Unit 10: Introduction to Quadratic Functions Foundations of Mathematics 1 Ms. C. Taylor.

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Unit 10: Introduction to Quadratic Functions Foundations of Mathematics 1 Ms. C. Taylor

Transcript of Unit 10: Introduction to Quadratic Functions Foundations of Mathematics 1 Ms. C. Taylor.

Page 1: Unit 10: Introduction to Quadratic Functions Foundations of Mathematics 1 Ms. C. Taylor.

Unit 10: Introduction to Quadratic FunctionsFoundations of Mathematics 1

Ms. C. Taylor

Page 2: Unit 10: Introduction to Quadratic Functions Foundations of Mathematics 1 Ms. C. Taylor.

Warm-Up

Page 3: Unit 10: Introduction to Quadratic Functions Foundations of Mathematics 1 Ms. C. Taylor.

Quadratic FunctionA function of the form

y=ax2+bx+c where a≠0 making a u-shaped graph called a parabola.

Example quadratic equation:

Page 4: Unit 10: Introduction to Quadratic Functions Foundations of Mathematics 1 Ms. C. Taylor.

Vertex-

The lowest or highest point

of a parabola.

Vertex

Axis of symmetry-

The vertical line through the vertex of the parabola.

Axis ofSymmetry

Page 5: Unit 10: Introduction to Quadratic Functions Foundations of Mathematics 1 Ms. C. Taylor.

Standard Form Equationy=ax2 + bx + c

If a is positive, u opens up

If a is negative, u opens down The x-coordinate of the vertex is at To find the y-coordinate of the vertex, plug the x-

coordinate into the given eqn. The axis of symmetry is the vertical line x= Choose 2 x-values on either side of the vertex x-

coordinate. Use the eqn to find the corresponding y-values.

Graph and label the 5 points and axis of symmetry on a coordinate plane. Connect the points with a smooth curve.

a

b

2

a

b

2

Page 6: Unit 10: Introduction to Quadratic Functions Foundations of Mathematics 1 Ms. C. Taylor.

Example 1: Graph y=2x2-8x+6

a=2 Since a is positive the parabola will open up.

Vertex: use b=-8 and a=2

Vertex is: (2,-2)

a

bx

2

24

8

)2(2

)8(

x

26168

6)2(8)2(2 2

y

y

• Axis of symmetry is the vertical line x=2

•Table of values for other points: x y

0 6 1 0 2 -2 3 0 4 6

* Graph!x=2

Page 7: Unit 10: Introduction to Quadratic Functions Foundations of Mathematics 1 Ms. C. Taylor.

Now you try one!

y=-x2+x+12

* Open up or down?* Vertex?

* Axis of symmetry?* Table of values with 5

points?

Page 8: Unit 10: Introduction to Quadratic Functions Foundations of Mathematics 1 Ms. C. Taylor.

(-1,10)

(-2,6)

(2,10)

(3,6)

X = .5

(.5,12)

Page 9: Unit 10: Introduction to Quadratic Functions Foundations of Mathematics 1 Ms. C. Taylor.

Example 2: Graphy=-.5(x+3)2+4 a is negative (a = -.5), so parabola opens down. Vertex is (h,k) or (-3,4) Axis of symmetry is the vertical line x = -3 Table of values x y

-1 2

-2 3.5

-3 4

-4 3.5

-5 2

Vertex (-3,4)

(-4,3.5)

(-5,2)

(-2,3.5)

(-1,2)

x=-3

Page 10: Unit 10: Introduction to Quadratic Functions Foundations of Mathematics 1 Ms. C. Taylor.

Now you try one!

y=2(x-1)2+3

Open up or down?Vertex?

Axis of symmetry?Table of values with 5 points?

Page 11: Unit 10: Introduction to Quadratic Functions Foundations of Mathematics 1 Ms. C. Taylor.

(-1, 11)

(0,5)

(1,3)

(2,5)

(3,11)

X = 1

Page 12: Unit 10: Introduction to Quadratic Functions Foundations of Mathematics 1 Ms. C. Taylor.

Example 3: Graph y=-(x+2)(x-4)Since a is negative,

parabola opens down.

The x-intercepts are (-2,0) and (4,0)

To find the x-coord. of the vertex, use

To find the y-coord., plug 1 in for x.

Vertex (1,9)

2

qp

12

2

2

42

x

9)3)(3()41)(21( y

•The axis of symmetry is the vertical line x=1 (from the x-coord. of the vertex)

x=1

(-2,0) (4,0)

(1,9)

Page 13: Unit 10: Introduction to Quadratic Functions Foundations of Mathematics 1 Ms. C. Taylor.

Now you try one!

y=2(x-3)(x+1)

Open up or down?X-intercepts?

Vertex?Axis of symmetry?

Page 14: Unit 10: Introduction to Quadratic Functions Foundations of Mathematics 1 Ms. C. Taylor.

(-1,0) (3,0)

(1,-8)

x=1

Page 15: Unit 10: Introduction to Quadratic Functions Foundations of Mathematics 1 Ms. C. Taylor.

5 4 3 2 1 0 1 2 3 4 5

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x2

3x2

5x2

x2

2 x2

x

Quadratic of the form f(x) = ax2

Key Features

Symmetry about x =0

Vertex at (0,0)

The bigger the value

of a the steeper the curve.

-x2 flips the curve about x - axis

Page 16: Unit 10: Introduction to Quadratic Functions Foundations of Mathematics 1 Ms. C. Taylor.

5 4 3 2 1 0 1 2 3 4 5

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x2

4

x2

1

3x2

1

x2 3

2 x2

2

x

Quadratic of the form f(x) = ax2 + c Key Features

Symmetry about x = 0

Vertex at (0,C)

a > 0 the vertex (0,C) is a minimum turning point.

a < 0 the vertex (0,C) is a maximum turning point.

Page 17: Unit 10: Introduction to Quadratic Functions Foundations of Mathematics 1 Ms. C. Taylor.

8 7 6 5 4 3 2 1 0 1 2 3 4 5 6

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x 3( )2

x 4( )2

x 1( )2

2 x 3( )2

x 1( )2

x

Quadratic of the form f(x) = a(x - b)2

Key Features

Symmetry about x = b

Vertex at (b,0)Cuts y - axis at x =

0

a > 0 the vertex (b,0) is a minimum turning point.

a < 0 the vertex (b,0) is a maximum turning point.

Page 18: Unit 10: Introduction to Quadratic Functions Foundations of Mathematics 1 Ms. C. Taylor.

Roots & Zeros of Polynomials I

How the roots, solutions, zeros, x-intercepts and factors of a polynomial function are related.

Page 19: Unit 10: Introduction to Quadratic Functions Foundations of Mathematics 1 Ms. C. Taylor.

Polynomials

A Polynomial Expression can be a monomial or a sum of monomials. The Polynomial Expressions that we are discussing today are in terms of one variable.

In a Polynomial Equation, two polynomials are set equal to each other.

Page 20: Unit 10: Introduction to Quadratic Functions Foundations of Mathematics 1 Ms. C. Taylor.

Factoring PolynomialsTerms are Factors of a Polynomial if, when

they are multiplied, they equal that polynomial:

2 2 15 ( 3)( 5)x x x x (x - 3) and (x + 5) are

Factors of the polynomial 2

2 15x x

Page 21: Unit 10: Introduction to Quadratic Functions Foundations of Mathematics 1 Ms. C. Taylor.

Since Factors are a Product...

…and the only way a product can equal zero is if one or more of the factors are zero…

…then the only way the polynomial can equal zero is if one or more of the factors are zero.

Page 22: Unit 10: Introduction to Quadratic Functions Foundations of Mathematics 1 Ms. C. Taylor.

Solving a Polynomial Equation

The only way that x2 +2x - 15 can = 0 is if x = -5 or x = 3

Rearrange the terms to have zero on one side: 2 22 15 2 15 0x x x x

Factor: ( 5)( 3) 0x x

Set each factor equal to zero and solve: ( 5) 0 and ( 3) 0

5 3

x x

x x

Page 23: Unit 10: Introduction to Quadratic Functions Foundations of Mathematics 1 Ms. C. Taylor.

Solutions/Roots a Polynomial

Setting the Factors of a Polynomial Expression equal to zero gives the Solutions to the Equation when the polynomial expression equals zero. Another name for the Solutions of a Polynomial is the Roots of a Polynomial !

Page 24: Unit 10: Introduction to Quadratic Functions Foundations of Mathematics 1 Ms. C. Taylor.

Zeros of a Polynomial Function

A Polynomial Function is usually written in function notation or in terms of x and y.

f (x) x2 2x 15 or y x2 2x 15

The Zeros of a Polynomial Function are the solutions to the equation you get when you set the polynomial equal to zero.

Page 25: Unit 10: Introduction to Quadratic Functions Foundations of Mathematics 1 Ms. C. Taylor.

Zeros of a Polynomial Function

The Zeros of a Polynomial Function ARE the Solutions to the Polynomial Equation when the polynomial equals zero.

Page 26: Unit 10: Introduction to Quadratic Functions Foundations of Mathematics 1 Ms. C. Taylor.

Graph of a Polynomial FunctionHere is the graph of our polynomial function:

The Zeros of the Polynomial are the values of x when the polynomial equals zero. In other words, the Zeros are the x-values where y equals zero.

y x2 2x 15

Page 27: Unit 10: Introduction to Quadratic Functions Foundations of Mathematics 1 Ms. C. Taylor.

y x2 2x 15

x-Intercepts of a PolynomialThe points where y = 0 are called the x-intercepts of the graph.

The x-intercepts for our graph are the points...

and(-5, 0) (3, 0)

Page 28: Unit 10: Introduction to Quadratic Functions Foundations of Mathematics 1 Ms. C. Taylor.

x-Intercepts of a PolynomialWhen the Factors of a Polynomial Expression are set equal to zero, we get the Solutions or Roots of the Polynomial Equation.

The Solutions/Roots of the Polynomial Equation are the x-coordinates for the x-Intercepts of the Polynomial Graph!

Page 29: Unit 10: Introduction to Quadratic Functions Foundations of Mathematics 1 Ms. C. Taylor.

Factors, Roots, Zeros

y x2 2x 15For our Polynomial Function:

The Factors are: (x + 5) & (x - 3)

The Roots/Solutions are: x = -5 and 3

The Zeros are at: (-5, 0) and (3, 0)