Many technical, economic, and social issues are predicted using curves. The most used type among them is parabola, or rather, half of it. An important component of any parabolic curve is its vertex, the determination of the exact coordinates of which sometimes plays a key role not only in the display of the process, but also for subsequent conclusions. How to find its exact coordinates will be discussed in this article ....
Before proceeding to the search for the coordinates of the vertex of a parabola, we will familiarize ourselves with the definition itself and its properties. In the classical sense, a parabola is an arrangement of points that removed at the same distance from a specific point (focus, point F), as well as from a straight line that does not pass through point F. Let us consider this definition in more detail in Figure 1.
Figure 1. Classic view of a parabola
The figure shows the classic shape. The focus is point F. In this case, the directrix is considered to be a line parallel to the Y axis (highlighted in red). From the definition you can make sure that absolutely any point of the curve, not counting the focus, has a similar on the other hand, removed at the same distance from the axis of symmetry as itself. Moreover, the distance from any of the points on the parabola equal to the distance to the director. Looking ahead, we say that the center of the function does not have to be at the origin, and the branches can be directed in different directions.
Parabola, like any other function, has its own entry in the form of a formula:
In the indicated formula, the letter “s” denotes the parabola parameter, which is equal to the distance from the focus to the directrix. There is also another form of recording, indicated by the GMT, which has the form:
This formula is used in solving problems from the field of mathematical analysis and is used more often than traditional (due to convenience). In the future, we will focus on the second record.
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The numerical value of the coordinate of the vertex on the abscissa
If the parabola equation is given in the classical form (1), then the value of the abscissa at the desired point will be equal to half the value of the parameter s (half the distance between the director and the focus). If the function is presented in the form (2), then x zero is calculated by the formula:
That is, looking at this formula, it can be argued that the vertex will be in the right half relative to the y axis if one of the parameters a or b is less than zero.
The directrix equation is defined by the following equation:
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The vertex value on the y-axis
The numerical value of the vertex location for formula (2) on the ordinate axis can be found by the following formula:
From this we can conclude that if a & lt, 0, then the top of the curve will be in the upper half-planeotherwise, at the bottom. In this case, the points of the parabola will have the same properties that were mentioned earlier.
If the classical form of writing is given, then it will be more rational to calculate the value of the location of the vertex on the abscissa axis, and through it the subsequent ordinate value. Note that for the recording form (2), the axis of symmetry of the parabola, in the classical representation, will coincide with the ordinate axis.
Important! When solving tasks using the parabola equation, first of all, highlight the main values that are already known. Moreover, it will be useful if the missing parameters are determined. Such an approach will give a greater “room for maneuver” and a more rational solution in advance. In practice, try to use the notation (2). It is simpler to perceive (you don’t have to “turn over the coordinates of Descartes), moreover, the overwhelming majority of tasks are adapted precisely for this form of writing.
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Construction of a curve of parabolic type
Using a common notation, before constructing a parabola, you need to find its vertex. Simply put, you must run the following algorithm:
- Find the coordinate vertex on the x axis.
- Find the coordinate of the vertex location on the Y axis.
- Substituting different values of the dependent variable X, find the corresponding values of Y and construct a curve.
Those. the algorithm is nothing complicated, the main emphasis is on how to find the top of the parabola. The further construction process can be considered mechanical.
Provided that three points are given whose coordinates are known, it is first necessary to draw up the equation of the parabola itself, and then repeat the procedure described previously. Because in equation (2) there are 3 coefficients, then, using the coordinates of the points, we calculate each of them:
In the formulas (5.1), (5.2), (5.3), respectively, those points are used that are known (for example, A (, B (, C (. This way we find the parabola equation by 3 points. From the practical side, this approach is not the most pleasant ”, however, it gives a clear result, on the basis of which the curve itself is subsequently built.
When building a parabola always there must be an axis of symmetry. The formula of the axis of symmetry for writing (2) will look like this:
Those. finding the axis of symmetry with which all points of the curve are symmetric is not difficult. More precisely, it is equal to the first coordinate of the vertex.
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Example 1. Suppose we have the parabola equation:
It is required to find the coordinates of the vertex of the parabola, and also to check whether the point D (10, 5) belongs to this curve.
Solution: First of all, we verify that the mentioned point belongs to the curve itself
Whence we conclude that the indicated point does not belong to the given curve. Find the coordinates of the vertex of the parabola. From formulas (4) and (5) we obtain the following sequence:
It turns out that the coordinates on the top, at point O, are as follows (-1.25, -7.625). This suggests that our parabola originates in the 3rd quarter of the Cartesian system coordinates.
Example 2. Find the vertex of a parabola, knowing three points that belong to it: A (2,3), B (3,5), C (6,2). Using formulas (5.1), (5.2), (5.3), we find the coefficients of the parabola equation. We get the following:
Using the obtained values, we obtain the following equation:
In the figure, the specified function will look like this (Figure 2):
Figure 2. A graph of a parabola passing through 3 points
Those. the parabola chart, which runs along three given points, will have a vertex in the 1st quarter. However, the branches of this curve are directed downwards, i.e. there is a parabola offset from the origin. Such a construction could be foreseen by paying attention to the coefficients a, b, c.
In particular, if a & lt, 0, then the branches will be directed down. With a & gt, 1 the curve will be stretched, and if less than 1, it will be compressed.
The constant c is responsible for the “movement” of the curve along the ordinate axis. If c & gt, 0, then the parabola creeps upotherwise, down. Regarding coefficient b, it is possible to determine the degree of influence only by changing the form of the equation, leading it to the following form:
If the coefficient is b & gt, 0, then the coordinates of the vertex of the parabola will be shifted to the right by b units, if less, then by b units to the left.
Important! The use of techniques for determining the displacement of a parabola on the coordinate plane sometimes helps to save time in solving problems or to learn about the possible intersection of a parabola with another curve even before construction. Usually they look only at the coefficient a, since it is he who gives a clear answer to the question posed.
Useful video: how to find the top of the parabola
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Such as an algebraic process, such as determining the vertices of a parabola, is not complicated, but at the same time rather laborious. In practice, they try to use the second form of recording in order to facilitate understanding of the graphic solution and the solution as a whole. Therefore, we strongly recommend that you use this approach, and if you do not remember the vertex coordinate formulas, then at least have a cheat sheet.
When examining a quadratic function whose graph is a parabola, in one of the points it is necessary to find the coordinates of the vertex of the parabola. How to do this analytically using the equation given for the parabola?
Hope, the article above just describes how to find the coordinates of the vertex of a parabola. The abscissa is found by the formula x0 = -b / 2a. To find the ordinate, it is enough to substitute the found value x0 in place of each x in the formula of the function and calculate.