College Level Math – Vertex, Axis of Symmetry and Maximum/Minimum points

Find the vertex, the axis of symmetry, the maximum or minimum value of the quadratic function.

The above equation is written in the standard quadratic equation form which is:

It is important to be able to calculate the vertex, line of symmetry, and the maximum or minimum points from a quadratic equation. These crucial points can assist in identifying the characteristics of the resulting parabola and thus being able to sketch the resulting graph.

It is important to note that the sign of the (a) term will determine the orientation of the parabola. A positive (a) term – an (a) term greater than zero - will indicate that the parabola will open upwards, while a negative (a) term – an (a) term less than zero – will indicate that the parabola will open downward.

A vertex is either the highest or lowest point of a parabola depending on its orientation. Once calculated, the vertex is written in an (x ,y) ordered pair.

Therefore, it is easy to determine the following characteristics about the parabola presented in this problem. Those characteristics are:

1) This parabola opens upwards since the (a) term is positive or greater than zero.

2) This parabola will have a vertex at its lowest point.

3) This parabola will have a minimum point.

4) The axis of symmetry will be the vertical line passing through the x-coordinate of the vertex and dividing the parabola into two equal parts.

The following steps detail how the vertex is calculated for this problem:

Vertex

Axis of Symmetry

The axis of symmetry is the line that splits the parabola into two identical halves. It passes through the vertex.
Since this quadratic opens upwards, that means that the axis of symmetry is the x co-ordinate of the vertex. Therefore, the axis of symmetry is the vertical line that passes through x = 5

Maximum or Minimum points

The vertex of a parabola is usually considered as either a maximum or a Minimum value.

We determine if the point is a maximum or minimum by looking at the a co-efficient of the a-term of the quadratic equation. Since the a-term in this quadratic equation is greater than zero, then the vertex is a minimum point because the parabola opens upwards.

Therefore for this quadratic the minimum value point occurs at (5,-28)
If a > 0, the vertex is a minimum point and the minimum value of the quadratic function f is equal to the y-co-ordinate of the vertex. That means the minimum point for this quadratic = -28
This minimum value occurs at x = = -b/2a which is the x-co-ordinate of the vertex. For this quadratic the minimum value = 5



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