Direction: _____________Width: ______________AOS: _________________Set of corresponding points: _______________Vertex: _______________

Max or Min? __________

y – int: _____________x – int: _____________Function? __________Domain: ___________Range: _____________Rising: _____________Falling: ____________

Do Now:Find all properties:

Opens downNarrow

(-0.5, -7) & (2.5, -7)

x = 1

(1, 0)

Maximum(0, -3)(1, 0)

Yes(-, )

(-, 0](-, 1)(1, )

Answers to Homeworkx x2 – 4x – 5 0 -5

1 -8

2 -9

3 -8

4 -5

Direction: Opens UpWidth: StandardAOS: x = 2Corresponding Point: (6, 7)Vertex: (2, -9); Minimumy – intercept: (0, -5)x – intercept: (-1, 0) & (5, 0)Function? YesDomain: (-, )Range: [-9, )Rising: (2, )Falling: (-, 2)

Answers to Homeworkx -2(x+2)2 + 8-4 0

-3 6

-2 8

-1 6

0 0

Direction: Opens DownWidth: NarrowAOS: x = -2Corresponding Point: (1, -10)Vertex: (-2, 8); Maximumy – intercept: (0, 0)x – intercept: (-4, 0) & (0, 0)Function? YesDomain: (-, )Range: (-, 8]Rising: (-, -2)Falling: (-2, )

Answers to Homeworkx ½(x+4)2 -6 2

-5 ½

-4 0

-3 ½

-2 2

Direction: Opens UpWidth: WideAOS: x = -4Corresponding Point: (-6, 2)Vertex: (-4, 0); Minimumy – intercept: (0, 8)x – intercept: (-4, 0)Function? YesDomain: (-, )Range: [0, )Rising: (-4, )Falling: (-, -4)

Answers to Homeworkx -3x2 + 6x - 4-1 -13

0 -4

1 -1

2 -4

3 -13

Direction: Opens DownWidth: NarrowAOS: x = 1Corresponding Point: (4, -28)Vertex: (1, -1); Maximumy – intercept: (0, -4)x – intercept: Function? YesDomain: (-, )Range: (-, -1]Rising: (-, 1)Falling: (1, -)

Homework

Need Help? Look in textbook inSection 5.1: Modeling Data w/ Quadratic FunctionsSection 5.2: Properties of ParabolasSection 5.5: Quadratic EquationsSection 5.8: The Quadratic Formula

Worksheet: Properties of Parabolas

Unit 4: QuadraticsDay 14: Finding Properties of Parabolas Using Algebra

Unit 5: Quadratics

Objectives:To identify properties of parabolas using algebra

Properties of Parabolas

Begin with the equation in standard form.

Standard form of a quadratic equation:

2y ax bx c

Properties of ParabolasDirection: Parabolas open up or open downDirection is determined by the sign of “a”

Open “up”a is positive

Open “down”a is negative

y = ax2 + bx +c

2 4 42f x x x 2 6 5y x x

Properties of ParabolasWidth: Parabolas can be narrow, standard or wideWidth is determined by the value of a (not including the sign)

Narrow|a| > 1

Standard|a| = 1

Wide|a| < 1

y = ax2 + bx +c

24y x 2y x 21

4f x x

Properties of ParabolasAxis of Symmetry: The line that divides the parabola into two parts that are mirror imagesAOS is found using the formula:

Equation: a = 1, b = 4, c = 1

AOS: x = -2

y = ax2 + bx +c

2

bx

a

2 4 1y x x

2

bx

a

4

2 1

4

2

2

Properties of ParabolasVertex: The point where the parabola passes through the AOSVertex is found by plugging the AOS into the equation.

Equation: AOS: x = -2

Vertex: (-2, -3)Vertex is a minimum because a is

positive and parabola opens up.

22 4 2 1y

2 4 1y x x

4 8 1y 3y

y = ax2 + bx +c

Properties of Parabolasy – intercept: The point on the graph where the parabola intersects the y-axis.y – intercept is found by, making x = 0 and solving for y

Y – intercept will be “c” value

Equation:

y -intercept: (0, 1)

2 4 1y x x

20 4 0 1y

1y

y = ax2 + bx +c

Properties of ParabolasNumber of Real Solutions: The number of times the parabola intersects the x-axis on the real coordinate plane. Use the disriminant to determine the number of solutionsThe discriminant is b2 – 4ac

2 Real Rootsb2 – 4ac > 0

2 4 1y x x 24 4 1 1 12

1 Real Rootb2 – 4ac = 0

2 4 4y x x 24 4 1 4 0

0 Real Roots2 Imaginary Roots

b2 – 4ac < 0

2 4 5y x x 24 4 1 5 4

Properties of Parabolasx – intercept(s): The point(s) on the graph where the parabola intersect the x - axis. Other names include: roots, zeroes and solutions.To find x – intercepts, make y = 0 and solve.Solve quadratics by taking square roots, factoring or using the quadratic equation.

y = ax2 + bx +c

Equation:

x -intercept: (-2+√3, 0) & (-2-√3, 0)

2 4 1y x x 20 4 1x x

214

2 1

4 4 1x

4 12

2x

4 2 3

2

2 3x

Properties of ParabolasFunction: Yes, passes the VLTDomain: The domain of a parabola is (-, )Range: Depends on how parabola opens, includes max or min and infinity. Always use bracket w/ #.

Use y – value of vertex and direction to determine range.

(-, )

Equation: Opens UpVertex: (-2, -3)

Function? Domain: Range:

2 4 1y x x

y = ax2 + bx +c

Yes

[-3, )

Properties of ParabolasIntervals of Rising/Falling: The interval of the domain where the graph is rising or falling as x increasesUse x – value of vertex and direction to find intervals

Rising: ______________

Falling: _____________ , 2

2,

Equation: Opens Up Vertex: (-2, -3)

2 4 2y x x

Find all properties: Direction: __________Width: _____________AOS: ______________Vertex: _____________

Max or Min? __________

y – int: _____________# of Real Solutions: ___ x – int: _____________Function? __________Domain: ___________Range: _____________Rising: _____________Falling: ____________

Opens upStandard

x = -2(-2, -9)

Minimum(0, -5)

(-5, 0) & (1, 0)Yes

(-, )[-9, )(-2, )(-, -2)

2 4 5y x x a is positive a =1

2

bx

a

4

2 1

2

22 4 2 5y

4 8 5y 9

20 4 0 5y 5y

20 4 5x x 0 5 1x x

5 0x 1 0x

5x 1x

2

24 4(1)( 5) 362 4b ac

4 36

2 1x

4 6

2x

Or

Properties of Parabolas

Direction: _____________Width: ______________AOS: _________________Vertex: _______________

Max or Min? __________

y – int: _____________# of Real Solutions: ____

x – int: _____________Function? __________Domain: ___________Range: _____________Rising: _____________Falling: ____________

Find all properties: Opens DownNarrow

x = -1(-1, -2)

Maximum(0, -5)

Yes(-, )

(-, -2](-, -1)(-1, )

23 6 5y x x a is negative a = 3

2

bx

a

6

2 3

1

23 1 6 1 5y 3 6 5y 2

23 0 6 0 5y 5

20 3 6 5x x 6 24

6x

6 2 6

6

i

6

13

i

6

6

6 61 ,0 1 ,0

3 3

i i

26 4 3 5 242 4b ac

0

Properties of Parabolas

Did you meet today’s objective?Name three of the properties you learned about today and how to find them.