ParabolaConic section

Quadratic FunctionsThe graph of a quadratic function is a parabola.

If the parabola opens up, the lowest point is called the vertex.

If the parabola opens down, the vertex is the highest point.

y

x

Vertex

Vertex

Standard Form

y = ax2 + bx + c

The parabola will open down when the a value is negative.

The parabola will open up when the a value is positive.

y

x

The standard form of a quadratic function is

a = positive

a = negative

Example

Graph the quadratic equation by factoring

1. y = x2 + 2x - 15

y

x

(x - 3) ( x + 5)

Parabola opens

upward

x = 3 , x =-5

Example

Graph the quadratic equation by factoringy

x

Parabola opens

downward

2. y = -(x2 - 6x + 8)

(x - 4) ( x - 2)x = 4 , x = 2

Graphing with vertex

Vertex = (__, __)x y

(__, __)0 –4

Formula in finding the vertex

x = –b_ 2a

y = substitute the value of x

y = ax2 + bx + c

Example

Find the vertex of y = -3x2 + 6x

+ 5

Formula in finding the vertex

x = –b_ 2a

y = substitute the value of x

x = –b_ 2a

x = –6 2(-3)

x = –6 –6

x = 1

y = substitute the value of x

y = -3x2 + 6x + 5

y = -3(1)2 + 6(1) + 5

y = -3 + 6 + 5

y = 8 Vertex = (1, 8)

y

x

Graphing: y = ax2 + c4. y = x2 – 2

Vertex = (0, –2)

5. y = x2 + 2

Vertex = (0, 2)y

x

Graphing: y = ax2 + c5. y = x2 + 2

Vertex = (0, 2)y

x

6. y = –x2 + 2

Vertex = (0, 2)y

x

Comparing Parabolay

x x - axis

y - axis

Green y = 5x2

Purple y = ½x2

Blue y = ¼ x2

Smaller coefficient = Wider parabola