One occupation is a rule that relates each element of a set A to a single element of a set B. This rule is usually achieved through a algebraic expression much like a equation and, depending on the degree of this algebraic expression and the number of variables it has, it is possible to construct its graph.
Chart definition
O graphic of a occupation is the set of points (x, y) of the Cartesian plane which satisfy the following condition: y = f(x). In other words, for each value of x, there is a single value of y relative to it, obtained by the law of formation of the occupation.
You graphics most important ones studied in elementary school belong to the first degree function It's from second degree. In high school, the graphicsgivesoccupation logarithmic, exponential, trigonometric etc. In this article, we'll discuss a technique that can be used to build the graphic of a occupation of seconddegree.
Second degree function graph
One occupation of seconddegree is one that can be written as follows:
f(x) = ax2 + bx + c
where a, b and c are real numbers, called coefficients, with a always non-zero, and x is the independent variable.
O graphic of these functions is always a parable which can be constructed from three points that belong to it: vertex and the two roots, or vertex and two “random” points.
1 – Finding the vertex of the parabola
At parables that can be used as graphic of a occupation of seconddegree they must have their concavity facing up or down. In the first case, the parabola has a lower point, where the function is no longer decreasing and becomes increasing. In the second case, the parabola has a higher point, where the function stops being increasing and becomes decreasing. This point is called vertex.
To find the coordinates of the vertex V = (xvyv), we can use the following formulas:
xv = - B
2nd
and
yv = – Δ
4th
2 – Finding the two roots of the parable
The roots of a function are the points at which the graphic of that occupation finds the x-axis of the Cartesian plane. In the case of the functions of the seconddegree, the number of roots can be 0, 1 or 2. If the function has two roots, the best thing to do is to use them in the construction of the graph.
To find the roots of a occupationofseconddegree, use the Bhaskara's formula. First, determine the discriminating of the function:
Δ = b2 – 4ac
Then substitute it in Bhaskara's formula, as well as the coefficients:
x = – b ± √?
2nd
The coordinates of the roots of the function will be: A = (x’, 0) and B = (x’’, 0). From these three points, the two roots and the vertex, just place them on the Cartesian plane and connect them by means of a parable. In this process, notice that the parabola will have the concavity facing downwards if the vertex is above the x-axis, or it will have the concavity facing upwards if the vertex is below the x-axis.
In the image above, note that the first parable it has a vertex below the x-axis and its concavity faces upwards. The opposite happens to the second parabola, which has the vertex above the x-axis and the concavity facing downwards.
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Example:
build the graphic gives occupation: f(x) = x2 + 2x – 8.
The first step is to find the vertex of this occupation. Using the studied formulas, we will have:
xv = - B
2nd
xv = – 2
2
xv = – 1
yv = – Δ
4th
yv = - (B2 – 4ac)
4th
yv = – (22 – 4·1·[– 8])
4
yv = – (4 + 32)
4
yv = – (4 + 32)
4
yv = – (36)
4
yv = – 9
Thus, the coordinates of the vertex of that parable are: V = (– 1, –9).
Note that we already know the discriminant value of this occupation, which was made to find yv. Δ = 36. Using Bhaskara's formula to find the roots, we will have:
x = – b ± √?
2nd
x = – 2 ± √36
2
x = – 2 ± 6
2
x’ = – 2 – 6 = – 8 = – 4
2 2
x’’ = – 2 + 6 = 4 = 2
2 2
So the roots can be found at the points: A = (–4, 0) and B = (2, 0). Marking these three points on the Cartesian plane, and then building the parable that passes through them, we will have:
Vertex + random points
This construction is valid when the occupation does it have two real and distinct roots, that is, when? > 0. when the occupation has only one real root, or has none, it makes no sense to try to find your roots to build your graphic.
In this case, we will first find the coordinatesofvertex, then, given xv the x-coordinate of the vertex, we'll choose the x-valuesv + 1 and xv – 1 as points “random” and we'll find the value of y related to each of these points. The results of this will be points V, A, and B, just like the roots, with the difference that points A and B are no longer on the x-axis.
For example, graph the function: f (x) = x2 + 4.
That occupation has no roots, because the value of? is less than zero. In this case, we will find the coordinates of the vertex and calculate the points “random”, previously proposed:
xv = - B
2nd
xv = – 0
2
xv = 0
yv = – Δ
4th
yv = - (B2 – 4ac)
4th
yv = – (02 – 4·1·4)
4
yv = – (– 16)
4
yv = 16
4
yv = 4
Thus, V = (0, 4).
taking xv = 0, we will do: xv + 1 = 0 + 1 = 1. Replacing this value in the occupation, to find y relative to it, we will have:
f(x) = x2 + 4
f(1) = 12 + 4
f(1) = 5
Therefore, point A will be: A = (1, 5).
taking xv = 0, we will also do: xv – 1 = 0 – 1 = – 1. Therefore:
f(x) = x2 + 4
f(– 1) = (– 1)2 + 4
f(- 1) = 1 + 4
f(- 1) = 5
Therefore, point B will be: B = (–1, 5).
So, the graphic of that occupation it will be:
By Luiz Paulo Moreira
Graduated in Mathematics
