**Quadratic Functions**

Graphing a Quadratic Function

Discriminant

Word Problem

The height in feet of a bottle rocket is given by h(t)= 160t-16t where t is the time in seconds. How long will it take for the rocket to return to the ground? What is the height after 2 seconds?

Finding Roots

**By: Sam Baker and Allan Andrews**

**Vocabulary**

Axis of Symmetry

The line that goes through the middle of a parabola and through the vertex to create two equal sections.

Discriminant

b

2

- 4ac

When the letters from

ax

2

+bx+ c = 0

are pulled out into

the discriminant, the out come determines how many solutions the equation has.

Maximum/ Minimum Point

Quadratic Formula

Standard Form

Vertex

Vertex Form

Zeros/ Roots/ Intercepts

2

Using Factoring

Using the Quadratic Formula

Opens up or down

Axis of Symmetry

Vertex

Writing a Quadratic Equation

Given the roots

Standard form to vertex form

Vertex form to standard form

The highest point of the graph is called the maximum point. This is caused from the coefficient being greater then zero.

The lowest point of the graph is called the minimum point. This is caused from the coefficient being less then zero.

Formula to solve quadratic equations.

ax

2

+

The vertex is the highest or lowest point in a parabola including the y-coordinate and x- coordinate.

X=

____

-b

2a

The zeros are when y=0 the two x intercepts are the zeros.

These are also the roots to the quadratic equation.

At 10 seconds the rocket will hit the ground

After 2 seconds the height of the bottle rocket will be 256.

x

2

+ 3x+ 2=0

2

1

(x+2)(x+1)=0

x+2=0

x+1=0

The roots are -2 and -1

x

2

+ 3x+ 2=0

a=1

b=3

c=2

Plug in the numbers into the formula.

The roots are -2 and -1

x

2

+3x+2=0

Plug the numbers into the discriminant.

The discriminant equals 1. There for there is 2 real roots.

x=-2

x=-1

(x+2)(x+1)=0

Now foil

x

2

+3x+2

no real solutions

one real solutions

two real solutions

ax

2

2

is the quadratic term

is the linear term

is the constant term

bx

bx

+

c

c

=0

b

2

-4

a

c

a=1

b=3

c=2

1(x+1.5)

2

+.25

y=x

2

+3x+2

x

2

+3x +2

1(x-1.5)

2

+.25

To determine if a parabola opens up or down you look at your quadratic term.

y=x

2

y=-x

2

If the parabola opens up the quadratic term is positive.

If the parabola opens down the quadratic term is negative

f(x)= x

2

+2x+1

(x+1)

(x+1)

x=-1

f(x)= -1

2

-2+1

x=-1

To find the vertex just plug the axis of symmetry back into your original problem