The Vertex of a Parabola

Finding the Maximum or Minimum

The Graph of a Parabola

Parabolas all have the same shape but they can be oriented in different directions, depending on the equation.

This is the general equation of parabolas that open up or down.2y ax bx c

Parabola Opening Upwards

If a is positive, the graph opens upwards.

2y ax bx c

23 4 1y x x

3 is positive – graph opens upwards

Parabola Opening Downwards

If a is negative, the graph opens downwards.

2y ax bx c

3 is negative – graph opens downwards

23 2 1y x x

The Vertex of a Parabola

The vertex of a parabola is the low point if it opens upward.

Vertex

This is called a minimum.

The Vertex of a Parabola

The vertex of a parabola is the high point if it opens downward.

Vertex

This is called a maximum.

Axis of Symmetry of a Parabola

The axis of symmetry of a parabola passes through the vertex.

Axis of Symmetry of a Parabola

The axis of symmetry of a parabola passes through the vertex.

If we folded the parabola across the axis of symmetry, the left side and the right side would match up.

We can say that the graph is symmetric with respect to the line.

Axis of Symmetry Formula

The formula for the axis of symmetry of a parabola that opens up or down is:

2Parabola:

Axis of symmetry: 2

y ax bx c

bx

a

The axis of symmetry is a vertical line.We’re expecting an equation in the form x = a number.

Axis of Symmetry of a Parabola

Let’s look at an example:

23 6 9

6

2 36

61

y x x

x

x

x

2Parabola:

Axis of symmetry: 2

y ax bx c

bx

a

Axis of symmetry

Vertex of a Parabola

The vertex is on the axis of symmetry.

To find the y-coordinate of the vertex, substitute the value you found for x into the equation of the parabola.

2

2

3 6 9

3( 1) 6( 1) 9

3(1) 6 9

3 6 9

12

y x x

y

y

y

y

23 6 9

6

2 36

61

y x x

x

x

x

Vertex: (-1, -12)

Remember

a is positive a is negative

graph opens up graph opens down

vertex is a minimum vertex is a maximum

2Parabola:

x-coordinate of vertex: 2

To calculate the y-coordinate of the vertex,

plug that x value into the equation of the parabola.

y ax bx c

bx

a