Maximum point and minimum point

One high school function is occupation which can be written in the form: f (x) = ax² + bx + c, where a ≠ 0. All high school function can be graphically represented by a parable. There are some cases in which this parable may be facing upwards, thus having a minimum point, and others in which it can be turned down, thus having a Scoreinmaximum.

the candidate for Scoreinmaximum (or minimum) in the graph of a parable it's called vertex, therefore, finding the coordinates of the vertex is equivalent to finding the localizationofScoreinmaximum or from the minimum of the parable. If V(x_vy_v) is the vertex with its coordinates, so the formulas that can be used to find those coordinates are:

x_v = - B
2nd

y_v = – Δ
4th

Minimum point

It is not necessary to build the parable to observe your Scoreinmaximum. From the function of the second degree, it is possible to obtain all the necessary information algebraically. It's just not possible to see the location of that point.

All parable/second degree function has vertex. That

vertex is the point of Minimum if the coefficient a > 0. This causes the parabola to have a concavity facing upwards and thus have a “minimum value”, as shown in the following figure.

Looking at the drawing, it is possible to see that "below" the minimum point there are no other points in the parable. However, it is more correct to say that the smallest y-coordinate of some point belonging to a parabola, with a > 0, is the coordinate of the ScoreinMinimum.

maximum point

All parable/occupation of seconddegree with the maximum coordinate, as its concavity is turned downwards and, therefore, it has a point that is the “highest of all”.

Again, it is correct to say that there is no point belonging to this parabola with a y coordinate greater than this same coordinate of the vertex.

The following image shows a parabola with a concavity facing downwards and its point of maximum.

It is possible to determine whether the vertex of a occupation it's point of maximum or of Minimum just checking the value of the coefficient a. If a > 0, the function has a minimum point, and if a

Another method for finding vertex coordinates

when the occupation has roots, we can find the function vertex coordinates as follows:

1 – Find the roots of the function.

2 – Find the Scoreaverage between the roots. This value is the x-coordinate of the vertex.

3 – Find the Imagegivesoccupation related to the value found in step 2 for the x of the vertex. This will be the y value of the vertex.

Example

Determine the coordinates of the vertex of the occupation f(x) = x² – 16.

Solution 1 - Using the formulas

x_v = - B
2nd

x_v = – 0
2·1

x_v = 0
2

x_v = 0

y_v = – Δ
4th

y_v = - (B² – 4ac)
4th

y_v = – (0 – 4·1·[– 16])
4

y_v = – (– 4·1·[– 16])
4

y_v = – (64)
4

y_v = – 16

Solution 2 – Finding the midpoint of the roots and the function image relative to it

The roots of this function can be obtained by Bhaskara's formula. However, we will use another method to find them.

f(x) = x² – 16

0 = x² – 16

x² = 16

√x² = ± √16

x = ± 4

The midpoint of the roots is x_v:

x_v = 4 – 4 = 0 = 0
2 2

Replacing 0 in occupation to find y_v, we will have:

f(x) = x² – 16

f (0) = 0² – 16

f (0) = – 16

Therefore, the coordinates of the vertex are: V(0, – 16).