Chapter 6: Quadratic Functions

VoglerAlgebra II

Simplifying Quadratics: Factoring

Simplifying Quadratics: Factoring

Simplifying quadratics makes for easier graphing(x+y)2=x2+2xy+y2

(x-y)2=x2-2xy+y2

(2x+3)2=4x2+12x+9You can apply this property to any

expanded quadratic with perfect squares on x and y

Simplifying quadratics makes for easier graphing(x+y)2=x2+2xy+y2

(x-y)2=x2-2xy+y2

(2x+3)2=4x2+12x+9You can apply this property to any

expanded quadratic with perfect squares on x and y

Simplifying Quadratics: Factoring

Simplifying Quadratics: Factoring

FOILFirst Outer Inner Last

(2x+4)(x-3)2x•x (First)2x•-3 (Outer)4•x (Inner)4•-3 (Last)2x2-6x+4x-12Simplify: 2x2-2x-12

FOILFirst Outer Inner Last

(2x+4)(x-3)2x•x (First)2x•-3 (Outer)4•x (Inner)4•-3 (Last)2x2-6x+4x-12Simplify: 2x2-2x-12

Simplifying Quadratics: Factoring

Simplifying Quadratics: Factoring

Not all quadratics are easily factorable:3x2+5x+2Set up a gridFind a x c (3 x 2)Find all the factorsof 6 that sum to 5Fill in the gridFind the GCF of each column and rowWrite the expression

(3x+2)(x+1)

Not all quadratics are easily factorable:3x2+5x+2Set up a gridFind a x c (3 x 2)Find all the factorsof 6 that sum to 5Fill in the gridFind the GCF of each column and rowWrite the expression

(3x+2)(x+1)

3x2

2

3x

2

X 1

3x

2x

Graphing: TranslationsGraphing: Translations

In ax2+bx+c, c refers to the y-interceptx2+4 is 4 units up

from x2

y=x2+k moves a graph up k units

y=x2-k also moves a graph down k units

In ax2+bx+c, c refers to the y-interceptx2+4 is 4 units up

from x2

y=x2+k moves a graph up k units

y=x2-k also moves a graph down k units

0

1

2

3

45

6

7

8

9

-4 -2 0 2 40

1

2

3

45

6

7

8

9

-4 -2 0 2 4

Graphing: TranslationsGraphing: Translations

In y=(x-h)2

h is the distance moved left or right

-h moves right+h moves left

(x+3)2 is 3 units left of x2

x=h is the line of symmetry

In y=(x-h)2

h is the distance moved left or right

-h moves right+h moves left

(x+3)2 is 3 units left of x2

x=h is the line of symmetry

0

2

4

6

810

12

14

16

18

-4 -2 0 20

2

4

6

810

12

14

16

18

-4 -2 0 2

Graphing: TranslationsGraphing: Translations

In ax2+bx+c:+a opens up-a opens down

To find the line of symmetry:Find -b/2a

In ax2+bx+c:+a opens up-a opens down

To find the line of symmetry:Find -b/2a

0

0.5

1

1.5

22.5

3

3.5

4

4.5

-4 -2 0 2 40

0.5

1

1.5

22.5

3

3.5

4

4.5

-4 -2 0 2 4

Solving Quadratics: Factoring

Solving Quadratics: Factoring To solve by factoring, make ax2+bx+c equal to 0:

0= 3x2+5x+2 Then factor:

0=(3x+2)(x+1) Make each binomial equal to zero and solve:

0=3x+2 0=x+1-2=3x -1=x (second solution)-2/3=x (first solution)

The two solutions are: -2/3 and -1 Solutions to quadratics refer to the x-intercepts

In other words, if the equation is not equal to 0, then we have to make it equal to 0 to solve it

To solve by factoring, make ax2+bx+c equal to 0:0= 3x2+5x+2

Then factor:0=(3x+2)(x+1)

Make each binomial equal to zero and solve:0=3x+2 0=x+1-2=3x -1=x (second solution)-2/3=x (first solution)

The two solutions are: -2/3 and -1 Solutions to quadratics refer to the x-intercepts

In other words, if the equation is not equal to 0, then we have to make it equal to 0 to solve it

Solving Quadratics: Completing the Square

Solving Quadratics: Completing the Square

X2+10=39 Draw a square with

area x2

Add a rectangle of length 10

Split the rectangle Find the area

Add the area to 39 Find the square root of

the answer (64) Solve x+5=8 and

x+5=-8 X=3 and -13

X2+10=39 Draw a square with

area x2

Add a rectangle of length 10

Split the rectangle Find the area

Add the area to 39 Find the square root of

the answer (64) Solve x+5=8 and

x+5=-8 X=3 and -13

x

x

10

=39

x

x

5

5 25

=39+25

=64+

+

Graphing: Quadratic modeling

Graphing: Quadratic modeling

Coordinates:(0,5), (1, 10), (2,

19)

Use a basic formula:

5=a(0)2+b(0)+c10=a(1)2+b(1)+c19=a(2)2+b(2)+c

Coordinates:(0,5), (1, 10), (2,

19)

Use a basic formula:

5=a(0)2+b(0)+c10=a(1)2+b(1)+c19=a(2)2+b(2)+c

02468

101214161820

0 1 2 302468

101214161820

0 1 2 3

Graphing: Quadratic modeling

Graphing: Quadratic modeling

Set up and solve a system for your three equations:

5=a(0)2+b(0)+c10=a(1)2+b(1)+c19=a(2)2+b(2)+ca=2; b=3; c=5So y=2x2+3x+5

Set up and solve a system for your three equations:

5=a(0)2+b(0)+c10=a(1)2+b(1)+c19=a(2)2+b(2)+ca=2; b=3; c=5So y=2x2+3x+5

Graphing: Quadratic modeling

Graphing: Quadratic modeling

A vehicle’s braking distance is found for the following three coordinates:

Find the equationfor this vehicle’sbraking distance

A vehicle’s braking distance is found for the following three coordinates:

Find the equationfor this vehicle’sbraking distance

Speed (MPH)

Distance (ft.)

0 0

10 19

40 116

Graphing: Quadratic modeling

Graphing: Quadratic modeling

A ball’s trajectory can be found using the equation:

h(t) = -4.9t2 + vot + ho

The ball has an initial velocity of 14 m/sec and was thrown from a height of 30 meters.

A ball’s trajectory can be found using the equation:

h(t) = -4.9t2 + vot + ho

The ball has an initial velocity of 14 m/sec and was thrown from a height of 30 meters.

Solving Quadratics: Quadratic EquationSolving Quadratics: Quadratic Equation

The quadratic formula only works when y=0 in y=ax2+bx+c

So, in 10x2-13x-3=0,

The quadratic formula only works when y=0 in y=ax2+bx+c

So, in 10x2-13x-3=0,

€

x =−b ± b2 − 4ac

2a

€

x =13± −132 − 4(10)(−3)

2(10)

Solving Quadratics: Quadratic EquationSolving Quadratics: Quadratic Equation

Simplify: do what is under the radical sign first:

Both add and subtract from 13

Simplify:

X=3/2 and -1/5

Simplify: do what is under the radical sign first:

Both add and subtract from 13

Simplify:

X=3/2 and -1/5

€

x =13± 289

20

€

x =13+17

20

€

x =13−17

20

€

x =30

20

€

x =−4

20

Solving Quadratics: Quadratic EquationSolving Quadratics: Quadratic Equation

A cat is dropped from a height of 40 feet. Use the formula h=-16t2+44t+40.

1. When does it hit the ground?2. Does it land on it’s feet?

A cat is dropped from a height of 40 feet. Use the formula h=-16t2+44t+40.

1. When does it hit the ground?2. Does it land on it’s feet?

Solving Quadratics: Quadratic EquationSolving Quadratics: Quadratic Equation

Sometimes we want to solve for equations that do not equal 0. So, make them equal to 0:

A baseball is hit from home plate into the outfield. If the ball is hit at a height of 5 ft. and an initial velocity of 147 ft/sec, when will it reach a height of 10 ft?

Write an equation: 10=-16t2+147t+5 Get the equation in terms of 0:

0=-16t2+147t-5 Solve for t. T=.03 and 9.15 Which value makes more sense? Why?

Sometimes we want to solve for equations that do not equal 0. So, make them equal to 0:

A baseball is hit from home plate into the outfield. If the ball is hit at a height of 5 ft. and an initial velocity of 147 ft/sec, when will it reach a height of 10 ft?

Write an equation: 10=-16t2+147t+5 Get the equation in terms of 0:

0=-16t2+147t-5 Solve for t. T=.03 and 9.15 Which value makes more sense? Why?

DiscriminantDiscriminantAll quadratics

have two solutions, but not all solutions are real:

Discriminant:b2-4ac>0, two real

solutionsb2-4ac=0, one real

solutionb2-4ac<0, two

complex (imaginary) solutions

All quadratics have two solutions, but not all solutions are real:

Discriminant:b2-4ac>0, two real

solutionsb2-4ac=0, one real

solutionb2-4ac<0, two

complex (imaginary) solutions

0

1

2

3

45

6

7

8

9

-4 -2 0 2 40

1

2

3

45

6

7

8

9

-4 -2 0 2 4

Imaginary NumbersImaginary Numbers

All numbers have square roots, even negative numbers:√4=2, -2√-4=2i

Imaginary number: i…for imaginaryi= √-1, so i2=-1

X2=-100 √x2= √-100X= √100• √-1X=10i and -10i

All numbers have square roots, even negative numbers:√4=2, -2√-4=2i

Imaginary number: i…for imaginaryi= √-1, so i2=-1

X2=-100 √x2= √-100X= √100• √-1X=10i and -10i

(√-25)(2i)√25• √-1(2 √-1)5(2) √-12

10(-1)=-10

Complex NumbersComplex Numbers

Complex numbers: a+bi4+2iThe conjugate (opposite) is 4-2i

(4+2i)+(3+i)Only combine like terms: 2i and i are like

terms7+3i

2i(4-7i)8i-14i2=8i-14(-1)=8i+14

Complex numbers: a+bi4+2iThe conjugate (opposite) is 4-2i

(4+2i)+(3+i)Only combine like terms: 2i and i are like

terms7+3i

2i(4-7i)8i-14i2=8i-14(-1)=8i+14

Complex NumbersComplex Numbers

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2 − 3i

1+ 4i

To convert to a+bi form:

Multiply by 1 (the conjugate of the bottom number)

Simplify

To convert to a+bi form:

Multiply by 1 (the conjugate of the bottom number)

Simplify€

2 − 3i

1+ 4i•1− 4i

1− 4i

€

2 −11i +12i2

1−16i2

€

14 −11i

−15