Unit 10: Introduction to Quadratic FunctionsFoundations of Mathematics 1

Ms. C. Taylor

Warm-Up

Quadratic FunctionA function of the form

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

Example quadratic equation:

Vertex-

The lowest or highest point

of a parabola.

Vertex

Axis of symmetry-

The vertical line through the vertex of the parabola.

Axis ofSymmetry

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

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

Now you try one!

y=-x2+x+12

* Open up or down?* Vertex?

* Axis of symmetry?* Table of values with 5

points?

(-1,10)

(-2,6)

(2,10)

(3,6)

X = .5

(.5,12)

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

Now you try one!

y=2(x-1)2+3

Open up or down?Vertex?

Axis of symmetry?Table of values with 5 points?

(-1, 11)

(0,5)

(1,3)

(2,5)

(3,11)

X = 1

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)

Now you try one!

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

Open up or down?X-intercepts?

Vertex?Axis of symmetry?

(-1,0) (3,0)

(1,-8)

x=1

5 4 3 2 1 0 1 2 3 4 5

20

18

16

14

12

10

8

6

4

2

2

4

6

8

10

12

14

16

18

20

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

5 4 3 2 1 0 1 2 3 4 5

20

18

16

14

12

10

8

6

4

2

2

4

6

8

10

12

14

16

18

20

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.

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

20

18

16

14

12

10

8

6

4

2

2

4

6

8

10

12

14

16

18

20

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.

Roots & Zeros of Polynomials I

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

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.

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

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.

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

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 !

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.

Zeros of a Polynomial Function

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

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

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)

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!

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)