Chapter 5wellsmat.startlogic.com/sitebuildercontent/sitebuilderfiles/alg2_ch5... · Lesson 5-5,...

Chapter 5

Quadratic Equations and Functions

Lesson 5-1

Modeling Data with Quadratic

Functions

Quadratic Function

Standard Form of a Quadratic Function

f x ax bx c2( ) Quadratic

Term

Linear

Term

Constant

Term

Example 1 – Page 237, #2

Determine whether each function is linear or quadratic.

Identify the quadratic, linear and constant terms.

y x x22 (3 5)

y x x22 3 5

quadratic

x x22 , 3 ,5


Determine whether each function is linear or quadratic.

Identify the quadratic, linear and constant terms.

y x x x2(1 ) (1 )

y x x x

x

2 21

1

Linear

x, 1none,

Parabola

Axis of symmetry

Vertex

Minimum Value

y x2

Maximum Value

y x2

Parabola

Vertex is the point at which the

parabola intersects the axis of

symmetry

Axis of symmetry is the line that

divides a parabola into two parts that

are mirror images.

Example – Page 237, #12

Identify the vertex and the axis of symmetry of each parabola.

Vertex: ( 1, 4)

Axis of Symmetry: x 1


For each parabola, identify points corresponding to P and Q

P '(1,5)

Q'( 2,8)

Lesson 5-2, Part 1

Properties of Parabola

Graphing Parabolas

cxax by 2

Step 1 – Determine the direction of the parabola

a is positive it opens up

a is negative it opens down

Step 2 – Find the y-intercept (0,?).

Substitute x = 0 into the quadratic and find the y-value.

Graphing Parabolas

cxax by 2

xb

a2

Step 3 – Find the axis of symmetry (x = ?)

Step 4 – Find the vertex (x, ?)

Substitute the x-value from step 3 into the quadratic and

find the y-value.

Graphing Parabolas

cxax by 2

Step 5 – Graph the quadratic

Get additional points if needed


Graph each function.

y x25 12

Step 1

y

y

25 12

12

(0,12)

0

Opens down

Step 2


cxax by 2 xb

a2

y x25 12

Step 3

x

x

0

2 5

0


y x25 12

Step 3

x 0

Step 4

y

y

25( ) 12

12

(0,

0

12)


y x25 12

Step 5

y

y

25(2) 12

8

(2, 8)


y x25 12


Graph each function. Label the vertex and axis of symmetry

y x x24 12 9

Step 1

y

y

20 04 12 9

9

(0,9)

Opens up

Step 2


cxax by 2 xb

a2

y x x24 12 9

Step 3

x

x

12

2 4

1.5


y x x24 12 9

Step 3

x 1.5

Step 4

y

y

21.5 1.4( ) 12 9

0

(1.5,0

5

)


y x x24 12 9 Step 5

Vertex (1.5, 0)

Axis of Symmetry

x = 1.5

Lesson 5-2, Part 2

Properties of Parabola

Example 2 – 4, #22

Graph each function. If a > 0 find the minimum value.

If a < 0 find the maximum value.

y x x2 2 5

Step 1

Opens down

Step 2

y

y

2( ) 2( ) 5

5

(0,5)

0 0


cxax by 2

y x x2 2 5

Step 3

xb

a2

x

x

21

2( 1)

1


y x x2 2 5

Step 3

x 1

Step 4

y

y

2( ) 2( ) 5

6

(1,6)

1 1


y x x2 2 5

Step 5

y

y

2( ) 2( ) 5

3

4

(4,

4

3)


y x x2 2 5

max, 6


A model for a company’s revenue is

where p is the price in dollars of the company’s product.

What price will maximize revenue? Find the maximum revenue.

R p p215 300 12,000,

Find the vertex

xb

a2

x

x

30010

2( 15)

10

R

R

215(10) 300(10) 12,000

13500

$10 will maximum the revenue

and the maximum revenue

is $13,500

Lesson 5-3, Part 1

Translating Parabolas

Vertex Form of Quadratic

Function

Parent Function

y ax2

y a x h k2( )

Translated Function

Vertex Form

y x2

Graph of a Quadratic

Function in Vertex Form

h units moves horizontally

h is positive the graph shifts right

h is negative the graph shifts left

k units moves vertically

k is positive the graph shifts up

k is negative the graph shifts down

Vertex is (h, k)

Axis of symmetry is x = h

y a x h k2( )


Graph each function

y x 2( 3) 4 y a x h k2( )

Step 1 – Find the vertex.

h 3 k 4

vertex: ( 3, 4)


y x 2( 3) 4

y x2

vertex: ( 3, 4)


y x 2( 3) 4

Step 2 – Find the y-intercept

y

y

2(0 3) 4

5

(0,5)


Graph each function

y x2

4 8 6

Step 1 – Find the vertex

( 8, 6)


y x2

4 8 6

Step 2 – Find the y-intercept

Too big

Step 3 – Get additional points

y

y

24( 8) 6

10

( 7,

7

10)

Example 2, Page 251, #16

Write the equation of each parabola in vertex form.

( 2,0)

( 3, 1)

y a x h k2( )

y xa 2( 2) 0

a 21 ( 3 2)

a

a

a

1 1

1

1

y x 2( 2)

Lesson 5-3, Part 2

Translating Parabolas

Example 3, Page 252, #26

Identify the vertex and the y-intercept of the graph of

each function.

y x 2( 125) 125

(125,125)

Vertex

y a x h k2( )

y-intercept

y

y

2(0 125) 125

15750


Write each function in vertex form.

y x x22 8 3

Find the vertex.

cxax by 2 xb

a2

x8

22( 2)

y

y

22( ) 8( ) 3

11

(

2 2

2,11)

y x 22( 2) 11

y a x h k2( )

Lesson 5-4, Part 1

Factoring Quadratic Expression


Find the GCF of each expression. Then factor the

expression

p p227 9

GCF: p9

p p9 (3 1)


Factor each expression.

x x2 65

x xx

x

x

x

x x2

2

2 6

( 3)(

( 3) ( 3)

3

2)

x x2 26 6

x x x

x x x

23 2 6

3 2 5



x x2 10 24 x x2 224 24

x x x

x x x

26 ( 4 ) 24

6 ( 4 ) 10

x

x

x x

x

x

x

x

2

( 6) 4

( 4)(

( 6

2

)

4 46

6)



c c2 62 3 C c2 2( 63) 63

c c c

c c c

29 ( 7 ) 63

9 ( 7 ) 2

c

c

c c

c

c

c

c

2

( 7) 9

( 9)(

( 7

6

)

9 37

7)



t t2 47 4 t t2 2( 44) 44

211 4 44

11 4 7

t t t

t t t

t

t

t t

t

t

t

t

2

( 11)

4

4

( 4)(

4411

( 11)

11)

Factoring

x x2 65 x x( 3)( 2)

x x2 10 24 x x( 4)( 6)

c c2 62 3 c c( 9)( 7)

t t2 47 4 t t( 4)( 11)

Last number is positive – both negative or positive

Last number is negative – a negative and a positive



xx2 192 24 2 224 42 8xx

216 ( 3 ) 48

16 ( 3 ) 19

x x x

x x x

x

x

x

x

x

x

x

x

2

( 82 3

(2

2 16

3)(

)

4

(

3 2

8)

8)

Lesson 5-4, Part 2

Factoring Quadratic Expression



yy 2 125 32 2 2( 32) 165 0y y

220 ( 8 ) 160

20 8 12

y y y

y y y

y

y

y

y

y

y

y

y

2

( 45 8

(5

5 20

8)(

)

2

(

8 3

4)

4)

Factoring

Perfect Square Trinomial

Difference of Two Squares

a ab b a b a b a b2 2 22 ( )( ) ( )

a ab b a b a b a b2 2 22 ( )( ) ( )

a b a b a b2 2 ( )( )



nn2 204 25 n n n2

(2 5)(2 5) 2 5



c2 64 c c( 8)( 8)


Find the area of rectangular cloth is cm².

The length is cm. Find the width.

x x2(6 19 85)

x(2 5)

x26

85

x2

5

x3 17

x cm2(3 17)x34

x15


Factor each expression completely

x x212 36 27 x x23(4 12 9)

x x x 23(2 3)(2 3) 3(2 3)


2 5 4x x

Factor each expression completely.

21( 5 4)x x 1( 1)( 4)x x


Factor each expression completely

x21 1

2 2 x x

11 1

2 x21

12

Lesson 5-5, Part 1

Quadratic Equations

Standard Form

Standard form of a quadratic equation

ax bx c2 0


Solve each equation by factoring.

x x2 18 9

x x2 9 18 0

Step 1 – Write in standard form


x x2 9 18 0

Step 2 – Factor

x x( 3)( 6) 0

x 3 0 x 6 0

x 3 x 6


Solve each equation by finding square roots.

x2 4 0

x2 4

x2 4

x 2

x or x2 2


Solve each equation by finding square roots.

x25 40 0

x25 40

x2 8

x 2 2

x2 8

8 2 4 2 2

x or x2 2 2 2


Solve each equation by factoring or by taking square roots.

x x26 4 0

x x2 3 2 0

x2 0 x3 2 0

x 0 x

x

3 2

2

3


Solve each equation by factoring or by taking square roots.

x24 80 0

x24 80

x2 20

x2 20

20 4 5 2 5

x 2 5

x or x2 5 2 5

Lesson 5-5, Part 2

Quadratic Equations

Solving Quadratics

Not all quadratics can be factored.

Zero of a function is called the solution of the quadratic

It is where the quadratic crosses the x-axis.

Solve quadratics by

Factoring

Square Roots

Graphing (TI-Calculator)

Completing the Square (Lesson 5-7)

Quadratic Formula (Lesson 5-8)


Solve each equation by factoring, by taking the square

roots, or by graphing (not factorable)

x x x2 2 6 6

x x x2 2 6 6 0

x x2 8 6 0

Not factorable


Solve each equation by factoring, by taking the square

roots, or by graphing (not factorable)

x x22 6 8

x x22 6 8 0

x x2( 1)( 4) 0

x 1 0 x 4 0

x 1 x 4

22( 3 4) 0x x

Lesson 5-6, Part 1

Complex Numbers

Complex Number System

Real Numbers Imaginary

Numbers

Rational Numbers Irrational Numbers

Rational Numbers: 8

5, 0, , 93

Irrational Numbers: 3, 5

Imaginary Numbers: 4 , 3 2 , 2 2i i i

Imaginary Numbers

a i a i a2

Imaginary number is a number whose square is -1. So

that i² = -1. Imaginary number is any form a + bi


Simplify each number by using the imaginary number i.

72 i 272 i i9 8 3 8 i i3 4 2 3 2 2

i6 2

i 72

Complex Number

Complex numbers are imaginary numbers and real numbers.

a bi

Real Part Imaginary Part


Write each number in the form a + bi.

8 8 i8 4 2 i8 2 2 8 8

Absolute Value of a

Complex Number

a bi a b2 2


Find the absolute value of each complex number.

i1 4 2 21 ( 4) 1 16 17


Find the additive inverse of each number.

i9 i9


Simplify each expression

i i3 5 4 2

i i3 4 5 2

i1 7



i6 8 3

i6 8 3

i2 3

Lesson 5-6, Part 2

Complex Numbers



i i(4 3 )(5 2 )

i i 220 7 6

i20 7 6( 1)

i20 7 6

i26 7

i i i 220 8 15 6


Solve each equation.

x 25 3 0

x 25 3

x 23

5

x 23

5

x i3

5

3 3 15

55 5

5

5

x i15

5


Simplify each expression.

10 9 2 25

i i10 3 2 5

i8 2

10 9 2 25


Simplify each expression.

1 4 3 25

i i 23 10

i3 10 1

i13

i i

i i i 21 2 3 5

3 5 6 10

Lesson 5-7, Part 1

Completing the Square


If an quadratic equations is not a perfect square

trinomial (a + b)², you can convert it into a perfect square

trinomial by rewriting the constant term.


Solve each equation.

x x2 168 16

9

x 2 16( 4)

9

x2 164

9

x4

43

x4

43

x

x

443

16

3

x

x

443

8

3


Complete the square.

x x2 20 ____

2

10

1

20

2

0 100

100


Solve each quadratic equation by completing the square.

x x2 3 4 0

Step 1 – Get all terms containing x to one side.

x x2 3 ____ 4 ____


x x2 3 ____ 4 ____

Step 2 – Complete the square

23 3 9

2 2 4

9

4

9

4


x x2 3 ____ 4 ____

Step 3 – Factor the perfect square.

9

4

9

4

x2

3 25

2 4


Step 4 – Solve for x.

x2

3 25

2 4

x2

3 25

2 4

x3 5

2 2

x3 5

2 2

x3 5

12 2

x3 5

42 2


Solve each quadratic equation by completing the square.

x x2 6 22


x x2 6 ____ 22 ____


x x2 6 ____ 22 ____

Step 2 – Complete the square

263 9

2

9 9


x x2 6 9 22 9

Step 3 – Factor the perfect square.

x23 13



x23 13

x23 13

x i3 13

x i3 13

Lesson 5-7, Part 2



Solve each quadratic equation by completing the square

2 2 5x x


Step 2 – Leading coefficient x² is 1.

2 2 5x x

2 51 2x x

2 2 5x x


Step 3 – Complete the square.

2 2 5x x

2 2 ____ 5 ____x x

221 1

2

1 1

2 2 1 4x x

Step 4 – Factor the perfect Square.

21 4x



21 4x

21 4x

1 2x i

1 2x i


Solve each quadratic equation by completing the square

29 12 5 0x x


Step 2 – Leading coefficient x² is 1.

29 12 5x x

219 12 5

9x x

2 12 5

9 9x x


Step 3 – Complete the square.

2 4 5

3 9x x

2 4 5____ ____

3 9x x

24

2 432 3 9

4

9

2 4 4 1

3 9 9x x

4

9


Step 4 – Factor the perfect Square.

2 4 4 1

3 9 9x x

22 1

3 9x


22 1

3 9x


22 1

3 9x

2 1

3 3x i

2 1

3 3x i


Rewrite each equation in vertex form.

2 4 7y x x

2 ____( 4 ) ___7 _y x x

242 4

2

4 4

2 4 4 11y x x

22 11y x

Lesson 5-8, Part 1

Quadratic Formula

Quadratic Formula

x xa cb2 0

x

a

cb ab 2 4

2


Solve each equation using the Quadratic Formula.

x x28 2 3 0

a

b

c

8

2

3

b b acx

a

2 4

2

x

22 2 4 8 3

2 8

x2 4 96

16


x2 4 96

16

x2 100 2 10

16 16

x2 10 12 3

16 16 4

x2 10 8 1

16 16 2


Solve each equation using the Quadratic Formula.

x x22 7 8

a

b

c

2

7

8

b b acx

a

2 4

2

x

27 7 4 2 8

2 2

x7 49 64

4

x x22 7 8 0


x7 49 64

4

ix7 15 7 15

4 4

ix7 15

4 4

Lesson 5-8, Part 2

Quadratic Formula


Solve each quadratic using the Quadratic Formula. Find the

exact solutions. Then approximate any radical solutions.

Round to the nearest hundredth

x x23 4 3 0

a

b

c

3

4

3

b b acx

a

2 4

2

x

24 4 4 3 3

2 3

x

4 16 36

6


x

4 16 36

6

x4 52

6

x4 4 13 4 2 13

6 6

x4 2 13 2 13 2 13

6 6 3 3 3

x2 13

0.543

x2 13

1.873

Discriminant

x xa cb2 0

x

a

cb ab 2 4

2

Discriminant

2 Real Solutions

2 x-intercepts

b ac2 4 0

2 Imaginary Solutions

no x-intercepts

1 Real Solutions

1 x-intercepts

b ac2 4 0 b ac2 4 0


Evaluate the discriminant of each equation. Tell how many

solutions each equation has and whether the solutions are

imaginary or real.

x x22 28 0

a

b

c

2

1

28

b ac2 4

21 4 2 28

1 224

223 0

2 imaginary solutions


Evaluate the discriminant of each equation. Tell how many

solutions each equation has and whether the solutions are

imaginary or real.

x x2 12 36 0

a

b

c

1

12

36

b ac2 4

2

12 4 1 36

144 (144)

0 0

1 real solution

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