quadratic functions chapter 7. vertex form vertex (h, k)

Quadratic Functions

Chapter 7

Vertex Form

• Vertex (h, k)

khxaxf 2)()(

)8,5(

8)5(2 2 x

)2,7(

2)7(3 2

x

)6,0(

65 2 x

)0,0(

3 2x

)0,9(

)9(2 2x

Vertex Form

• a > 0, opens upward

• a < 0, opens downward

• the larger│a│is the narrower the parabola

• the closer a is to zero the wider the parabola

khxaxf 2)()(

Stretching the Unit Quadratic

2)( xxf 22)( xxf

2

2

1)( xxf

Reflecting Across the x-axis

2)( xxf

2)( xxf

Translating Graphs Up/Down

2)( xxf

2)( 2 xxf

2)( 2 xxf

Translating Graphs Right/Left

2)( xxf

2)3()( xxf

2)4()( xxf

Graphing a Quadratic Function

• First graph vertex

• Find a point

1)3(2)( xxf

)1,2(

1)2(

12)2(

1)1(2)2(

1)1(2)2(

1)32(2)2(2

2

f

f

f

f

f

)1,3(

• Draw axis of symmetry through vertex

• Reflect point over axis

Graphing a Quadratic Function1)3(2)( xxf

)1,4(

3x

Finding a Quadratic Model

• Create a scattergram• Select a vertex (Doesn’t have to be data

point)• Select non-vertex point• Plug vertex in for h and k, and the nonvertex

point for x and f(x)/y into a standard equation

• Solve for a• Then substitute a into the standard equation

Graph Quadratic Model

• Pick vertex– (70, 5)

• Pick point– (40, 9)

x f(x)

1930 (30) 12

1940 (40) 9

1950 (50) 7

1960 (60) 6

1970 (70) 5

1980 (80) 6

1990 (90) 7

2000 (100) 10

900

4

9004

5590059

5)30(9

5)7040(9

5)70(

)(

)()(

2

2

2

2

2

a

a

a

a

a

xay

khxay

khxaxf5)70(

900

4)( 2 xxf

7.2 Graphing Quadratics in Standard Form

Quadratic in Standard Form

• Find y-intercept (0, c)

• Find symmetric point

• Use midpoint formula of the x-coordinates of the symmetric points to find the x-coordinate of the vertex

• Plug x-coordinate of the vertex into equation for x

cbxaxxf 2)(

Graphing Quadratics

• Y-intercept– (0, 7)

• Symmetry Point

76)( 2 xxxf

)7,6)(7,0(

6,0

06,0

)6(0

60

77677

767

2

2

2

xorx

xorx

xx

xx

xx

xx

Graphing Quadratics

• (0, 7) (6, 7)• Midpoint

76)( 2 xxxf

32

)6(0

16)3(

7189)3(

7)3(6)3()3( 2

f

f

f

Vertex formula

• vertex formula

x-coordinate

• y-coordinate

cbxaxxf 2)(

a

bx

2

a

bf

2

a

bf

a

b

2,

2

Vertex Formula

3

23

3

11

3

12

3

2

3

1

43

12

9

13

43

12

3

13

3

1

3

1

)3(2

2

423)(

2

2

f

x

xxxf

3

23,

3

1

Maximum/Minimum

• For a quadratic function with vertex (h, k)

• If a > 0, then the parabola opens upward and the vertex is the minimum point (k minimum value)

• If a < 0, then the parabola opens downward and vertex is the maximum point (k maximum value)

cbxaxxf 2)(

Maximum Value Model

• A person plans to use 200 feet of fencing and a side of her house to enclose a rectangular garden. What dimensions of the rectangle would give the maximum area? What is the area?

22200

)2200(

2200

2002

wwA

wwA

wl

lwA

lw

100

100200

)50(2200

2200

504

200

)2(2

200

l

l

l

wl

w

Maximum area would be 50 x 100 = 5000

7.3 Square Root Property

Product/Quotient Property for Square Roots

• For a ≥ 0 and b ≥ 0,

• For a ≥ 0 and b > 0,

• Write radicand as product of largest perfect-square and another number

• Apply the product/quotient property for square roots

baab

b

a

b

a

Simplifying Radical Expressions

• No radicand can be a fraction

• No radicand can have perfect-square factors other than one

• No denominator can have a radical expression

Examples

52

54

54

20

53

59

59

45

5

3

25

3

25

3

6

14

2

2

23

7

29

7

18

7

Square Root Property• Let k be a nonnegative constant. Then,

is equivalent to kx 2 kx

3

3

5

15

5

5

785

2

2

2

x

x

x

x

24

173

8

173

8

173

8

17)3( 2

x

x

x

x

4

3412

4

34

4

12

4

343

2

2

22

173

x

x

x

x

Imaginary Numbers

• Imaginary unit, (i), is the number whose square is -1.

• Square root of negative number– If n is a positive real number,

12 i 1i

nin

Complex Numbers

• A complex number is a number in the form

• Examples

• Imaginary number is a complex number, where a and b are real numbers and b ≠ 0

bia

i73 i35 ii 330 606 i

Solving with Negative Square Roots

ix

ix

x

x

6

36

36

362

24

216

32

322

ix

ix

x

x

7.4 Completing the Square

Perfect Square Trinomial

• For perfect square trinomial in the form

dividing by b by 2 and squaring the result gives c:

cbxx 2

cbxx 2

cb

2

2 c

c

cxx

16

9

4

3

2

1

2

3

2

3

2

2

2

Examples

34

34

3)4(

3)4(8

32

88

38

2

22

22

2

x

x

x

xx

xx

xx

5

2

5

1

5

2

5

2

10

2

5

2

5

2

2

1

5

2

5

2

5

2

5

2

225

0225

22

22

22

2

2

2

xx

xx

xx

xx

xx

xx

5

101

5

10

5

1

5

5

5

2

5

1

5

2

5

1

5

2

5

12

x

x

x

x

x

7.5 Quadratic Formula

Quadratic Formula

• The solutions of a quadratic equation in the form are given by the quadratic formula:

02 cbxax

a

acbbx

2

42

Determining the Number of Real-Number Solutions

• The discriminant is and can be used to determine the number of real solutions

• If the discriminant > 0, there are two real-number solutions

• If the discriminant = 0, there in one real-number solution

• If the discriminant < 0, there are two imaginary-number solutions (no real)

acb 42

Quadratic Formula

8

1284

8

144164

)4(2

)9)(4(4)4()4(

944

2

2

xx

22

18

284

8

2644

i

i

i

Examples

0169 2 xx

0

3636

)1)(9(462

0852 2 xx

39

6425

)8)(2(4)5( 2

One real-number solution Two imaginary-number solutions

Intersections with y = n lines/points at a certain height

12

02432

y

xx

2

573

2

4893

)1(2

)12)(1(4)3()3(

0123

12243

2

2

2

xx

xx

Note if the discriminant is < 0, then there are no intersections