chapter 9: quadratic equations and functionsteachers.sduhsd.net/tsargent/images/alg...

Sargent Fall 2010 Algebra 1 Notes: Chapter 9

Name:

Date:__________ Period: __________

CHAPTER 9: Quadratic Equations and Functions

Notes #23

9-1: Exploring Quadratic Graphs

A. Graphing 2y ax

A ____________________ is a function that can be written in the form 2y ax bx c where

a, b, and c are real numbers and a 0.

Examples: 25y x 2 7y x 2 3y x x

The graph of a quadratic function is a U-shaped curve called a ________________. When

graphed it will look like: OR

You can fold a parabola so that the two sides match exactly. This property is called:

_____________.

The highest or lowest point of the parabola is called the ________________, which is on the

axis of symmetry.

B. Identifying a Vertex

Identify the vertex of each graph. Tell whether it is a minimum or a maximum.

1.) 2.)

Vertex: ( , ) _________ Vertex: ( , ) _________

3.) 4.)

Vertex: ( , ) _________ Vertex: ( , ) _________


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Graph each function. State the domain, the vertex (min/max point), the range, the x-intercepts,

and the axis of symmetry.

5.) f(x)= x2 – 4

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Domain: ________

Range: ________

Vertex:________

Max or min?________

x-intercepts: ____________

Axis of symmetry: ________

6.) h(x) = -2x2

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x

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Domain: ________

Range: ________

Vertex:________

Max or min?________



7.) f(x) = 21

2x

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Domain: ________

Range: ________

Vertex:________

Max or min?________




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8.) k(x) = x2 + 2x + 1

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Domain: ________

Range: ________

Vertex:________

Max or min?________



9.) f(x) = x2 – x – 6

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Domain: ________

Range: ________

Vertex:________

Max or min?________



C. Comparing Widths of parabolas

The value of a, the coefficient of the 2x term in a quadratic function, affects the width of the parabola

as well as the direction in which it opens.

When 1,a then the parabola is steeper, (or _________) than y = x2

When 1,a then the parabola is not as steep, (or _________) than y = x2

Order each group of quadratic functions from widest to narrowest graph:

10.) 2 2 21( ) 3 , ( ) 4 , ( )

2f x x f x x f x x 11.) 2 2 25

2 , , 4

y x y x y x


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D. Applications

12.) A monkey drops an banana from a branch 64 feet above the ground. Gravity causes the banana to

fall. The function 216 64h t gives the height of the banana, h, in feet, after t seconds.

a) Graph this quadratic function b) When does the banana hit the ground?

feet

time (sec)

70

60

50

40

30

20

10

321

13) A bungee jumper dives from a platform. The function h = -16t2 + 160 describes her height, h,

after t seconds in the air.

a) What will her height be after 1 second? b) what will her height be after 2 seconds?

c) How far did she fall between 1 and 2 seconds in the air?

t 2( ) 16 64h t t (t, h(t))


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Notes #24

9-2: Quadratic Functions

y = ax2 + bx + c

One key characteristic of a parabola is its vertex (min/max point). Yesterday we found the vertex after

we graphed the function. It would help to find the vertex first.

Vertex

- find x = 2

b

a

- plug this x-value into the function (table)

- this point (___, ___) is the vertex of the parabola

Graphing

- put the vertex you found in the center of

your x-y chart.

- choose 2 x-values less than and 2 x-values more

than your vertex.

- plug in these x values to get 4 more points.

- graph all 5 points

Find the vertex of each parabola. Graph the function and find the requested information

1.) f(x)= -x2 + 2x + 3 a = ____, b = ____, c = ____

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Vertex: _______

Max or min? _______

Direction of opening? _______

Wider or narrower than y = x2 ?

_______________

Domain: ________

Range: ________



2.) h(x) = 2x2 + 4x + 1

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Vertex: _______

Max or min? _______



_______________

Domain: ________

Range: ________




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3.) k(x) = 2 – x –1

2x

2

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Vertex: _______

Max or min? _______



_______________

Domain: ________

Range: ________



Without graphing the quadratic functions, complete the requested information:

4.) 2( ) 3 7 1f x x x

What is the direction of opening? _______

Is the vertex a max or min? _______

Wider or narrower than y = x2 ? ___________

5.) 25( ) 3

4g x x x




6.) 2211

3y x




7.) 20.6 4.3 9.1y x x




B. Application

8.) Suppose a particular star is projected from an aerial firework at a starting height of 520 feet with an

initial upward velocity of 88 ft/s. How long will it take for the star to reach its maximum height? How

far above the ground will it be? The equation 216 88 520h t t gives the star’s height h in feet at

time t in seconds.


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Notes #25

9-3: Finding and Estimating Square Roots

A. Finding Square Roots

The expression __________ means the positive, or __________ square root.

The expression __________ means the negative square root.

The expression ___________ means both the ____________ and _____________ square root

Simplify each expression.

1.) 64 2.) 100 3.) 9

16

4.) 0 5.) 25 6.) 0.09

7) 27 8.) 72 9.) 108

10.) 4

5 11.)

45

4 12.) 2.25

B. Rational and Irrational Square Roots

Tell whether each expression is rational or irrational.

13.) 144 14.) 1

5 15.)

1

9 16.) 7

17.) Between what two consecutive integers is 28.34 ?


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C. Application: Pythagorean Theorem (Review)

Use the Pythagorean theorem (_________________) to solve for the missing side of the right

triangle.

20.) 21.)

x8

4

y

46

9-4: Solving Quadratic Equations

A. Solving Quadratic Equations by Graphing

The solutions of a quadratic equation and the related x-intercepts are often called _______ of

the equation or _______ of the function.

1.) The function f(x) = x2 + x – 6 is graphed to the left.

a) Circle and name the zeros of the function graphed here.

( , ) and ( , )

b) Use this graph to solve the equation: x2 + x – 6 = 0

(This is asking: “At what x-values does y = 0?”)


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B. Solving by Graphing

Solve each equation by graphing the related function:

Find the vertex and 4 other points on the parabola; graph.

Find the x-intercepts from the graph. These are the _______ or _______.

2.) x2 – 4 = 0

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3.) 2x2 – 2 = 0

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4.) x2 + 6 = 0

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C. Solving Quadratic Equations Using Square Roots

Isolate the variable or expression being squared (get it ______________)

Square root both sides of the equation (include + and – on the right side!)

This means you have _____________ equations to solve!!

Solve for the variable (make sure there are no roots in the denominator)

5.) x2 = 25 6.) 3x

2 = 48

7.) 4x2 – 1 = 0 8.) 3m

2 – 5 = 0

9.) 2y2 – 81 = 0 10.) 36b

2 – 7 = 0

11.) (x – 1)2 = 4 12.) (2y + 3)

2 = 49

13.) (r + 5)2 = 12 14.) (3m – 1)

2 = 20


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If the left side is not already factored or squared, _______________ it!

15.) x2 + 2x + 1 = 8 16.) n

2 – 14n + 49 = 3

17.) w2 + 22w + 121 = 169 18.) g

2 + 10g + 25 = 18

D. Application

19.) A museum is planning an exhibit that will contain a large globe. The surface area of the globe will

be 100 ft2. Find the radius of the sphere producing this surface area. Use the equation 24S r ,

where S is the surface area and r is the radius.


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Notes #26

9-5: Solving Quadratic Equations by Factoring

A. Solving Quadratic Equations

Zero Product Property

List some pairs of numbers that multiply to zero:

(___)(___) = 0 (___)(___) = 0 (___)(___) = 0 (___)(___) = 0

What did you notice? _______________________________________________

Use this pattern to solve for the variable:

1. get = 0 and factor (sometimes this is done for you)

2. set each ( ) = 0 (this means to write two new equations)

3. solve for the variable (you sometimes get more than 1 solution)

1.) (3)(x) = 0 2.) (2)(x + 1) = 0 3.) -2y(y – 7) = 0

4.) (m + 1)(5m – 3) = 0 5.) 3

2 9 05

w w 6.) 2 6 4 8

03 7 5 9

x x

7.) x2 – 4x – 5 = 0 8.) y

2 + 6 + 5y = 0 9.) 4v

2 – 9 = 0


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10.) 2 42 0x x 11.) 23 2 21x x 12.) 22 15x x

13.) v(v + 3) = 10 14.) b(b – 2) = 3(b + 2)

B. Solving Word Problems with Quadratics

Steps:

1. Draw a picture and define your variable (let statement)

2. Write an equation

3. Get = 0 (bring all variables and numbers to one side)

4. Factor completely and solve

5. Do all the answers make sense?

6. Write your answer in a complete sentence

Translate and solve:

15.) The square of a positive number minus twice the number is 48. Find the number.

Let n = _____________ _________ - _________ = ______

16.) One more than a negative number times one less than that number is 8. Find the number.

Let n = ______________ (_________)(_________) = _____


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17.) The product of two consecutive integers is 12. Find the integers.

Let x = 1st integer

______ = 2nd

integer

18.) The product of two consecutive odd integers is 35. Find the integers.

Let x = 1st odd integer

______ = 2nd

odd integer

19.) The length of a rectangle is 3ft greater than its width. The area of the rectangle is 54ft2. Find the

length and the width of the rectangle.

20.) The area of a square is 5 more square inches than there are inches in the square’s perimeter. Find

the length of a side of the square.


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21.) Two less than the square of a number is equal to the number. Find the number.

___________ - _________ = _______

22.) The sum of the square of a number and three times the number is the same as one less than the

number. Find the number.

__________ + _________ = _________ - _________

Notes #27

9-6: Completing the Square

So far in this course, we have solved quadratics by _______________, __________________ and

___________________. We will eventually learn two more ways to solve quadratics.

Solve these equations. What makes these quadratics “easy” to solve?

a) (x – 1)2 = 9 b) (k + 2)

2 = 12

Solving quadratics by _________________ ______ _______________ helps us turn all quadratics

into this form.

Complete the square:

Take half the b (the x coefficient)

Square this number (no decimals – leave as a fraction!)

Add this number to the expression

Factor – it should be a binomial, squared ( )2

1.) x2 + 6x + _____ 2.) m

2 – 14m + _______

( )( )

( )2


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Find the value of n such that each expression is a perfect square trinomial:

3.) w2 + 7w + n 4.) k

2 – 5k + n

5.) j2 – j + n 6.) y

2 + 18y + n

Solving by Completing the Square:

Collect variables on the left, numbers on the right

Divide ALL terms by a; leave as fractions (no decimals!!)

Complete the square on the left – add this number to BOTH sides

Square root both sides (include a ______ and _______ equation!)

Solve for the variable (simplify all roots)

7.) x2 + 4x – 5 = 0 8.) x

2 – 6x – 11 = 0

9.) k2 – 4k – 7 = 0 10.) m

2 – 5m + 1 = 0

11.) 2y2 + 6y – 18 = 0 12.) 2x

2 – 3x – 1 = 0


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Cumulative Review: Solving Quadratics

Solve by factoring:

1.) 12k2 – 5k = 2 2.) 49m

2 – 16 = 0

Solve by using square roots:

3.) 4w2 = 18 4.) 3y

2 – 8 = 0

5.) 5m2 – 16 = 0 6.) (2x – 1)

2 = 20

Solve by completing the square:

7.) x2 – 10x + 7 = 0 8.) 4m

2 + 12m – 7 = 0

9.) 3y2 – 2y – 1 = 0 10.) 2x

2 – 20x = -50


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11.) 5x2 + 13x + 7 = 0 12.) ax

2 + bx + c = 0

Notes #28

9-7: Using the Quadratic Formula

A. Review of Simplifying Radicals and Fractions

Simplify expression under the radical sign, reduce

Reduce only from ALL terms of the fraction

1.) 6 18

2

2.)

5 20

2

3.) 4 20

4

4.) 8 27

2

5.) 29 ( 5) (4)(2)(3)

4

6.)

29 (6) 4( 3)( 3)

4


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B. Solving Quadratics using the Quadratic Formula

So far, we have solved quadratics by: (1) _______________, (2) ______________,

(3) ___________________, and (4) _________________

The final method for solving quadratics is to use the quadratic formula.

Solving using the quadratic formula:

Put into standard form (ax2 + bx + c = 0)

List a = , b = , c =

Plug a, b, and c into

2 4

2

b b acx

a

Simplify all roots, reduce

Solve by using the quadratic formula:

1.) x2 + x = 12

2 4

2

b b acx

a

(std. form):

a = _____

b = _____

c = _____

2.) 5x2 – 8x = -3

2 4

2

b b acx

a

(std. form):

a = _____

b = _____

c = _____


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3.) 2x2 = 4 – 7x 4.) 3x

2 – 8 = 10x

5.) -x2 + x = -1 6.) 3x

2 = 7 – 2x

Review of Solving Quadratics:

Solve by factoring:

7.) 4m2 +5m – 6 = 0 8.) 3x

3 – 27x = 0


9.) 4b2 – 1 = 0 10.) 3y

2 – 36 = 0 11.) (3x + 1)

2 = 18


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12.) x2 – 10x + 9 = 0 12.) x

2 – 7x – 18 = 0

13.) 4m2 + 12m + 5 = 0 14.) 3y

2 + 2y – 1 = 0

Solve by using the quadratic formula:

15.) x2 – 20 = 0 16.) x

2 – 6x + 9 = 0


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Notes #29

9-8: Using the Discriminant

Quadratic equations can have two, one, or no solutions. You can determine how many solutions

a quadratic equation has before you solve it by using the ________________.

The discriminant is the expression under the radical in the quadratic formula:

2 4

2

b b acx

a

Discriminant = b2 – 4ac

If b2 – 4ac = 0, then the equation has 1 solution

If b2 – 4ac < 0, then the equation has 0 real solutions

If b2 – 4ac > 0, then the equation has 2 solutions

A. Finding the number of x-intercepts

Determine whether the graphs intersect the x-axis in zero, one, or two points.

1.) 24 12 9y x x 2.) 23 13 10y x x

B. Finding the number of solutions

Find the number of solutions for the following:

3.) 23 5 1x x 4.) 2 3 7x x

5.) 9x2 – 6x = 1 6.) 4x

2 = 5x + 3


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C. Review of Solving Quadratics

Solve by factoring:

7.) 2x2 + 12x = -10 8.) 16(x – 1) = x(x + 8)


9.) 3b2 – 1 = 7 10.) (3x + 1)

2 = 18


11.) x2 – 10x – 11 = 0 12.) x

2 – 3x – 6 = 0


- 24 -

Solve using the quadratic formula:

13.) 6x2 + 7x = 5 14.) x

2 = 8 – 6x

Graph the quadratic. Name the vertex, axis of symmetry, x-intercepts, domain, and range.

13.) f(x)= x2 – 9

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Domain: ________

Range: ________

Vertex:________

Max or min?________




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Notes #30: Solving Radical Equations with Quadratics (Section 10.4)

Solving Radical Equations:

Isolate the _________________

______________ both sides. If one side is a binomial, be sure to use ___________ to

square it.

Get all terms to one side to = 0

Solve the quadratic using: factoring, quadratic formula, or completing the square.

Check your solution by ___________________ into the original equation. Check for

extraneous roots.

1.) 2 3 1x

2.) 4 2 5x

3.) 2x x 4.) 35 2x x

5.) 6 9x x 6.) 11 28x x


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7.) 2 7 2x x 8.) 3 2 4x x

9.) Graph the quadratic. Find the requested

information: f(x)= -x

2 – 3x + 4

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Domain: ________

Range: ________

Vertex:________

Max or min?________




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Chapter 9 Review (Classwork #31) Solve by factoring: 1.) 4x2 – 12x = 0 2.) 6x2 + 15x = 21 Solve by square rooting: Solve and check:

3.) 5y2 – 6 = 12 4.) 1 1x x

Solve by completing the square: Solve using the quadratic formula: 5.) x2 – 6x – 27 = 0 6.) x2 – x – 1 = 0 Find the discriminant; find the Define a variable, write an equation, number of real solutions: and solve: 7.) 4x2 – 6x + 3 = 0 8.) The width of a rectangle is two inches less

than its length. The area of the rectangle is 63in2. Find the dimensions of the rectangle.

Find the vertex of each parabola. Graph the function and find the requested information 9.) f(x)= x

2 – 2x – 3 a = ____, b = ____, c = ____

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x

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Vertex: _______

Max or min? _______



_______________

Domain: ________

Range: ________




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Algebra 1: Chapter 9 Study Guide Name: ___________________________

Match each graph with its function

1.) 2.) 3.)

Without graphing, provide the requested information about each parabola:

4.) a = ______ b=______ c=________

vertex _________

equation of the axis of symmetry ________

max or min? _______

direction of opening? _______

wider or narrower than ______

5.) a = ______ b=______ c=________

vertex _________

equation of the axis of symmetry ________

max or min? _______

direction of opening? _______

wider or narrower than ______

Graph the function. State the domain, the vertex (min/max point), the range, the

x-intercepts, and the axis of symmetry.

6.) y = x2 – 4x + 3

Domain: ________

Range: ________

Vertex:________

Max or min?________



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7.) y = -x2 – 6x – 5

Domain: ________

Range: ________

Vertex:________

Max or min?________



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y

Solve by factoring

8.) k2 – 3k – 10 = 0 9.) 3m

2 – 2m = 21

Solve by using square roots

10.) 3m2 – 108 = 0

11.) 4w2 – 75 = 0

Solve by completing the square

12.) x2 + 4x – 5 = 0 13.) x

2 – 6x – 3 = 15

14.) 2x2 + 6x – 18 = 0

Solve by using the quadratic formula: Write down the formula:

15.) 2x2 + 5x + 3 = 0 16.) x

2 – 2x = 2


- 30 -

Write the expression for the discriminant. Use this to find the number of real solutions for each

equation: 17.) 3x

2 + 6x + 7 = 0 18.) x

2 + 5x = -2 19.) x

2 – 4x + 4 = 0

Solve each equation. Check for extraneous solutions.

20.) 1 1x x 21.) 3 2x x

Translate and solve:

22.) One more than a positive number times one

less than that number is 8. Find the number.

23.) The product of two consecutive odd integers

is 63. Find the numbers.

24.) The length of a rectangle is one inch longer than twice the width. The area of the rectangle is

36in2. Find the length and width of the rectangle.

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