QUADRATIC FUNCTIONS AND EQUATIONS Student Pages for …Solving Quadratic Equations 1 2.1 The Square...

Name ___________________________ Period __________ Date ___________

Quadratic Functions and Equations Unit (Student Pages) QUAD2 – SP

QUAD2.1 The Square Root Property • Solve quadratic equations using the square root property • Understand that if a quadratic function is set equal to zero, then

the result is a quadratic equation whose roots are equal to the x-intercepts of the function.

1

QUAD2.2 The Zero Product Property • Solve quadratic equations using the zero product property • Understand that if a quadratic function is set equal to zero, then

the result is a quadratic equation whose roots are equal to the x-intercepts of the function.

10

QUAD2.3 The Rabbit Pen • Model a situation using mathematics • Use quadratic functions and equations to solve a problem.

16

QUAD2.4 Vocabulary, Skill Builders, and Review 22

QUAD2 STUDENT PAGES

QUADRATIC FUNCTIONS AND EQUATIONS Student Pages for Packet 2: Solving Quadratic Equations 1

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http://en.wikipedia.org/wiki/Quadratic_equation


Solving Quadratic Equations 1

Quadratic Functions and Equations Unit (Student Pages) QUAD2 – SP0

WORD BANK (QUAD2)

Word Definition Example or Picture

difference of two squares

maximum point

minimum point

perfect square trinomial

quadratic equation

quadratic regression

square root property

zero product property

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Solving Quadratic Equations 1 2.1 The Square Root Property


THE SQUARE ROOT PROPERTY

Ready (Summary) We will learn to use the square root property to solve certain quadratic equations. We will connect this procedure to some special factoring cases and to the graph of quadratic functions.

Set (Goals) • Solve quadratic equations using the

square root property • Understand that if a quadratic function is

set equal to zero, then the result is a quadratic equation whose roots are equal to the x-intercepts of the function.

Go (Warmup)

Solve each equation. 1. x – 16 + 2x – 17 = 0 2. -4 + 2x + 8 = 3(x + 3) – (x + 5)

3. x – 9 = x + 9 4. 1 3 2 22 4

x x− = +

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RADICALS REVISITED

A square root of a number n is a number whose square is equal to n, that is, a solution of the equation 2 = x n . The positive square root of a number n, written n , is the positive number whose square is n. The negative square root of a number n, written - n , is the negative number whose square is n.

Estimate each expression as a decimal to the nearest tenth. 1. 30 2. 5 30+ 3. 5 30−

4. 6 5. 8 6− + 6. 8 6− −

A (square root) radical expression is considered simplified if: 1. There are no perfect squares under the radical. 2. There are no fractions under the radical. 3. There are no radicals in the denominator.

Simplify each expression. 7. 25 8. 25± 9. 25−

10. 2

(25)− 11. 36 64+ 12. 36 64+

13. 20 14. 34

15. 23

16. −9 4 17. 4 8+ 18. 4 82+ Sam

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FACTORING: SPECIAL CASES REVISITED Factor. Look for patterns. 1. a. x2 + 10x + 25 = __________________

b. x2 – 14x + 49 = _________________

c. x2 + 2x + 1 = __________________

d. x2 – 6x + 9 = __________________

2. a. x2 – 25 = ________________ b. x2 – 49 = ________________ c. x2 – 1 = _________________

d. x2 – 9 = __________________

Describe the pattern and write it in a general form. This pattern is called a __________ __________ ___________

Describe the pattern and write it in a general form. This pattern is called the _____________ ___ ______ _________

Factor the following using any method. Put a star by the perfect square trinomials. Put a happy face by the difference of two squares. 1. x2 + 10x + 25 2. x2 – 16

3. x2 + 13x + 36 4. x2 – 8x + 16

5. x2 – 81 6. x2 + 6x – 16

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EXPLORING THE SQUARE ROOT PROPERTY 1. Fill in the t-table and draw the graph for this quadratic function.

2. Write the x-intercepts for the graph as coordinate pairs. 3. If y = x2 – 9 and y = 0, then 0 = x2 – 9 (which can be written: x2 = 9).

What value(s) of x make this equation true?

(_______)2 = 9

_______ = _______

(_______)2 = 9

_______ = _______

4. How are the zeros (sometimes called solutions) of quadratic equation and the graph of the quadratic function related?

x y = x2 – 9

4

3

2

1

0

-1

-2

-3

-4

y

x

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EXPLORING THE SQUARE ROOT PROPERTY (continued) 1. If x2 = 9, then x = _______ or x = _______ (sometimes written x = _______ ) 2. If x2 = 49, then x = _______ or x = _______ (sometimes written x = _______ ) 3. If x2 = a2, then x = _______ or x = _______ (sometimes written x = _______ )

The square root property states that if x2 = a2, then x = a±

4. Build the equation x2 + 10x + 25 = 100 using algebra pieces. Arrange both sides of the equation into squares. Then use the square root property to solve this quadratic equation:

Write the equation. x2 + 10x + 25 = 100

Write the quadratic polynomial as a perfect

square.

(_________)2 = 100

Apply the square root property. (_________) = 10±

Solve for x.

_______ = 10 or _______ = -10 x = _____ or x = _____

Check both solutions in the original equation.

For x = _____ , check:

(___)2 + 10(____) + 25 = 100


(___)2 + 10(____) + 25 = 100

Remember: two solutions ( )±

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EQUATION SOLVING PRACTICE 1

1. Restate the square root property in your own words.

Solve. Check your solution(s) in the original equation. 2. x2 = 9 3. (x + 3)2 = 9

4. (x – 10)2 = 81 5. x2 + 6x + 9 = 64

6. 3x + 1 = 121

7. x2 + 4x + 4 = 36

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MORE ABOUT THE SQUARE ROOT PROPERTY The square root property applies when constant expression is not a perfect square. 1. If x2 = 5, then x = _______ or x = _______ (sometimes written x = _______ ) 2. If x2 = 20, then x = _______ or x = _______ (sometimes written x = _______ ) 3. If x2 = a, then x = _______ or x = _______ (sometimes written x = _______ )

Alternate Statement of the Square Root Property The square root property states that if x2 = a, then x = .a±

Example: Solve 2 ( 2) 12x + =

a. Find solutions in simplest radical form.

+ =

+ = ±

= − ±

= − ±

= − ±

= − ±

= − +

2 ( 2) 12

2 12

2 12

2 4 3

2 4 3

2 2 3

2 2 3 or

x

x

x

x

x

x

x x = − −2 2 3

b. Estimate solutions as decimals. We know that 2 12x = − ± 9 12 16< <

3 12 4< < An estimate for 12 is 3.5

If = − ±2 3.5x , then

x ≈ −5.5 or x ≈1.5

Solve. Write solutions in simplest radical form and estimate solutions as decimals. 4. x2 = 7

5. (x + 3)2 = 28

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EQUATION SOLVING PRACTICE 2

Solve. Check your solution(s) in the original equation.

1. (x + 4)2 = 16 2. x2 - 11 = -11

3. (x – 7)2 = 50 4. x2 + 2x + 3 = 2

5. x2 - 7 = 0 6. -3(x – 7) = 2(x + 6) + x + 9

7. (x + 3)2 = 14

8. x2 = -9

9. Describe the kind of equations that can be solved using the square root property. Sam

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FIND THE MISTAKE

Jenna tried to solve this equation and explain what she did, but she thinks she made a mistake. Put an X next to step that is incorrect. Jenna’s Work Explanations

1. x2 + 6x + 7 = 18 +2 +2

original equation

2. addition property of equality

3. x2 + 6x + 9 = 20 arithmetic

4. (x + 3)2 = 20 factor left side

5.

x + 3( )2 = 20 Take square root of both sides

6. (x + 3) = 20 arithmetic

7. x + 3 = 2 5 arithmetic

8. x + 3 = 2 5 – 3 – 3

arithmetic

9. multiplication property of equality

10. x = -3 + 2 5 arithmetic

11. x ≈5.8 arithmetic

Now rework the problem above, beginning from the step where the first error occurred. Your corrected work Explanations

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Solving Quadratic Equations 1 2.2 The Zero Product Property


THE ZERO PRODUCT PROPERTY

Ready (Summary) We will learn to use the zero product property to solve certain quadratic equations by factoring. We will connect this procedure to the graphs of quadratic functions.

Set (Goals) • Solve quadratic equations using the

zero product property • Understand that if a quadratic function is

set equal to zero, then the result is a quadratic equation whose roots are equal to the x-intercepts of the function.

Go (Warmup)

Solve each equation. 1. x2 = 144 2. (x – 5)2 = 4 3. x2 + 8x + 16 = 25

4. Factor x2 + 2x – 3 using any method.

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EXPLORING THE ZERO PRODUCT PROPERTY 1. Fill in the t-table and draw the graph for this quadratic function.

2. Write the x-intercepts for the graph as coordinate pairs. 3. If y = x2 + 2x – 3 and y = 0, then 0 = x2 + 2x – 3 (which can be written x2 + 2x – 3 = 0).

Write the left side of this quadratic equation in factored form.

(_________)(_________) = 0

4. What value(s) of x make this equation true? Why?

5. How are the solutions (also known as zeros) of a quadratic equation and the graph of the quadratic function related?

x y = x2 + 2x – 3

4

3

2

1

0

-1

-2

-3

-4

y

x

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EXPLORING THE ZERO PRODUCT PROPERTY (continued)

6. 5 • _____ = 0 7. _____ • 9 = 0 8. 0 • _____ = 0

The zero product property states that if ab = 0, then a = 0 or b = 0.

9. If ab = 0, then what do we know about a and b? 10. Use the zero product property to solve this quadratic equation: x2 – x = 2

Write the equation. x2 – x = 2

Rewrite so that the quadratic polynomial is

equal to zero. _____ – _____ – _____ = 0

Factor the polynomial, (_________)(_________) = 0

Apply zero product property. ________ = 0 or ________ = 0

Solve for x. x = ________ or x = ________

Check both solutions in the original equation.


(___)2 – (____)= 2


(___)2 – (____)= 2

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EQUATION SOLVING PRACTICE

1. Restate the zero product property in your own words. Solve using the zero product property. 2. (x – 5)(x + 4) = 0 3. (x + 1)(x + 6) = 0

4. x2 + 9x + 14 = 0 5. x2 + 2x – 15 = 0

6. x2 – 9x = -20 7. x2 = 8x + 48

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EQUATION SOLVING PRACTICE (continued) Solve each equation. Placing them in factored form first may be helpful.

8. x2 + 11x + 24 = 0 9. x2 – 25 = 0

10. x2 – 10x = -21 11. x2 + 9x = 22

12. x2 – 8x = 0

13. x = 3x + 40

14. Describe the kind of equations that can be solved using the zero product property. Sam

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FIND THE MISTAKES 1. Matt tried to solve the equation below. Circle the step where he made an error. Then rework the problem below, beginning from the first step in which an error is found.

Matt’s Work Corrected work

1. x2 + 8x + 12 = -3

2. (x + 2)(x + 6) = -3

3. x + 2 = -3 or x + 6 = -3

-2 -2 or -6 -6

4.

5. x = -5 or x = -9

Explain Matt’s error. Do you promise to never do this? 2. Blakely tried to solve the equation below. Circle the step where he made an error. Then rework the problem below, beginning from the first step in which an error is found.

Blakely’s Work Corrected work

1. x2 + x - 12 = 0

2. (x + 4)(x – 3) = 0

3. x = 4 or x = - 3

Explain Blakely’s error. Do you promise to never do this?

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Solving Quadratic Equations 1 2.3 The Rabbit Pen


THE RABBIT PEN

Ready (Summary) We will use our knowledge of length, area, and quadratic functions and equations to find the largest area possible, given a fixed perimeter.

Set (Goals) • Model a situation using mathematics. • Solve a problem involving maximization. • Use quadratic functions and equations

to solve a problem.

Go (Warmup)

Find the missing measurements for each rectangle. (Note: Diagrams are not to scale.)

1. A = ______ P = ______

2.

A = _____ P = _____

3.

A = _____ P = ______

4. A = 36 yd2 P = ______

5. A = ______ P = 16 mm

6.

A = 5x cm2 P = ______

7 cm

7 cm

9 in

3 in

6 ft

6 yd

2 mm

5 cm

x ft

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MAKING A RABBIT PEN

1. You have 40 feet of wire mesh to make a rectangular rabbit pen. Sketch several possible rectangles that you could create. Label the length and width of each. (Let L be the length of the horizontal side and W be the length of the vertical side.)

2. For each rectangle, what do you notice about L+W? 3. What is the maximum area that you found for a rabbit pen? Sam

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THE RABBIT PEN: LENGTHS AND AREAS 1. Use the pictures and data you created on

the previous page to complete the table at the right. Recall that you have 40 feet of wire mesh. Complete the table in any order.

2. Write a formula to find area of a rectangle:

A = ________________ 3. Write two different formulas to find

perimeter of a rectangle:

P = _________________

P = _________________ 4. What is the perimeter of each rabbit pen? 5. What must be the sum of one length and

one width?

L + W = ______.

Length in ft. (L)

Width in ft. (W)

Area in sq. ft (A)

0

2

4

6 14 84

8

10

12

14

16

18

20

L

6. What is strange about the first row of the table? In other words, what seems problematic

about the area measurement associated with L = 0? 7. Why can there not be a rectangle with L = 22? 8. What is the maximum area that you found in the table? _______ 9. What is special or different about this rectangle?

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Wid

th

Length

RABBIT PENS: WIDTH VS. LENGTH 1. Graph the data for width vs. length from the previous table. Scale the axes appropriately. 2. Find an equation for the graph using your

knowledge of linear functions. 3. Find a function for the graph using known

facts about the rabbit pen.

a. Write a general formula for the perimeter of a rectangle.

b. Substitute the known value for

perimeter for the rabbit pen. c. Solve for W in terms of L.

4. Do the two equations agree? Explain. 5. Interpret your graph.

a. What is the domain?_____________ What is the range?_____________

b. Why are they restricted?

c. What is the x-intercept? _________ What is the y-intercept?____________

d. What do these points represent in the context of the rabbit pen problem? Sam

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Are

a

Length

RABBIT PENS: AREA VS. LENGTH 1. Graph the data for area vs. length from the previous table. Scale the axes appropriately.

2. Find a function for the graph using known

facts about the rabbit pen.

a. First recall the formula for W in terms of L from the previous page.

b. Write the general formula for the area of a rectangle.

c. Make a substitution for W so that your

function for area is in terms of L. (This expression will be in factored form.)

d. Write this function as a sum of terms in standard form.

3. Find a function for the rabbit pen data using a graphing calculator or a web-based quadratic

regression application. 4. Do the two functions for area vs. length agree? Explain.

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RABBIT PENS: AREA VS. LENGTH (continued) 5. Interpret your graph.

a. What is the domain?_____________ What is the range?_____________ Why are they restricted?

b. What are the x-intercepts ? _____________ What is the y-intercept?________ What do these points represent in the context of the rabbit pen problem?

c. Is there a maximum or a minimum?__________________ If so, what is it, and what does it represent in the context of the rabbit pen problem.

d. What (if any) of these points is the vertex? 6. Explore the quadratic equation obtained by setting the quadratic function equal to zero.

a. Write this quadratic equation in factored form.

b. Find the solutions to this quadratic equation.

c. Interpret the solutions to the quadratic equation in the context of the rabbit pen problem and the area vs. length graph.

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Solving Quadratic Equations 1 2.4 Vocabulary, Skill Builders, and Review


FOCUS ON VOCABULARY (QUAD2) Match each phrase with its explanation or example. _______1.

square root property a. An equation of the form + + =2 0ax bx c

_______2.

zero product property b. A coordinate pair that has the greatest y-value for a function

_______3.

difference of two squares c. if =2 2x a then = ±x a .

_______4.

perfect square trinomial d. 2 9 ( 3)( 3)x x x− = + −

_______5.

maximum point of a function e. A process by which an equation for a parabola that best fits data is found

_______6.

minimum point of a function f. another name for x-intercept

_______7.

quadratic equation g. A coordinate pair that has the least y-value for a function.

_______8.

quadratic regression h. if = 0ab , then = 0a or = 0b .

_______9.

root of an equation i. − + = −2 26 9 ( 3)x x x

______10.

zeroes of a function j. a solution of an equation

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SKILL BUILDER 1 Simplify each expression. If it is already in simplest form, circle the expression. 1. 9 2. 23 3. 3 • 3 4. 2( 3 )

5. 3 6. x + 3 7. 16 25

8. 1625

9. 11 25

10. 1125

11. 115

12. 2511

13. 7 – 54

14. 5− 15. -2 + 9 4

16. -2 – 9 4

Estimate each expression to the nearest tenth.

17. 20 18. 5 20 + 19. 5 20 − 20. 5 20

2−

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x

y

SKILL BUILDER 2

Fill in the t-table and draw the graph for each equation.

1.

x y = x2 + 2 y = x2 – 2

2

1

0

-1

-2

2. How are the parabolas on the right similar?

How are they different?

It’s time to review your skills in scientific notation from earlier this year. Your future science teacher is going to appreciate your effort here! Make sure your answer is in proper scientific notation. An example is done for you.

Written in Proper Scientific Notation

Written without Scientific Notation

Is this number really small or really large?

Ex: (21050) ÷ (41040) 2.41014 240,000,000,000,000 Really Large

3. (5107)(3106)

4. (210−17)(4.1106)

5. (2.310−2)(810−5)

6. (21050) ÷ (41040) Sample

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x

y

SKILL BUILDER 3

Fill in the t-table and draw the graph for each equation. 1. 2.

x y = 3x2 y = 3x2 + 4

2

1

0

-1

-2

3. Multiply (x – 6)(x + 9)

Simplify. 4. 2(x2 – x) – 5(x – 2)

5. 3(x + 1)(x + 12) – 4x(x – 7)

Factor. 6. x2 – 11x + 18 7. x2 – x – 42

8. x2 – 100 9. 2 + 6 + 5 • 5 x x Sam

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x

y

SKILL BUILDER 4

Fill in the t-table and draw the graph for each equation. 1.

x y = 2x2 y = 12

x2

2

1

0

-1

-2

2. How are the parabolas on the right similar? How are they different?

Solve for x. 3. (x – 1)2 = 81

4. x2 – 2x – 15 = 0

5. x2 + 16x + 64 = 9

6. x 2 = -6x + 40

7. x 2 – 18 = -x – 6 8. x 2 + 4x = 4(x + 4) Sam

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SKILL BUILDER 5 Which of the following input-output tables illustrate functions? If NO, state why not. 1.

x (input) -3 -2 -1 0 1 2 3 y (output) 12 7 4 3 4 7 12

2.

x (input) 3 2 1 0 1 2 3 y (output) 9 4 1 0 -1 -4 -9

Which of the following ordered pairs illustrate functions? If NO, state why not. 3. (1,1), (2,2), (3,3), (4,4)

4. (10,8), (-10,-8), (8,10), (-8,-10) 5. (0,10), (0.5,10), (1,10), (1.5,10) Which of the following mapping diagrams illustrate functions? If NO, state why not.

0 6 2 4

8 10 12

5 5 7

-2 6 -9 11

6. 7.

8. 9.

a b c

a d b c

-1 -3 -5

-13 -1 -7 Sam

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SKILL BUILDER 6

On each graph below, the x-axis is the horizontal axis and the y-axis is the vertical axis. Which of the following graphs illustrate functions of x? If it is not a function, state why not. If it is a function, state the domain and range. 1. 2.

3. 4.

5. Sketch a nonlinear graph that

represents a function 6. Sketch a linear graph that does NOT

represent a function

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x

y

SKILL BUILDER 7

1. Graph the lines y1 = − 12

x – 2 and y2 = 2x + 4 using two different colored pencils.

2. Find the intercepts for each function.

y1 = − 12

x – 2 y2 = 2x + 4

x-intercept ( _____ , _____ ) ( _____ , _____ )

y-intercept ( _____ , _____ ) ( _____ , _____ )

3. Graph a parabola by multiplying y-coordinates from the lines. Use a third color. 4. Find an equation for the parabola. y3 = y1 • y2 = _______________

5. Identify the following coordinates for the parabola:

a. x-intercept(s)

b. y-intercept(s)

c. vertex

6. Is the vertex a minimum or a maximum point? What are its coordinates?

7. What is the domain of the parabola’s function?

8. What is the range of the parabola’s function? Sam

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SKILL BUILDER 8

Write the following numbers without exponents.

1. 82 2. (-8)2 3. -82

Evaluate the following expressions for a = 2, b = -5, and c = -3.

4. -b 5. a - c 6. a − c

b

7. b2

8. - 4ac

9. b2 – 4ac

10. 12 ± −20b

4

11. 12 ± −20b

4

12. b2 ± −20b

4

13. Joey and Maria are trying to solve x2 – 9 = 0.

Joey’s work Maria’s work

x2 – 9 = 0

x2 = 9

x = ±3

x2 – 9 = 0

(x – 3)(x + 3) = 0

x – 3 = 0 x + 3 = 0

x = 3 x = -3

What property did Joey use in his work to solve the equation?

What property did Maria use in her work to solve the equation?

Whose method would you use to solve x2 – 16 = 0? _____________ Why? Sample

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TEST PREPARATION (QUAD2) Show your work on a separate sheet of paper and choose the best answer. 1. Circle the letter that represents the complete solution(s) to: (x + 3)2 = 17

A. x = 3 + 17 B. x = ± 14 C. 3 ± 17 D. None of these

2. Which quadratic function has a graph with x-intercepts at x = -3 and x = 5?

A. f(x) = x2 + 2x + 15 B. f(x) = x2 + 8x + 15

C. f(x) = x2 - 2x - 15 D. f(x) = x2 - 8x + 15

3. Circle the statement(s) that are NOT true about the function f(x) = x2 + 7x – 8.

A. The factored form is f(x) = (x + 8)(x – 1). B. The graph of f(x) has an x-intercept when x = -8.

C. The graph of f(x) has an x-intercept when x = 8.

D. The graph of f(x) has an x-intercept when x = 1.

4. Circle all of the following expressions that are perfect trinomials? A. x2 + 4x + 8 B. x2 + 8x + 16 C. x2 + 6x + 9 D. x2 – 12x + 144

5. Which of the following expressions represents the zero product property rule?

A. a + 0 = a B.

a0

is undefined.

C. If ab = 0, then a = 0 and/or b = 0. D. If ab = 6, then a = 2 and b = 3.

6. Farmer Wayne has 100 feet of fencing to use for his pigpen. What size and shape

should he make his pen to maximize the area?

A. A rectangle with a length of 30 feet and a width of 20 feet.

B. A square with sides of 50 feet.

C. A square with sides of 25 feet. D. A rectangle with a length of 70 feet and a width of 30 feet. Sam

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KNOWLEDGE CHECK (QUAD2)

Show your work on a separate sheet of paper and write your answers on this page. QUAD 2.1 The Square Root Property 1. Solve using the square root property: x2 = 25 2. Solve using the square root property: (x - 3)2 = 50 3. Solve using the square root property: x2 + 10x + 25 = 100

QUAD 2.2 The Zero Product Property 4. Solve using the zero product property: (x + 3)(x – 4) = 0 5. Solve using the zero product property: x2 + 10x = 11

QUAD 2.3 The Rabbit Pen 6. Given a fixed perimeter, what kind of quadrilateral has a maximum area?

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HOME-SCHOOL CONNECTION (QUAD2) Here are some questions to review with your young mathematician. Solve the following equations using the square root property: 1. (x + 7)2 = 81 2. x2 – 6x + 9 = 49 Solve the following equations using the zero product property: 3. (x + 7)(x – 8) = 0 4. x2 + 2x = 8 5. At what values of x does the graph of f(x) = x2 + 3x – 4 cross the x-axis? Parent (or Guardian) signature ____________________________

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COMMON CORE STATE STANDARDS FOR MATHEMATICS A-SSE-1b* Interpret expressions that represent a quantity in terms of its context: Interpret complicated

expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P.

A-SSE-3a* Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression: Factor a quadratic expression to reveal the zeros of the function it defines.

A-CED-2* Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

A-CED-3* Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

A-CED-4* Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.

A-REI-1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

A-REI-4b Solve quadratic equations in one variable: Solve quadratic equations by inspection (e.g., for x2= 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.

F-IF-4* For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

F-IF-5* Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.

F-IF-7a* Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases: Graph linear and quadratic functions and show intercepts, maxima, and minima.

F-IF-8a Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function: Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

STANDARDS FOR MATHEMATICAL PRACTICE

MP1 Make sense of problems and persevere in solving them. MP2 Reason abstractly and quantitatively. MP3 Construct viable arguments and critique the reasoning of others. MP4 Model with mathematics. MP5 Use appropriate tools strategically. MP6 Attend to precision. MP7 Look for and make use of structure. MP8 Look for and express regularity in repeated reasoning.

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QUADRATIC FUNCTIONS AND EQUATIONS Student Pages for …Solving Quadratic Equations 1 2.1 The Square...

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