Chapter 8 Review Quadratic Functions § 8.3 Graphing Quadratic Equations in Two Variables

Chapter 8Review

Quadratic Functions

§ 8.3

Graphing Quadratic Equations in Two

Variables

Martin-Gay, Developmental Mathematics 3

We spent a lot of time graphing linear equations in chapter 3.

The graph of a quadratic equation is a parabola.

The highest point or lowest point on the parabola is the vertex.

Axis of symmetry is the line that runs through the vertex and through the middle of the parabola.

Graphs of Quadratic Equations


x

y

Graph y = 2x2 – 4.

x y

0 –4

1 –2

–1 –2

2 4

–2 4

(2, 4)(–2, 4)

(1, –2)(–1, – 2)

(0, –4)


Example


Although we can simply plot points, it is helpful to know some information about the parabola we will be graphing prior to finding individual points.

To find x-intercepts of the parabola, let y = 0 and solve for x.

To find y-intercepts of the parabola, let x = 0 and solve for y.

Intercepts of the Parabola


If the quadratic equation is written in standard form, y = ax2 + bx + c,

1) the parabola opens up when a > 0 and opens down when a < 0.

2) the x-coordinate of the vertex is . a

b

2

To find the corresponding y-coordinate, you substitute the x-coordinate into the equation and evaluate for y.

Characteristics of the Parabola


x

yGraph y = –2x2 + 4x + 5.

x y

1 7

2 5

0 5

3 –1

–1 –1

(3, –1)(–1, –1)

(2, 5)(0, 5)

(1, 7)Since a = –2 and b = 4, the graph opens down and the x-coordinate of the vertex is 1

)2(2

4


Example


The Graph of a Quadratic Function

The vertical line which passes through the vertex is called the Axis of Symmetry or the Axis

Recall that the equation of a vertical line is x =c

For some constant c

x coordinate of the vertex

The y coordinate of the vertex is

a

b

2

a

bac

2

4 2

The Axis of Symmetry is the x =a

b

2


The Quadratic function

Opens up when a>o

axis of symmetry

opens down a < 0

Vertex


Identify the Vertex and Axis of Symmetry of a Quadratic Function

Vertex =(x, y). thusVertex =

Axis of Symmetry: the line x =

Vertex is minimum point if parabola opens up Vertex is maximum point if parabola opens down

a

bf

a

b

2,

2

a

b

2


Identify the Vertex and Axis of Symmetry

Vertex x =

y =

Vertex = (-1, -3)Axis of Symmetry is x =

)3(2

6

xxxf 63)( 2

a

b

2

= = -1

1

2f

a

bf -3

a

b

2

= -1


The number of real solutions is at most two.

8.5 Quadratic Solutions

No solutions One solution Two solutions


Example f(x) = x2 - 4

Identifying Solutions

Solutions are -2 and 2.


Now you try this problem.

f(x) = 2x - x2

Solutions are 0 and 2.

Identifying Solutions


The graph of a quadratic equation is a parabola.

The roots or zeros are the x-intercepts.

The vertex is the maximum or minimum point.

All parabolas have an axis of symmetry.

Graphing Quadratic Equations


One method of graphing uses a table with

arbitrary

x-values.Graph y = x2 - 4x

Roots 0 and 4 , Vertex (2, -4) , Axis of Symmetry x = 2


x y0 01 -32 -43 -34 0


Try this problem y = x2 - 2x - 8.

RootsVertexAxis of Symmetry


x y-2-1134


8.6 – Solving Quadratic Equations by Factoring

A quadratic equation is written in the Standard Form, 2 0ax bx c where a, b, and c are real numbers and .0a

Examples: 2 7 12 0x x

23 4 15x x 7 0x x

(standard form)


Zero Factor Property: If a and b are real numbers and if , 0ab

Examples:

7 0x x

then or . 0a 0b

0x 7 0x 7x 0x



Zero Factor Property: If a and b are real numbers and if , 0ab

Examples: 10 3 6 0x x

then or . 0a 0b

10 0x 3 6 0x

10x 3 6x 2x

10 10 01 0x 63 66 0x 3 6

3 3

x



Solving Quadratic Equations: 1) Write the equation in standard form.

4) Solve each equation.

2) Factor the equation completely.

3) Set each factor equal to 0.

5) Check the solutions (in original equation).



2 3 18 0x x

6 0x 3 0x 3x

6x 3x

2 3 18x x

18 :Factors of1,18 2, 9 3, 6

26 3 16 8

36 18 18

18 18

213 3 83

9 9 18 18 18

6x 0



3 18x x 18x

2 3 18x x

218 13 18 8

324 54 18

270 18

221 23 11 8

441 63 18 378 18

3 18x

3 183 3x 21x

If the Zero Factor Property is not used, then the solutions will be incorrect



2 4 5x x

1 0x 5 0x

1 5 0x x

1x 5x

4 5x x

2 4 5 0x x



23 7 6x x 3 0x 3 2 0x

3 3 2 0x x

3x 2

3x

3 7 6x x

23 7 6 0x x 3 2x

6 :Factors of2, 31, 6

3:Factors of1, 3



29 24 16x x 29 24 16 0x x

3 4 0x 3 4 3 4 0x x

4

3x

3 4x

9 16and are perfect squares



32 18 0x x 2x

2 0x

2x

3x 3 0x 3 0x

3x 0x

2 9x 0

3x 3x 0



23 3 20 7 0x x x

3x

3 0x

7x 7 0x 3 1 0x

1

3x

3x 3 1x

3:Factors of 1, 3 7 :Factors of 1, 7

7x 0 3 1x



0

A cliff diver is 64 feet above the surface of the water. The formula for calculating the height (h) of the diver after t seconds is: 216 64.h t How long does it take for the diver to hit the surface of the water?

0 0

2 0t 2 0t 2t 2t seconds

216 64t 16 2 4t

16 2t 2t

8.6 – Quadratic Equations and Problem Solving


2x

The square of a number minus twice the number is 63. Find the number.

7x

7x

x is the number.

2 2 63 0x x

7 0x 9 0x

9x

2x 63

63:Factors of 1, 63 3, 21 7, 9

9x 0



5 176w w

The length of a rectangular garden is 5 feet more than its width. The area of the garden is 176 square feet. What are the length and the width of the garden?

11w The width is w.

11 0w 11w

The length is w+5.l w A

2 5 176w w 2 5 176 0w w

16 0w 16w

11w 11 5l 16l

feet

feet

176 :Factors of1,176 2, 88 4, 44

8, 22 11,16

16w 0



x

Find two consecutive odd numbers whose product is 23 more than their sum?

Consecutive odd numbers: x

5x 5x 2 2 2 25x x x

2 25 0x 5x

5 0x 5 0x

5, 3 5, 7

5 2 3 5 2 7

2.x 2x 2x x 23

2 22 2 2 25xx x x x

2 25 2525x

2 25x

5x 0



a x

The length of one leg of a right triangle is 7 meters less than the length of the other leg. The length of the hypotenuse is 13 meters. What are the lengths of the legs?

12a

.Pythagorean Th

22 27 13x x

5x

5

meters

7b x 13c

2 2 14 49 169x x x 22 14 120 0x x

22 7 60 0x x

2

5 0x 12 0x 12x

12 7b meters

2 2 2a b c

60 :Factors of 1, 60 2, 303, 20 4,15 5,12

5x 12x 0

6,10


§ 8.7

Solving Quadratic Equations by the Square

Root Property


Square Root Property

We previously have used factoring to solve quadratic equations.

This chapter will introduce additional methods for solving quadratic equations.

Square Root PropertyIf b is a real number and a2 = b, then

ba


Solve x2 = 49

2x

Solve (y – 3)2 = 4

Solve 2x2 = 4

x2 = 2

749 x

y = 3 2

y = 1 or 5

243 y


Example


Solve x2 + 4 = 0 x2 = 4

There is no real solution because the square root of 4 is not a real number.


Example


Solve (x + 2)2 = 25

x = 2 ± 5

x = 2 + 5 or x = 2 – 5

x = 3 or x = 7

5252 x


Example


Solve (3x – 17)2 = 28

72173 x

3

7217 x

7228 3x – 17 =


Example

§ 8.8

Solving Quadratic Equations by Completing

the Square


In all four of the previous examples, the constant in the square on the right side, is half the coefficient of the x term on the left.

Also, the constant on the left is the square of the constant on the right.

So, to find the constant term of a perfect square trinomial, we need to take the square of half the coefficient of the x term in the trinomial (as long as the coefficient of the x2 term is 1, as in our previous examples).

Completing the Square


What constant term should be added to the following expressions to create a perfect square trinomial?

x2 – 10xadd 52 = 25

x2 + 16xadd 82 = 64

x2 – 7x

add 4

49

2

72


Example


We now look at a method for solving quadratics that involves a technique called completing the square.

It involves creating a trinomial that is a perfect square, setting the factored trinomial equal to a constant, then using the square root property from the previous section.


Example


Solving a Quadratic Equation by Completing a Square

1) If the coefficient of x2 is NOT 1, divide both sides of the equation by the coefficient.

2) Isolate all variable terms on one side of the equation.

3) Complete the square (half the coefficient of the x term squared, added to both sides of the equation).

4) Factor the resulting trinomial.

5) Use the square root property.



Solve by completing the square.

y2 + 6y = 8y2 + 6y + 9 = 8 + 9

(y + 3)2 = 1

y = 3 ± 1

y = 4 or 2

y + 3 = ± = ± 11

Solving Equations

Example



y2 + y – 7 = 0

y2 + y = 7

y2 + y + ¼ = 7 + ¼

2

29

4

29

2

1y

2

291

2

29

2

1 y

(y + ½)2 = 429

Solving Equations

Example



2x2 + 14x – 1 = 0

2x2 + 14x = 1

x2 + 7x = ½

2

51

4

51

2

7x

2

517

2

51

2

7 x

x2 + 7x + = ½ + = 4

49

4

49

4

51

(x + )2 = 4

51

2

7

Solving Equations

Example

§ 8.9

Solving Quadratic Equations by the

Quadratic Formula


The Quadratic Formula

Another technique for solving quadratic equations is to use the quadratic formula.

The formula is derived from completing the square of a general quadratic equation.


A quadratic equation written in standard form, ax2 + bx + c = 0, has the solutions.

a

acbbx

2

42



Solve 11n2 – 9n = 1 by the quadratic formula.

11n2 – 9n – 1 = 0, so

a = 11, b = -9, c = -1

)11(2

)1)(11(4)9(9 2

n

22

44819

22

1259

22

559


Example


)1(2

)20)(1(4)8(8 2

x

2

80648

2

1448

2

128 20 4 or , 10 or 22 2

x2 + 8x – 20 = 0 (multiply both sides by 8)

a = 1, b = 8, c = 20

8

1

2

5Solve x2 + x – = 0 by the quadratic formula.


Example


Solve x(x + 6) = 30 by the quadratic formula.

x2 + 6x + 30 = 0

a = 1, b = 6, c = 30

)1(2

)30)(1(4)6(6 2

x

2

120366

2

846

So there is no real solution.


Example


The expression under the radical sign in the formula (b2 – 4ac) is called the discriminant.

The discriminant will take on a value that is positive, 0, or negative.

The value of the discriminant indicates two distinct real solutions, one real solution, or no real solutions, respectively.

The Discriminant


Use the discriminant to determine the number and type of solutions for the following equation.

5 – 4x + 12x2 = 0

a = 12, b = –4, and c = 5

b2 – 4ac = (–4)2 – 4(12)(5)

= 16 – 240

= –224

There are no real solutions.

The Discriminant

Example


Solving Quadratic Equations

Steps in Solving Quadratic Equations1) If the equation is in the form (ax+b)2 = c, use

the square root property to solve.

2) If not solved in step 1, write the equation in standard form.

3) Try to solve by factoring.

4) If you haven’t solved it yet, use the quadratic formula.


Solve 12x = 4x2 + 4.

0 = 4x2 – 12x + 4

0 = 4(x2 – 3x + 1)

Let a = 1, b = -3, c = 1

)1(2

)1)(1(4)3(3 2

x

2

493

2

53

Solving Equations

Example


Solve the following quadratic equation.

02

1

8

5 2 mm

0485 2 mm

0)2)(25( mm

02025 mm or

25

2 mm or

Solving Equations

Example



Solve for x by completing the square.

a

acbbx

2

42


Yes, you can remember this formula

Pop goes the Weaselhttp://www.youtube.com/watch?v=2lbABbfU6Zc&feature=related

Gilligan’s Islandhttp://www.youtube.com/watch?v=3CWTt9QFioY&feature=related

This one I can’t explainhttp://www.youtube.com/watch?v=haq6kpWdEMs&feature=related


How does it work

Equation:

1

5

3

0153 2

c

b

a

xx

a

acbbx

2

42


How does it work

Equation:

1

5

3

0153 2

c

b

a

xx

6

13

6

5

6

135

6

12255

32

13455 2

x

x

x

a

acbbx

2

42


The Discriminant

The number in the square root of the quadratic formula.

acb 42

12425

6145

0652

2

xxGiven


The Discriminant

The Discriminant can be negative, positive or zero

If the Discriminant is positive,

there are 2 real answers.

If the square root is not a prefect square

( for example ),

then there will be 2 irrational roots

( for example ).

25

52


The Discriminant

The Discriminant can be negative, positive or zero

If the Discriminant is positive,

there are 2 real answers.

If the Discriminant is zero,there is 1 real answer.

If the Discriminant is negative,there are 2 complex

answers.complex answer have i.


Solve using the Quadratic formula

3382 xx



12

331488

0338

338

2

2

2

x

xx

xx



32

6

2

148

112

22

2

1482

148

2

1968

12

331488

0338

338

2

2

2

x

x

x

x

xx

xx



0289342 xx



12

289143434

028934

2

2

x

xx



172

34

2

034

2

1156115634

12

289143434

028934

2

2

x

x

x

xx



0262 xx



732

72

2

6

2

286

2

8366

12

21466

026

2

2

x

x

x

xx



12

131466

0136

613

2

2

2

x

xx

xx



2

166

2

52366

12

131466

0136

613

2

2

2

x

x

xx

xx



ix

ii

x

x

x

xx

xx

23

2

4

2

6

2

46

2

166

2

52366

12

131466

0136

613

2

2

2


Describe the roots

Tell me the Discriminant and the type of roots 0962 xx


Describe the roots

Tell me the Discriminant and the type of roots

0, One rational root

0962 xx


Describe the roots



0962 xx

0532 xx


Describe the roots



-11, Two complex roots

0962 xx

0532 xx


Describe the roots




0962 xx

0532 xx

0482 xx


Describe the roots




80, Two irrational roots

0962 xx

0532 xx

0482 xx

Documents

Chapter 8 Review Quadratic Functions § 8.3 Graphing Quadratic Equations in Two Variables