- — [] quadratic function Function of the form y= ax2 + bx + c (a ? 0). Graph K.f. - a parabola, the vertex of which has coordinates [ b/ 2a, (b2 4ac) / 4a], with a>0 branches of the parabola ... ...
QUADRATIC FUNCTION, a mathematical FUNCTION whose value depends on the square of the independent variable, x, and is given, respectively, by a quadratic POLYNOMIAL, for example: f(x) = 4x2 + 17 or f(x) = x2 + 3x + 2. see also SQUARE THE EQUATION … Scientific and technical encyclopedic dictionary
Quadratic function- Quadratic function - a function of the form y= ax2 + bx + c (a ≠ 0). Graph K.f. - a parabola, the vertex of which has coordinates [ b/ 2a, (b2 4ac) / 4a], for a> 0 the branches of the parabola are directed upward, for a< 0 –вниз… …
- (quadratic) A function that has the following form: y=ax2+bx+c, where a≠0 and the highest degree of x is a square. Quadratic equation y=ax2 +bx+c=0 can also be solved using the following formula: x= –b+ √ (b2–4ac) /2a. These roots are real... Economic dictionary
An affine quadratic function on an affine space S is any function Q: S→K that has the form Q(x)=q(x)+l(x)+c in vectorized form, where q is a quadratic function, l is a linear function, c is a constant. Contents 1 Shifting the reference point 2 ... ... Wikipedia
An affine quadratic function on an affine space is any function that has the form in vectorized form, where is a symmetric matrix, a linear function, a constant. Contents... Wikipedia
A function on a vector space defined by a homogeneous polynomial of the second degree in the coordinates of the vector. Contents 1 Definition 2 Related definitions... Wikipedia
- is a function that in theory statistical solutions characterizes losses due to incorrect decision-making based on observed data. If the problem of estimating a signal parameter against a background of noise is being solved, then the loss function is a measure of the discrepancy... ... Wikipedia
objective function- - [Ya.N.Luginsky, M.S.Fezi Zhilinskaya, Yu.S.Kabirov. English-Russian dictionary of electrical engineering and power engineering, Moscow, 1999] objective function In extremal problems, a function whose minimum or maximum needs to be found. This… … Technical Translator's Guide
Objective function- in extremal problems, a function whose minimum or maximum needs to be found. This key concept optimal programming. Having found the extremum of C.f. and, therefore, having determined the values of the controlled variables that go to it... ... Economic and mathematical dictionary
Books
- Set of tables. Mathematics. Graphs of functions (10 tables), . Educational album of 10 sheets. Linear function. Graphical and analytical assignment of functions. Quadratic function. Graph Transformation quadratic function. Function y=sinx. Function y=cosx.…
- The most important function of school mathematics is quadratic - in problems and solutions, Petrov N.N.. The quadratic function is the main function of the school mathematics course. No wonder. On the one hand, the simplicity of this function, and on the other, the deep meaning. Many tasks of school...
In mathematics lessons at school, you have already become acquainted with the simplest properties and graph of a function y = x 2. Let's expand our knowledge on quadratic function.
Exercise 1.
Graph the function y = x 2. Scale: 1 = 2 cm. Mark a point on the Oy axis F(0; 1/4). Using a compass or a strip of paper, measure the distance from the point F to some point M parabolas. Then pin the strip at point M and rotate it around that point until it is vertical. The end of the strip will fall slightly below the x-axis (Fig. 1). Mark on the strip how far it extends beyond the x-axis. Now take another point on the parabola and repeat the measurement again. How far has the edge of the strip fallen below the x-axis?
Result: whatever point on the parabola y = x 2 you take, the distance from this point to the point F(0; 1/4) will be more distance from the same point to the x-axis always by the same number - by 1/4.
We can say it differently: the distance from any point of the parabola to the point (0; 1/4) is equal to the distance from the same point of the parabola to the straight line y = -1/4. This wonderful point F(0; 1/4) is called focus parabolas y = x 2, and straight line y = -1/4 – headmistress this parabola. Every parabola has a directrix and a focus.
Interesting properties of a parabola:
1. Any point of the parabola is equidistant from some point, called the focus of the parabola, and some straight line, called its directrix.
2. If you rotate a parabola around the axis of symmetry (for example, the parabola y = x 2 around the Oy axis), you will get a very interesting surface called a paraboloid of revolution.
The surface of the liquid in a rotating vessel has the shape of a paraboloid of rotation. You can see this surface if you stir vigorously with a spoon in an incomplete glass of tea, and then remove the spoon.
3. If you throw a stone into the void at a certain angle to the horizon, it will fly in a parabola (Fig. 2).
4. If you intersect the surface of a cone with a plane parallel to any one of its generatrices, then the cross section will result in a parabola (Fig. 3).
5. Amusement parks sometimes have a fun ride called Paraboloid of Wonders. It seems to everyone standing inside the rotating paraboloid that he is standing on the floor, while the rest of the people are somehow miraculously holding on to the walls.
6. In reflecting telescopes, parabolic mirrors are also used: the light of a distant star, coming in a parallel beam, falling on the telescope mirror, is collected into focus.
7. Spotlights usually have a mirror in the shape of a paraboloid. If you place a light source at the focus of a paraboloid, then the rays, reflected from the parabolic mirror, form a parallel beam.
Graphing a Quadratic Function
In mathematics lessons, you studied how to obtain graphs of functions of the form from the graph of the function y = x 2:
1) y = ax 2– stretching the graph y = x 2 along the Oy axis in |a| times (with |a|< 0 – это сжатие в 1/|a| раз, rice. 4).
2) y = x 2 + n– shift of the graph by n units along the Oy axis, and if n > 0, then the shift is upward, and if n< 0, то вниз, (или же можно переносить ось абсцисс).
3) y = (x + m) 2– shift of the graph by m units along the Ox axis: if m< 0, то вправо, а если m >0, then left, (Fig. 5).
4) y = -x 2– symmetrical display relative to the Ox axis of the graph y = x 2 .
Let's take a closer look at plotting the function y = a(x – m) 2 + n.
A quadratic function of the form y = ax 2 + bx + c can always be reduced to the form
y = a(x – m) 2 + n, where m = -b/(2a), n = -(b 2 – 4ac)/(4a).
Let's prove it.
Really,
y = ax 2 + bx + c = a(x 2 + (b/a) x + c/a) =
A(x 2 + 2x · (b/a) + b 2 /(4a 2) – b 2 /(4a 2) + c/a) =
A((x + b/2a) 2 – (b 2 – 4ac)/(4a 2)) = a(x + b/2a) 2 – (b 2 – 4ac)/(4a).
Let us introduce new notations.
Let m = -b/(2a), A n = -(b 2 – 4ac)/(4a),
then we get y = a(x – m) 2 + n or y – n = a(x – m) 2.
Let's make some more substitutions: let y – n = Y, x – m = X (*).
Then we obtain the function Y = aX 2, the graph of which is a parabola.
The vertex of the parabola is at the origin. X = 0; Y = 0.
Substituting the coordinates of the vertex into (*), we obtain the coordinates of the vertex of the graph y = a(x – m) 2 + n: x = m, y = n.
Thus, in order to plot a quadratic function represented as
y = a(x – m) 2 + n
through transformations, you can proceed as follows:
a) plot the function y = x 2 ;
b) by parallel translation along the Ox axis by m units and along the Oy axis by n units - transfer the vertex of the parabola from the origin to the point with coordinates (m; n) (Fig. 6).
Recording transformations:
y = x 2 → y = (x – m) 2 → y = a(x – m) 2 → y = a(x – m) 2 + n.
Example.
Using transformations, construct a graph of the function y = 2(x – 3) 2 in the Cartesian coordinate system – 2.
Solution.
Chain of transformations:
y = x 2 (1) → y = (x – 3) 2 (2) → y = 2(x – 3) 2 (3) → y = 2(x – 3) 2 – 2 (4) .
The plotting is shown in rice. 7.
You can practice graphing quadratic functions on your own. For example, build a graph of the function y = 2(x + 3) 2 + 2 in one coordinate system using transformations. If you have any questions or want to get advice from a teacher, then you have the opportunity to conduct free 25-minute lesson with an online tutor after . For further work with a teacher, you can choose the one that suits you
Still have questions? Don't know how to graph a quadratic function?
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As practice shows, tasks on the properties and graphs of a quadratic function cause serious difficulties. This is quite strange, because they study the quadratic function in the 8th grade, and then throughout the first quarter of the 9th grade they “torment” the properties of the parabola and build its graphs for various parameters.
This is due to the fact that when forcing students to construct parabolas, they practically do not devote time to “reading” the graphs, that is, they do not practice comprehending the information received from the picture. Apparently, it is assumed that, after constructing a dozen or two graphs, a smart student himself will discover and formulate the relationship between the coefficients in the formula and appearance graphic arts. In practice this does not work. For such a generalization, serious experience in mathematical mini-research is required, which most ninth-graders, of course, do not possess. Meanwhile, the State Inspectorate proposes to determine the signs of the coefficients using the schedule.
We will not demand the impossible from schoolchildren and will simply offer one of the algorithms for solving such problems.
So, a function of the form y = ax 2 + bx + c called quadratic, its graph is a parabola. As the name suggests, the main term is ax 2. That is A should not be equal to zero, the remaining coefficients ( b And With) can equal zero.
Let's see how the signs of its coefficients affect the appearance of a parabola.
The simplest dependence for the coefficient A. Most schoolchildren confidently answer: “if A> 0, then the branches of the parabola are directed upward, and if A < 0, - то вниз". Совершенно верно. Ниже приведен график квадратичной функции, у которой A > 0.
y = 0.5x 2 - 3x + 1
IN in this case A = 0,5
And now for A < 0:
y = - 0.5x2 - 3x + 1
In this case A = - 0,5
Impact of the coefficient With It's also pretty easy to follow. Let's imagine that we want to find the value of a function at a point X= 0. Substitute zero into the formula:
y = a 0 2 + b 0 + c = c. It turns out that y = c. That is With is the ordinate of the point of intersection of the parabola with the y-axis. Typically, this point is easy to find on the graph. And determine whether it lies above zero or below. That is With> 0 or With < 0.
With > 0:
y = x 2 + 4x + 3
With < 0
y = x 2 + 4x - 3
Accordingly, if With= 0, then the parabola will necessarily pass through the origin:
y = x 2 + 4x
More difficult with the parameter b. The point at which we will find it depends not only on b but also from A. This is the top of the parabola. Its abscissa (axis coordinate X) is found by the formula x in = - b/(2a). Thus, b = - 2ax in. That is, we proceed as follows: we find the vertex of the parabola on the graph, determine the sign of its abscissa, that is, we look to the right of zero ( x in> 0) or to the left ( x in < 0) она лежит.
However, that's not all. We also need to pay attention to the sign of the coefficient A. That is, look at where the branches of the parabola are directed. And only after that, according to the formula b = - 2ax in determine the sign b.
Let's look at an example:
The branches are directed upwards, which means A> 0, the parabola intersects the axis at below zero, that is With < 0, вершина параболы лежит правее нуля. Следовательно, x in> 0. So b = - 2ax in = -++ = -. b < 0. Окончательно имеем: A > 0, b < 0, With < 0.
A function of the form where is called quadratic function.
Graph of a quadratic function – parabola.
Let's consider the cases:
I CASE, CLASSICAL PARABOLA
That is , ,
To construct, fill out the table by substituting the x values into the formula:
Mark the points (0;0); (1;1); (-1;1), etc. on the coordinate plane (the smaller the step we take the x values (in this case, step 1), and the more x values we take, the smoother the curve will be), we get a parabola:
It is easy to see that if we take the case , , , that is, then we get a parabola that is symmetrical about the axis (oh). It’s easy to verify this by filling out a similar table:
II CASE, “a” IS DIFFERENT FROM UNIT
What will happen if we take , , ? How will the behavior of the parabola change? With title="Rendered by QuickLaTeX.com" height="20" width="55" style="vertical-align: -5px;"> парабола изменит форму, она “похудеет” по сравнению с параболой (не верите – заполните соответствующую таблицу – и убедитесь сами):!}
In the first picture (see above) it is clearly visible that the points from the table for the parabola (1;1), (-1;1) were transformed into points (1;4), (1;-4), that is, with the same values, the ordinate of each point is multiplied by 4. This will happen to all key points of the original table. We reason similarly in the cases of pictures 2 and 3.
And when the parabola “becomes wider” than the parabola:
Let's summarize:
1)The sign of the coefficient determines the direction of the branches. With title="Rendered by QuickLaTeX.com" height="14" width="47" style="vertical-align: 0px;"> ветви направлены вверх, при - вниз. !}
2) Absolute value coefficient (modulus) is responsible for the “expansion” and “compression” of the parabola. The larger , the narrower the parabola; the smaller |a|, the wider the parabola.
III CASE, “C” APPEARS
Now let's introduce into the game (that is, consider the case when), we will consider parabolas of the form . It is not difficult to guess (you can always refer to the table) that the parabola will shift up or down along the axis depending on the sign:
IV CASE, “b” APPEARS
When will the parabola “break away” from the axis and finally “walk” along the entire coordinate plane? When will it stop being equal?
Here to construct a parabola we need formula for calculating the vertex: , .
So at this point (as at point (0;0) new system coordinates) we will build a parabola, which we can already do. If we are dealing with the case, then from the vertex we put one unit segment to the right, one up, - the resulting point is ours (similarly, a step to the left, a step up is our point); if we are dealing with, for example, then from the vertex we put one unit segment to the right, two - upward, etc.
For example, the vertex of a parabola:
Now the main thing to understand is that at this vertex we will build a parabola according to the parabola pattern, because in our case.
When constructing a parabola after finding the coordinates of the vertex veryIt is convenient to consider the following points:
1) parabola will definitely pass through the point . Indeed, substituting x=0 into the formula, we obtain that . That is, the ordinate of the point of intersection of the parabola with the axis (oy) is . In our example (above), the parabola intersects the ordinate at point , since .
2) axis of symmetry parabolas is a straight line, so all points of the parabola will be symmetrical about it. In our example, we immediately take the point (0; -2) and build it symmetrical relative to the symmetry axis of the parabola, we get the point (4; -2) through which the parabola will pass.
3) Equating to , we find out the points of intersection of the parabola with the axis (oh). To do this, we solve the equation. Depending on the discriminant, we will get one (, ), two ( title="Rendered by QuickLaTeX.com" height="14" width="54" style="vertical-align: 0px;">, ) или нИсколько () точек пересечения с осью (ох) !} . In the previous example, our root of the discriminant is not an integer; when constructing, it doesn’t make much sense for us to find the roots, but we clearly see that we will have two points of intersection with the axis (oh) (since title="Rendered by QuickLaTeX.com" height="14" width="54" style="vertical-align: 0px;">), хотя, в общем, это видно и без дискриминанта.!}
So let's work it out
Algorithm for constructing a parabola if it is given in the form
1) determine the direction of the branches (a>0 – up, a<0 – вниз)
2) we find the coordinates of the vertex of the parabola using the formula , .
3) we find the point of intersection of the parabola with the axis (oy) using the free term, construct a point symmetrical to this point with respect to the symmetry axis of the parabola (it should be noted that it happens that it is unprofitable to mark this point, for example, because the value is large... we skip this point...)
4) At the found point - the vertex of the parabola (as at the point (0;0) of the new coordinate system) we construct a parabola. If title="Rendered by QuickLaTeX.com" height="20" width="55" style="vertical-align: -5px;">, то парабола становится у’же по сравнению с , если , то парабола расширяется по сравнению с !}
5) We find the points of intersection of the parabola with the axis (oy) (if they have not yet “surfaced”) by solving the equation
Example 1
Example 2
Note 1. If the parabola is initially given to us in the form , where are some numbers (for example, ), then it will be even easier to construct it, because we have already been given the coordinates of the vertex . Why?
Let's take quadratic trinomial and select a complete square in it: Look, we got that , . You and I previously called the vertex of a parabola, that is, now,.
For example, . We mark the vertex of the parabola on the plane, we understand that the branches are directed downward, the parabola is expanded (relative to ). That is, we carry out points 1; 3; 4; 5 from the algorithm for constructing a parabola (see above).
Note 2. If the parabola is given in a form similar to this (that is, presented as a product of two linear factors), then we immediately see the points of intersection of the parabola with the axis (ox). In this case – (0;0) and (4;0). For the rest, we act according to the algorithm, opening the brackets.
