solution

Have students try # 14 and #15 Answers should be: y=2-x2 and y = 2x + 4

14) NO solutions x2 + y2 = 16 and x-y=4

15) (0,-4) and (4,0)

xy = -6 and x+3y = 3

16) (-3,2) and (6,-1) If time you could have them do more practice problems from the Amsco book.

Homework – Day Three Course III – Math B

Solve the following systems of equations using any method you would like. Draw the graph on graph paper when indicated. 1)

2)

3)x2+y2=8 x+y = 4 Draw these graphs: 4) y = x2 –3 2x + y = -2

5) x2 + y 2 = 16 x – y = 4

KEY TO HOMEWORK – DAY THREE

1. solution: ( 1,0) solution: ( 4,3) 2. Solution: ( -4,-2) Solution: (2,4) 3. Solution: Solution: Solution: (2,2) 4.

Solution: (.414,-2.82) Solution: (-2.414,2.83) Or (-1 ± 2 , ± 2 2 ) 5.

Solution (4,0) and (0,-4)

Group Work

1. What are the domain and range of the function pictured below?

2. Give h(x) = x ! 5

a. Find the domain and range of h(x)

b. Is h(x) a one to one function? Why?

3. Which of the following would represent a number whose distance from 7 is 6 units?

a) x – 7 = 6 b) |x-7|=6 c) x-6 = 7 d) |x-6|=7

4. The average age of a student in Juanita’s class is 16 years old. However, the difference between a student’s age, ‘a’, and the average age 16, is half a year. Write an equation using absolute value that could be used to represent this information.

ANSWER KEY TO GROUP WORK

1. domain: { -8 ≤ x < 7} Range: {-8 ≤ y < 4} 2. Domain: x ≥ 5 Range: y > 0 b. Yes 3. b

4. |a-16| = ½

Lesson Plan – Day Four Quadratic Functions/Inequalities


Objectives: Review using the value of discriminant to determine the Nature of the Roots

Review finding the sum and product of roots; given a quadratic equation

Materials: Each student needs their TI-83 Plus calculators, Overhead and TI-83 Plus overhead unit, Warm up worksheet on previous days work, Previous night’s homework assignment, Amsco Course III Textbook, Review Worksheet for group work


Procedures: 10 min – go over homework and complete warm-up 5 min – Review Discriminant, Nature of Roots 5 min – Guided Practice p. 659 #2-12 even 5 min – Review Product and Sum of Roots 5 min – Guided Practice p. 666 #4-12 even 10 min – Guided Practice on Function Applications

Good Questions to Check for Understanding: When asked to state the number of and nature of roots you will use what? What is the formula for finding the sum of the roots and the product of the roots?


Homework: Group worksheet


37

Course III – Math B Warm-Up

What are the zeros for the following equations: 1. f(x) = x3 + 2x2 -1

2. f(x) = x2 + 10x + 18

38

Answers to warm – up

What are the zeros for the following equations: 1. f(x)=x3 + 2x2 –1

Solutions: x = -1 and .6180

2. f(x) = x2 + 10x + 18

Solutions: x = -7.6 and –2.35

39

Go through and solve x2 – 8x + 17 = 0 with the quadratic formula You get 4± i which means the roots are imaginary and conjugates For information about the roots of a quadratic equation: we can use the discriminant: b2 – 4ac b2 – 4ac > 0 and a perfect square the roots are real, rational and unequal b2 – 4ac > 0 and not a perfect square the roots are real, irrational and equal b2 – 4ac = 0 the roots are real, rational, and equal b2 – 4ac < 0 the roots are imaginary Give examples of each.

1. x2 – 2x – 3 = 0 16 so real rational and unequal 2. x2 – 4x +4 = 0 0 so real, rational and equal 3. x2 – 4x +5 = 0 -4 so imaginary Students try: 1. 2x2 +6x + 3 = 0 Real, Irrational and unequal 2. x2 + 3x – 5 = 0 Imaginary 3. x2 +3x +4 = 0 Imaginary 4. x2 – 10x +25 = 0 Real, rational and equal

Look at the graphs as well and have them describe what they see. Find the value(s) of K for which the given equation has one real double root.

4. 1) x2 +6x +k = 0 k = 9 check: x2 +6x– 2x +9 = 0 b2 – 4ac = 0 (x+3)(x+3) = 0 36 – 4k = 0 x = -3 -4k = -36

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Guided Practice : p . 659 #2- 12 even They have to evaluate the discriminant and tell the nature of roots.

2. x2 +3x – 10 = 0 real, rational, unequal

4. x2 –10x + 25 = 0 real, rational, equal

6. x2 + 7x – 30 = 0 real, rational, unequal

8. 2x2 +3x = 4 real, irrational, unequal

10. x2 +9 = 2x2 +x real, irrational, unequal

12. x2 +6 = 3x2 + x real, rational, unequal Now, talk about finding the sum and the product of the roots.

r1 + r2 = ! b

a

r1 • r2 = c

a

Try: Find the sum and product.

1) x2 – 2x -15 = 0 2/1 = sum -15 * 1 = product 2) 2x2 = 3x – 6 2x2 – 3x + 6 = 0 sum: 3/2 Product: 3 Have them try p. 666 # 4-12 even

4. 4x2 + x – 3= 0 sum: -1/4 Product: -3/4 6. 3x2 +9x = 2 Sum: -3 Product: -2/3 8. x2 +9 = 0 Sum: 0 Product: 9

10. 2m2 +2 = 5m Sum: 5/2 Product:1

12. y2 +2y = 2 Sum: -2 Product: -2 Give them group work:

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Group work:

7. Given: f(x) as shown in the accompanying diagram.

Find: a. f(2) b. x such that f(x) = 2 c. f-1 (2) d. the element in the domain that corresponds to 0 in the range

In 1-10, determine if each of the following relations is a function or not. Explain your answer. If the relation is a function, state its domain and range.

1. {(6, 3), (5, 8), (2, 7)}

2. {(5, 4), (5, 5), (5, 6)}

3. {(1, 2), (2, 2), (3, 2)}

4. y

6.

5.

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Answer key to group work:

1) yes, domain: {(6,5,2)} Range: {(3,8,7)} 2) not a function 3) yes domain: {(1,2,3)} Range: {(2,2,2)} 4) Not a function 5) Not a function 6) Not a function 7) a. 9 b. x = 1 c. 1 d. 1.5

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Lesson Plan – Day Five

Quadratic Functions/Inequalities Class: Course III – Math B

Objectives: Solve an inequality and graph on a number line

Materials: Each student needs their TI-83 Plus calculators, Overhead and TI-83 Plus overhead unit, Warm up worksheet on previous days work, Previous night’s homework assignment, Amsco Course III Textbook, Worksheet – Quadratic Inequalities, Review Worksheet for group work


Procedures: 10 min – go over homework and complete warm-up

10 min – Instruction on solving quadratic inequalities 10 min – Guided Practice – p. 677 #10-20 even, p. 264 #1,3,9,10,13,16

10 min. Functions Chapter Review Good Questions to Check for Understanding: What does critical value mean?


Homework: Worksheet


44


Warm-Up

For 1 – 3 Use 4x = 12 9x

x

!

1. The roots of this quadratic equation are:

a. Imaginary b. Real, rational, unequal c. Real, rational, equal d. Real, irrational, unequal

2. The sum of the roots is

3. The product of the roots is Answer key to warm-up

1. c 2. 3 3. 9/4

45

Solving Quadratic Inequalities Notes

We are looking for the points (zeros) on the x-axis that used to be solutions, but with inequalities they are ‘critical values’ One side of the critical value is a solution when we substitute the value for x or (t) the inequality is true, on one side of the critical value and false on the other. Ex) x2 – 2x – 3 < 0 Note: When we have an inequality with an exponent greater than one, we can solve it like an equation.

1) factor 2) quadratic formula if needed 3) calculator

x2 – 2x – 3 < 0 Factor x2 – 2x – 3 =0 Change < to = to find critical values, set = 0 & solve (x – 3) ( x+1) = 0 x = 3 x = -1 -1 0 3 Now graph : Ask yourself open circle or closed Which way to shade? Do a test point. Substitute a value that is between the two critical values. (0)2 – 2(0) – 3 < 0 Is this true? -3 < 0 Yes so shade in to include 0 Now write it in the correct notation. {x|-1<x<3} Now have students try p. 677 #10-20 even 10. 2x2 – 8 > 0 {x|x<-2 or x>2)} 12. x2 + 7x ≤ 0 {x| -7 ≤ x ≤ 0} 14. x2 – x – 6 > 0 {x|x<-2 or x>3}

46

16. x2 +8x + 7 ≥ 0 {x | x≤ -7 or x ≥ -1} 18. x2 +2x < 15 {x| -5 < x < 3} 20. 4x2 – 9 ≥ 0 {x |x ≤-3/2 or x ≥ 3/2} Hand out hw sheet

47

Course 3 Worksheet - Quadratic Inequalities

Graph the solution set of:

1. x 2 - 4 < 0 2. x 2 - 2 5 > 0

3. x2 - 4x ≥ 0 4. x2 - 7x + 10 < 0

5. x2 - 5x - 24 ≤ 0 6. x2 > 8x + 20

7. x2 + 27 < 12x 8. 2x2 - 11x + 5 ≥ 0

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Answer key to hw. 1. 2. -2<x<2 x<-5 or x > 5

3. 4. x ≤ 0 or x ≥ 4 2 < x < 5

5. 6. -3 ≤ x ≤ 8 x< -2 or x > 10

7. 8. 3 < x < 9 x ≤ .5 or x ≥ 5

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Functions Chapter Review – Group work

Felicia left home at 7:30 A.M. to walk to school. On the way, she stopped at her friend's house and had a glass of orange juice. Five minutes after she arrived, Felicia and her friend continued on to school, arriv-ing at 8:00 A.M. As soon as she got to her locker, Felicia realized that she left her book bag at her friend's house, and ran all the way back to pick it up, and then ran back to school. She got to school at 8:19 A.M., just in time for the bell, which rang at 8:20 A.M. Sketch a graph which would represent the distance, d(t), Felicia traveled from her home as a function of time, t. (Let t = 0 represent the time Felicia left her house.)

Given: h(x) = x ! 5 . a. Find the domain and range of h(x). b. Is h(x) a one-to-one function? Why?

Given f(x) and g(x) as pictured in the accompanying diagrams.

a. Find f((g(-2)). b. Find(g°f)(3). c. Find f-1(-3)

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In 23-28, select the numeral preceding the word or expression that best completes the statement or answers the question.

23. The function g(t) is defined as g(t) = t2 — 4 with the domain -6 ≤ t≤ 2. What is the smallest element in the range? (1) -6 (3) 0 (2) -4 (4) 32

24. Which of the following values is not in the domain of y = x x22 24! ! ?

(1) -5 (3) 6 (2) 2 (4) 8

25. If s(x) =| 3x - 4 |, then s(0) = (1) -4 (3) 2 (2) 0 (4) 4

26. What is the maximum value of the function y = -x2 + 3? (1) 0 (2) 2 (3) 3 (4) It has no maximum value.

27. For what values of x will the function f(x) — x + 4 be real? (1) {x \ x ≥ 0} (3) [ x \ x≤ 4} (2) {x | x ≥ 4} (4) {x I x ≥ -4)

28. For which value of x is the function g(x) = x

x

+

!

1

5 undefined?

(1) -5 (2) -1

In 19-22, find the inverse of each function.

19. {(a, b), (c, d), (e,f), (g, h)} 20. {(2, 8), (3, 27), (4, 64), (5,125)} 21. h(x) = 2 1

3x +

22. r(t) = ¾ t - 5

(3) 0 (4) 5

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D(t) Answer Key to group work: a. Domain x> 5 and Range y>0 b. Yes

a. –3 b. 2 c. 4

(t) 23. 3 24. 2 25. 4 26. 4 27. 4 28. 4 19. { (b,a), ( d,c), ( f,e) , ( h,g) } 20. { ( 8,2) , ( 27, 3) , ( 64, 4) , ( 125, 5) }

21. x31

2

!

22. 4 20

3

t +

52

Lesson Plan – Day Six Quadratic Functions/Inequalities


Objectives: Apply quadratic Inequalities to real life situations

Materials: Each student needs their TI-83 Plus calculators, Overhead and TI-83 Plus overhead unit, Warm up worksheet on previous days work, Previous night’s homework assignment, Application worksheet

Anticipatory Set: Go over previous day’s homework

Procedures: 10 min – go over homework

10 min – Work through guided practice questions slowly 10 min – Guided Practice

10 min. Review unit

Good Questions to Check for Understanding:


Homework: STUDY FOR UNIT TEST


53

Quadratic Inequalities - Applications

Island Rent-a-Car charges a car rental fee of $40 per day plus $5 per hour or fraction of an hour. Wayne's Wheels charges a car rental fee of $25 plus $7.50 per hour or fraction of an hour. Under what conditions does it cost less to rent from Island Rent-a-Car? A homeowner wants to increase the size of a rectangular deck that now measures 15 feet by 20 feet, but building code laws state that a home-owner cannot have a deck larger than 900 square feet. If the length and the width are to be increased by the same amount, find, to the nearest tenth, the maximum number of feet that the length of the deck may be increased in size legally. When a baseball is hit by a batter, the height of the ball, h(f), at time t, t > 0, is determined by the equation h(t) = -I6t2 + 64t + 4. For which interval of time is the height of the ball greater than or equal to 52 feet?

54

Answer key; Island Rent-a-Car charges a car rental fee of $40 per day plus $5 per hour or fraction of an hour. Wayne's Wheels charges a car rental fee of $25 plus $7.50 per hour or fraction of an hour. Under what conditions does it cost less to rent from Island Rent-a-Car?

A homeowner wants to increase the size of a rectangular deck that now measures 15 feet by 20 feet, but building code laws state that a home-owner cannot have a deck larger than 900 square feet. If the length and the width are to be increased by the same amount, find, to the nearest tenth, the maximum number of feet that the length of the deck may be increased in size legally. 15 * 2- = 300 sq. feet (15 + x) ( 20+x) < 900 300 + 35x + x2 < 900 x2 + 35 x – 600 < 0 12.6 Feet max. # of feet the deck may be increased by

55

When a baseball is hit by a batter, the height of the ball, h(f), at time t, t > 0, is determined by the equation h(t) = -I6t2 + 64t + 4. For which interval of time is the height of the ball greater than or equal to 52 feet? 1 ≤ t ≤ 3

Ask students if they have any questions with the test prep. Tell them to study and utilize the calculator.

exploring quadratic …math.buffalostate.edu/~it/projects/markle.pdfsolve systems of equations ......

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