Hmm, yoda_daro claims that this is a test to determine their sexuality...
Instruction The frames of the film merge for us into continuous movement due to ...
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- — [] quadratic function A function of the form y= ax2 + bx + c (a ? 0). Graph K.f. is a parabola whose vertex has coordinates [ b / 2a, (b2 4ac) / 4a], for 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- A quadratic function is a function of the form y= ax2 + bx + c (a ≠ 0). Graph K.f. is a parabola whose vertex has coordinates [b/ 2a, (b2 4ac) /4a], for a> 0 the branches of the parabola are directed upwards, for a< 0 –вниз… …
- (quadratic) A function that looks like this: y=ax2+bx+c, where a≠0 and the highest power of x is a square. The 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, and c is a constant. Contents 1 Transfer of the origin 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 given 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 the theory of statistical decisions, characterizes the losses due to incorrect decision making based on the observed data. If the problem of estimating the signal parameter against the background of interference 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 Industry, Moscow, 1999] objective function In extremal problems, a function whose minimum or maximum is to be found. This is… … Technical Translator's Handbook
objective function- in extremal problems, the function, the minimum or maximum of which is required to be found. This is the key concept of optimal programming. Having found the extremum of the C.f. and, therefore, by determining the values of the controlled variables that are to it ... ... Economic and Mathematical Dictionary
Books
- A set of tables. Mathematics. Function graphs (10 tables) , . Educational album of 10 sheets. Linear function. Graphical and analytical assignment of functions. Quadratic function. Converting the graph of a quadratic function. Function y=sinx. Function y=cosx.…
- The most important function of school mathematics - 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 - a deep meaning. Many tasks of the school ...
In the lessons of mathematics at school, you have already become acquainted with the simplest properties and the graph of a function y=x2. Let's expand our knowledge quadratic function.
Exercise 1.
Plot a function y=x2. Scale: 1 = 2 cm. Mark a point on the Oy axis F(0; 1/4). Using a compass or 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 this point so that it becomes vertical. The end of the strip will fall slightly below the x-axis (Fig. 1). Mark on the strip how far it goes beyond the x-axis. Take now another point on the parabola and repeat the measurement again. How much has the edge of the strip now dropped beyond the x-axis?
Result: no matter what point on the parabola y \u003d x 2 you take, the distance from this point to the point F (0; 1/4) will be greater than the distance from the same point to the x-axis always by the same number - by 1/4.
It can be said 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 line y = -1/4. This wonderful point F(0; 1/4) is called focus parabolas y \u003d x 2, and the straight line y \u003d -1/4 - headmistress this parabola. Each 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 line, called its directrix.
2. If you rotate a parabola around the axis of symmetry (for example, a parabola y \u003d x 2 around the Oy axis), you get a very interesting surface, which is called a paraboloid of revolution.
The surface of a liquid in a rotating vessel has the shape of a paraboloid of revolution. You can see this surface if you stir hard with a spoon in an incomplete glass of tea, and then remove the spoon.
3. If you throw a stone in the void at a certain angle to the horizon, then it will fly along a parabola (Fig. 2).
4. If you intersect the surface of the cone with a plane parallel to any one of its generators, then in the section you get a parabola (Fig. 3).
5. In amusement parks, they sometimes arrange a funny attraction called the Paraboloid of Wonders. To each of those standing inside the rotating paraboloid, it seems that he is standing on the floor, and the rest of the people, by some miracle, keep on the walls.
6. In reflecting telescopes, parabolic mirrors are also used: the light of a distant star, traveling in a parallel beam, falling on the telescope mirror, is collected in focus.
7. For spotlights, the mirror is usually made in the form 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.
Plotting a Quadratic Function
In the lessons of mathematics, you studied how to get graphs of functions of the form from the graph of the function y \u003d x 2:
1) y=ax2– expansion of the graph y = x 2 along the Oy axis in |a| times (for |a|< 0 – это сжатие в 1/|a| раз, rice. 4).
2) y=x2+n– graph shift by n units along the Oy axis, and if n > 0, then the shift is up, and if n< 0, то вниз, (или же можно переносить ось абсцисс).
3) y = (x + m)2– graph shift by m units along the Ox axis: if m< 0, то вправо, а если m >0, then to the left, (Fig. 5).
4) y=-x2- symmetrical display about the Ox axis of the graph y = x 2 .
Let's dwell on plotting a function graph in more detail. 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 \u003d a (x - m) 2 + n, where m \u003d -b / (2a), n \u003d - (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 notation.
Let be m = -b/(2a), a n \u003d - (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 get the function Y = aX 2 , whose graph is a parabola.
The vertex of the parabola is at the origin. x=0; Y = 0.
Substituting the coordinates of the vertex in (*), 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
by transformation, you can proceed as follows:
a) build a graph of 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 top of the parabola from the origin to the point with coordinates (m; n) (Fig. 6).
Write 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.
Decision.
Chain of transformations:
y=x2 (1) → y = (x - 3) 2 (2) → y = 2(x – 3) 2 (3) → y = 2(x - 3) 2 - 2 (4) .
The construction of the graph is shown in rice. 7.
You can practice quadratic function plotting by yourself. 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 registration . For further work with the teacher, you can choose the tariff plan that suits you.
Do you have any questions? Don't know how to graph a quadratic function?
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Everyone knows what a parabola is. But how to use it correctly, competently in solving various practical problems, we will understand below.
First, let us denote the basic concepts that algebra and geometry give to this term. Consider all possible types of this chart.
We learn all the main characteristics of this function. Let's understand the basics of constructing a curve (geometry). Let's learn how to find the top, other basic values of the graph of this type.
We will find out: how the required curve is correctly constructed according to the equation, what you need to pay attention to. Let's see the main practical application of this unique value in human life.
What is a parabola and what does it look like
Algebra: This term refers to the graph of a quadratic function.
Geometry: This is a second-order curve that has a number of specific features:
Canonical parabola equation
The figure shows a rectangular coordinate system (XOY), an extremum, the direction of the function drawing branches along the abscissa axis.
The canonical equation is:
y 2 \u003d 2 * p * x,
where the coefficient p is the focal parameter of the parabola (AF).
In algebra, it is written differently:
y = a x 2 + b x + c (recognizable pattern: y = x 2).
Properties and Graph of a Quadratic Function
The function has an axis of symmetry and a center (extremum). The domain of definition is all values of the x-axis.
The range of values of the function - (-∞, M) or (M, +∞) depends on the direction of the curve branches. The parameter M here means the value of the function at the top of the line.
How to determine where the branches of a parabola are directed
To find the direction of this type of curve from an expression, you need to specify the sign in front of the first parameter of the algebraic expression. If a ˃ 0, then they are directed upwards. Otherwise, down.
How to find the vertex of a parabola using the formula
Finding the extremum is the main step in solving many practical problems. Of course, you can open special online calculators, but it's better to be able to do it yourself.
How to define it? There is a special formula. When b is not equal to 0, we must look for the coordinates of this point.
Formulas for finding the top:
- x 0 \u003d -b / (2 * a);
- y 0 = y (x 0).
Example.
There is a function y \u003d 4 * x 2 + 16 * x - 25. Let's find the vertices of this function.
For such a line:
- x \u003d -16 / (2 * 4) \u003d -2;
- y = 4 * 4 - 16 * 2 - 25 = 16 - 32 - 25 = -41.
We get the coordinates of the vertex (-2, -41).
Parabola offset
The classic case is when in a quadratic function y = a x 2 + b x + c, the second and third parameters are 0, and = 1 - the vertex is at the point (0; 0).
Movement along the abscissa or ordinate axes is due to a change in the parameters b and c, respectively. The shift of the line on the plane will be carried out exactly by the number of units, which is equal to the value of the parameter.
Example.
We have: b = 2, c = 3.
This means that the classic view of the curve will shift by 2 unit segments along the abscissa axis and by 3 along the ordinate axis.
How to build a parabola using a quadratic equation
It is important for schoolchildren to learn how to correctly draw a parabola according to the given parameters.
By analyzing expressions and equations, you can see the following:
- The point of intersection of the desired line with the ordinate vector will have a value equal to c.
- All points of the graph (along the x-axis) will be symmetrical with respect to the main extremum of the function.
In addition, the intersections with OX can be found by knowing the discriminant (D) of such a function:
D \u003d (b 2 - 4 * a * c).
To do this, you need to equate the expression to zero.
The presence of parabola roots depends on the result:
- D ˃ 0, then x 1, 2 = (-b ± D 0.5) / (2 * a);
- D \u003d 0, then x 1, 2 \u003d -b / (2 * a);
- D ˂ 0, then there are no points of intersection with the vector OX.
We get the algorithm for constructing a parabola:
- determine the direction of the branches;
- find the coordinates of the vertex;
- find the intersection with the y-axis;
- find the intersection with the x-axis.
Example 1
Given a function y \u003d x 2 - 5 * x + 4. It is necessary to build a parabola. We act according to the algorithm:
- a \u003d 1, therefore, the branches are directed upwards;
- extremum coordinates: x = - (-5) / 2 = 5/2; y = (5/2) 2 - 5 * (5/2) + 4 = -15/4;
- intersects with the y-axis at the value y = 4;
- find the discriminant: D = 25 - 16 = 9;
- looking for roots
- X 1 \u003d (5 + 3) / 2 \u003d 4; (4, 0);
- X 2 \u003d (5 - 3) / 2 \u003d 1; (ten).
Example 2
For the function y \u003d 3 * x 2 - 2 * x - 1, you need to build a parabola. We act according to the above algorithm:
- a \u003d 3, therefore, the branches are directed upwards;
- extremum coordinates: x = - (-2) / 2 * 3 = 1/3; y = 3 * (1/3) 2 - 2 * (1/3) - 1 = -4/3;
- with the y-axis will intersect at the value y \u003d -1;
- find the discriminant: D \u003d 4 + 12 \u003d 16. So the roots:
- X 1 \u003d (2 + 4) / 6 \u003d 1; (1;0);
- X 2 \u003d (2 - 4) / 6 \u003d -1/3; (-1/3; 0).
From the obtained points, you can build a parabola.
Directrix, eccentricity, focus of a parabola
Based on the canonical equation, the focus F has coordinates (p/2, 0).
Straight line AB is a directrix (a kind of parabola chord of a certain length). Her equation is x = -p/2.
Eccentricity (constant) = 1.
Conclusion
We looked at the topic that students study in high school. Now you know, looking at the quadratic function of a parabola, how to find its vertex, in which direction the branches will be directed, whether there is an offset along the axes, and, having a construction algorithm, you can draw its graph.
Tasks on the properties and graphs of a quadratic function, as practice shows, cause serious difficulties. This is rather strange, because the quadratic function is passed in the 8th grade, and then the entire first quarter of the 9th grade is "extorted" by the properties of the parabola and its graphs are built for various parameters.
This is due to the fact that forcing students to build parabolas, they practically do not devote time to "reading" graphs, that is, they do not practice comprehending the information received from the picture. Apparently, it is assumed that, having built two dozen graphs, a smart student himself will discover and formulate the relationship between the coefficients in the formula and the appearance of the graph. In practice, this does not work. For such a generalization, serious experience in mathematical mini-research is required, which, of course, most ninth-graders do not have. Meanwhile, in the GIA they propose to determine the signs of the coefficients precisely according to the schedule.
We will not demand the impossible from schoolchildren and simply offer one of the algorithms for solving such problems.
So, a function of the form y=ax2+bx+c is called quadratic, its graph is a parabola. As the name suggests, the main component is ax 2. I.e a should not be equal to zero, the remaining coefficients ( b and with) can be equal to zero.
Let's see how the signs of its coefficients affect the appearance of the parabola.
The simplest dependence for the coefficient a. Most schoolchildren confidently answer: "if a> 0, then the branches of the parabola are directed upwards, and if a < 0, - то вниз". Совершенно верно. Ниже приведен график квадратичной функции, у которой a > 0.
y = 0.5x2 - 3x + 1
In this case a = 0,5
And now for a < 0:
y = - 0.5x2 - 3x + 1
In this case a = - 0,5
Influence of coefficient with also easy enough to follow. 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. I.e with is the ordinate of the point of intersection of the parabola with the y-axis. As a rule, this point is easy to find on the chart. And determine whether it lies above zero or below. I.e with> 0 or with < 0.
with > 0:
y=x2+4x+3
with < 0
y = x 2 + 4x - 3
Accordingly, if with= 0, then the parabola will necessarily pass through the origin:
y=x2+4x
More difficult with the parameter b. The point by 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 \u003d - b / (2a). Thus, b = - 2ax in. That is, we act as follows: on the graph we find the top of the parabola, 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, this is not all. We must also pay attention to the sign of the coefficient a. That is, to see where the branches of the parabola are directed. And only after that, according to the formula b = - 2ax in determine sign b.
Consider an example:
Branches pointing upwards a> 0, the parabola crosses the axis at below zero means 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.
