What is high school function?

One occupation is a rule that connects each element of a set A to a single element of a set B, respectively known as domain and counter-domain of the function. For the function to be called high school function, it is necessary that your rule (or law of formation) can be written in the following way:

f(x) = ax² + bx + c

or

y = ax² + bx + c

Furthermore, a, b and c must belong to the set of real numbers and a ≠ 0. Thus, they are examples of occupationofseconddegree:

a) f (x) = x² + x – 6

b) f (x) = – x²

Roots of high school function

the roots of a occupation are the values ​​assumed by x when f(x) = 0. So, to find them, just replace f (x) or y by zero in the occupation and solve the resulting equation. To resolve quadratic equations, we can use Bhaskara's formula, method of complete squares or any other method. Remember: how to occupation It's from seconddegree, she must have even two real roots different.

Example - The roots of the function f (x) = x² + x – 6 can be calculated as follows:

f(x) = x² + x – 6
0 = x² + x – 6
a = 1, b = 1 and c = – 6

? = b² – 4·a·c
? = 1² – 4·1·(– 6)
? = 1 + 24
? = 25

x = – b ± √?
2nd
x = – 1 ± √25
2
x = – 1 ± 5
2

x’ = – 1 + 5 = 4 = 2
2 2

x" = – 1 – 5 = – 6 = – 3
2 2

Hence, the roots of the function f (x) = x² + x – 6 are the coordinate points A = (2, 0) and B = (–3, 0).

Function vertex - Maximum or minimum point

O vertex is the point at which the function of the second degree reaches its value maximum or minimum. Its coordinates V = (x_vy_v) are given by the following formulas:

x_v = - B
2nd

and

y_v = – ?
4th

In the same example mentioned above, the vertex of the function f(x) = x² + x – 6 is obtained by:

x_v = - B
2nd

x_v = – 1
2·1

x_v = – 1
2

x_v = – 0,5

and

y_v = – ?
4th

y_v = – 25
4·1

y_v = – 25
4

y_v = – 6,25

Thus, the coordinates of the vertex of that occupation are V = (–0.5; – 6,25).

the y coordinate_v can also be obtained by substituting the value of x_v in the function itself.

Second degree function graph

O graphic of a occupationofseconddegree will always be a parable. There are some tricks involving this figure that can be used to make the graph easier. To illustrate these tricks, we will also use the function f (x) = x² + x – 6.

1 - The sign of the coefficient a is linked to the concavity of the parable. If a > 0 the concavity of the figure will face upwards, if a < 0 the concavity of the figure will be facing downwards.

So, in the example, as a = 1, which is greater than zero, the concavity of the parable which represents the function f(x) = x² + x – 6 will face up.

2 – The coefficient c is one of the coordinates of the meeting point of the parable with the y axis. In other words, the parabola always meets the y axis at point C = (0, c).

In the example, point C = (0, – 6). So the parable goes through that point.

3 – As in the study of the signs of equation of seconddegree, in the second degree functions, the sign of the determinant indicates the number of roots of the function:

If? > 0 the function has two distinct real roots.

If? = 0 the function has two equal real roots.

If? < 0 the function has no real roots.

Given these tricks, it will be necessary to find three points belonging to a occupationofseconddegree to build the graph. Then just mark these three points on the Cartesian plane and draw the parable that passes through them. Namely, the three points are:

• O vertex and the roots of function, if it has real roots;

or

• O vertex and any two other points, if the occupation not have real roots. In this case, one point must be to the left and another to the right of the vertex of the function in the Cartesian plane.

Note that one of these points can be C = (0, c), except in the case that point is the vertex itself.

In the example f(x) = x² + x – 6, we have the following graph:

By Luiz Paulo Moreira
Graduated in Mathematics

Source: Brazil School - https://brasilescola.uol.com.br/o-que-e/matematica/o-que-e-funcao-segundo-grau.htm