- — [] 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) \u003d 4x2 + 17 or f (x) \u003d 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 having the following form: y=ax2+bx+c, where a≠0 and highest degree 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. It… … Technical Translator's Handbook

objective function- in extremal problems, the function, the minimum or maximum of which is required to be found. it key concept 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. Maths. 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. This is not surprising. On the one hand - the simplicity of this function, and on the other - a deep meaning. Many tasks of the school ...

A quadratic function is a function of the form:
y=a*(x^2)+b*x+c,
where a is the coefficient at the highest degree of the unknown x,
b - coefficient at unknown x,
and c is a free member.
The graph of a quadratic function is a curve called a parabola. General form parabola is shown in the figure below.

Fig.1 General view of the parabola.

There are a few various ways plotting a quadratic function. We will consider the main and most general of them.

Algorithm for plotting a graph of a quadratic function y=a*(x^2)+b*x+c

1. Build a coordinate system, mark a single segment and label the coordinate axes.

2. Determine the direction of the branches of the parabola (up or down).
To do this, you need to look at the sign of the coefficient a. If plus - then the branches are directed upwards, if minus - then the branches are directed downwards.

3. Determine the x-coordinate of the top of the parabola.
To do this, you need to use the formula Tops = -b / 2 * a.

4. Determine the coordinate at the top of the parabola.
To do this, substitute the value of the Top found in the previous step in the equation of the Top = a * (x ^ 2) + b * x + c instead of x.

5. Put the resulting point on the graph and draw an axis of symmetry through it, parallel to the coordinate axis Oy.

6. Find the points of intersection of the graph with the x-axis.
This requires solving the quadratic equation a*(x^2)+b*x+c = 0 using one of the known methods. If the equation has no real roots, then the graph of the function does not intersect the x-axis.

7. Find the coordinates of the point of intersection of the graph with the Oy axis.
To do this, we substitute the value x = 0 into the equation and calculate the value of y. We mark this and the point symmetrical to it on the graph.

8. Find the coordinates of an arbitrary point A (x, y)
To do this, we choose an arbitrary value of the x coordinate, and substitute it into our equation. We get the value of y at this point. Put a point on the graph. And also mark a point on the graph that is symmetrical to the point A (x, y).

9. Connect the obtained points on the graph with a smooth line and continue the graph beyond the extreme points, to the end of the coordinate axis. Sign the graph either on the callout, or, if space permits, along the graph itself.

An example of plotting a graph

As an example, let's plot a quadratic function given by the equation y=x^2+4*x-1
1. Draw coordinate axes, sign them and mark a single segment.
2. The values ​​of the coefficients a=1, b=4, c= -1. Since a \u003d 1, which is greater than zero, the branches of the parabola are directed upwards.
3. Determine the X coordinate of the top of the parabola Tops = -b/2*a = -4/2*1 = -2.
4. Determine the coordinate At the top of the parabola
Tops = a*(x^2)+b*x+c = 1*((-2)^2) + 4*(-2) - 1 = -5.
5. Mark the vertex and draw an axis of symmetry.
6. We find the points of intersection of the graph of a quadratic function with the Ox axis. We solve the quadratic equation x^2+4*x-1=0.
x1=-2-√3 x2 = -2+√3. We mark the obtained values ​​on the graph.
7. Find the points of intersection of the graph with the Oy axis.
x=0; y=-1
8. Choose an arbitrary point B. Let it have a coordinate x=1.
Then y=(1)^2 + 4*(1)-1= 4.
9. We connect the received points and sign the chart.

The methodical material is for reference purposes and covers a wide range of topics. The article provides an overview of the graphs of the main elementary functions and considers the most important question – how to correctly and FAST build a graph. In the course of studying higher mathematics without knowledge of the graphs of basic elementary functions, it will be difficult, therefore it is very important to remember what the graphs of a parabola, hyperbola, sine, cosine, etc. look like, to remember some of the values ​​of the functions. We will also talk about some properties of the main functions.

I do not pretend to completeness and scientific thoroughness of the materials, the emphasis will be placed, first of all, on practice - those things with which one has to face literally at every step, in any topic of higher mathematics. Charts for dummies? You can say so.

By popular demand from readers clickable table of contents:

In addition, there is an ultra-short abstract on the topic
– master 16 types of charts by studying SIX pages!

Seriously, six, even I myself was surprised. This abstract contains improved graphics and is available for a nominal fee, a demo version can be viewed. It is convenient to print the file so that the graphs are always at hand. Thanks for supporting the project!

And we start right away:

How to build coordinate axes correctly?

In practice, tests are almost always drawn up by students in separate notebooks, lined in a cage. Why do you need checkered markings? After all, the work, in principle, can be done on A4 sheets. And the cage is necessary just for the high-quality and accurate design of the drawings.

Any drawing of a function graph starts with coordinate axes.

Drawings are two-dimensional and three-dimensional.

Let us first consider the two-dimensional case Cartesian coordinate system:

1) We draw coordinate axes. The axis is called x-axis , and the axis y-axis . We always try to draw them neat and not crooked. The arrows should also not resemble Papa Carlo's beard.

2) We sign the axes capital letters"x" and "y". Don't forget to sign the axes.

3) Set the scale along the axes: draw zero and two ones. When making a drawing, the most convenient and common scale is: 1 unit = 2 cells (drawing on the left) - stick to it if possible. However, from time to time it happens that the drawing does not fit on a notebook sheet - then we reduce the scale: 1 unit = 1 cell (drawing on the right). Rarely, but it happens that the scale of the drawing has to be reduced (or increased) even more

DO NOT scribble from a machine gun ... -5, -4, -3, -1, 0, 1, 2, 3, 4, 5, .... For the coordinate plane is not a monument to Descartes, and the student is not a dove. We put zero and two units along the axes. Sometimes instead of units, it is convenient to “detect” other values, for example, “two” on the abscissa axis and “three” on the ordinate axis - and this system (0, 2 and 3) will also uniquely set the coordinate grid.

It is better to estimate the estimated dimensions of the drawing BEFORE the drawing is drawn.. So, for example, if the task requires drawing a triangle with vertices , , , then it is quite clear that the popular scale 1 unit = 2 cells will not work. Why? Let's look at the point - here you have to measure fifteen centimeters down, and, obviously, the drawing will not fit (or barely fit) on a notebook sheet. Therefore, we immediately select a smaller scale 1 unit = 1 cell.

By the way, about centimeters and notebook cells. Is it true that there are 15 centimeters in 30 notebook cells? Measure in a notebook for interest 15 centimeters with a ruler. In the USSR, perhaps this was true ... It is interesting to note that if you measure these same centimeters horizontally and vertically, then the results (in cells) will be different! Strictly speaking, modern notebooks are not checkered, but rectangular. It may seem like nonsense, but drawing, for example, a circle with a compass in such situations is very inconvenient. To be honest, at such moments you begin to think about the correctness of Comrade Stalin, who was sent to camps for hack work in production, not to mention the domestic automotive industry, falling planes or exploding power plants.

Speaking of quality, or a brief recommendation on stationery. To date, most of the notebooks on sale, without saying bad words, are complete goblin. For the reason that they get wet, and not only from gel pens, but also from ballpoint pens! Save on paper. For clearance control works I recommend using the notebooks of the Arkhangelsk Pulp and Paper Mill (18 sheets, cage) or Pyaterochka, although it is more expensive. It is advisable to choose a gel pen, even the cheapest Chinese gel refill is much better than a ballpoint pen, which either smears or tears paper. The only "competitive" ballpoint pen in my memory is the Erich Krause. She writes clearly, beautifully and stably - either with a full stem, or with an almost empty one.

Additionally: the vision of a rectangular coordinate system through the eyes of analytical geometry is covered in the article Linear (non) dependence of vectors. Vector basis, detailed information about coordinate quarters can be found in the second paragraph of the lesson Linear inequalities.

3D case

It's almost the same here.

1) We draw coordinate axes. Standard: applicate axis – directed upwards, axis – directed to the right, axis – downwards to the left strictly at an angle of 45 degrees.

2) We sign the axes.

3) Set the scale along the axes. Scale along the axis - two times smaller than the scale along the other axes. Also note that in the right drawing, I used a non-standard "serif" along the axis (this possibility has already been mentioned above). From my point of view, it’s more accurate, faster and more aesthetically pleasing - you don’t need to look for the middle of the cell under a microscope and “sculpt” the unit right up to the origin.

When doing a 3D drawing again - give priority to scale
1 unit = 2 cells (drawing on the left).

What are all these rules for? Rules are there to be broken. What am I going to do now. The fact is that the subsequent drawings of the article will be made by me in Excel, and the coordinate axes will look incorrect in terms of proper design. I could draw all the graphs by hand, but it’s really scary to draw them, as Excel is reluctant to draw them much more accurately.

Graphs and basic properties of elementary functions

The linear function is given by the equation . Linear function graph is direct. In order to construct a straight line, it is enough to know two points.

Example 1

Plot the function. Let's find two points. It is advantageous to choose zero as one of the points.

If , then

We take some other point, for example, 1.

If , then

When preparing tasks, the coordinates of points are usually summarized in a table:

And the values ​​themselves are calculated orally or on a draft, calculator.

Two points are found, let's draw:

When drawing up a drawing, we always sign the graphics.

It will not be superfluous to recall special cases of a linear function:

Notice how I placed the captions, signatures should not be ambiguous when studying the drawing. AT this case it was extremely undesirable to put a signature next to the point of intersection of the lines, or at the bottom right between the graphs.

1) A linear function of the form () is called direct proportionality. For example, . The direct proportionality graph always passes through the origin. Thus, the construction of a straight line is simplified - it is enough to find only one point.

2) An equation of the form defines a straight line parallel to the axis, in particular, the axis itself is given by the equation. The graph of the function is built immediately, without finding any points. That is, the entry should be understood as follows: "y is always equal to -4, for any value of x."

3) An equation of the form defines a straight line parallel to the axis, in particular, the axis itself is given by the equation. The graph of the function is also built immediately. The entry should be understood as follows: "x is always, for any value of y, equal to 1."

Some will ask, well, why remember the 6th grade?! That's how it is, maybe so, only during the years of practice I met a good dozen students who were baffled by the task of constructing a graph like or .

Drawing a straight line is the most common action when making drawings.

The straight line is discussed in detail in the course of analytic geometry, and those who wish can refer to the article Equation of a straight line on a plane.

Quadratic function graph, cubic function graph, polynomial graph

Parabola. Graph of a quadratic function () is a parabola. Consider the famous case:

Let's recall some properties of the function.

So, the solution to our equation: - it is at this point that the vertex of the parabola is located. Why this is so can be learned from the theoretical article on the derivative and the lesson on the extrema of the function. In the meantime, we calculate the corresponding value of "y":

So the vertex is at the point

Now we find other points, while brazenly using the symmetry of the parabola. It should be noted that the function – is not even, but, nevertheless, no one canceled the symmetry of the parabola.

In what order to find the remaining points, I think it will be clear from the final table:

This construction algorithm can be figuratively called a "shuttle" or the "back and forth" principle with Anfisa Chekhova.

Let's make a drawing:

From the considered graphs, another useful feature comes to mind:

For a quadratic function () the following is true:

If , then the branches of the parabola are directed upwards.

If , then the branches of the parabola are directed downwards.

In-depth knowledge of the curve can be obtained in the lesson Hyperbola and parabola.

The cubic parabola is given by the function . Here is a drawing familiar from school:

We list the main properties of the function

Function Graph

It represents one of the branches of the parabola. Let's make a drawing:

The main properties of the function:

In this case, the axis is vertical asymptote for the hyperbola graph at .

Will be BAD mistake, if, when making a drawing, by negligence, we allow the graph to intersect with the asymptote .

Also one-sided limits, tell us that a hyperbole not limited from above and not limited from below.

Let's explore the function at infinity: , that is, if we start to move along the axis to the left (or right) to infinity, then the “games” will be a slender step infinitely close approach zero, and, accordingly, the branches of the hyperbola infinitely close approach the axis.

So the axis is horizontal asymptote for the graph of the function, if "x" tends to plus or minus infinity.

The function is odd, which means that the hyperbola is symmetrical with respect to the origin. This fact is obvious from the drawing, moreover, it can be easily verified analytically: .

The graph of a function of the form () represents two branches of a hyperbola.

If , then the hyperbola is located in the first and third coordinate quadrants(see picture above).

If , then the hyperbola is located in the second and fourth coordinate quadrants.

It is not difficult to analyze the specified regularity of the place of residence of the hyperbola from the point of view of geometric transformations of graphs.

Example 3

Construct the right branch of the hyperbola

We use the pointwise construction method, while it is advantageous to select the values ​​so that they divide completely:

Let's make a drawing:

It will not be difficult to construct the left branch of the hyperbola, here the oddness of the function will just help. Roughly speaking, in the pointwise construction table, mentally add a minus to each number, put the corresponding dots and draw the second branch.

Detailed geometric information about the considered line can be found in the article Hyperbola and parabola.

Graph of an exponential function

In this paragraph, I will immediately consider the exponential function, since in problems of higher mathematics in 95% of cases it is the exponent that occurs.

I remind you that - this is an irrational number: , this will be required when building a graph, which, in fact, I will build without ceremony. Three points probably enough:

Let's leave the graph of the function alone for now, about it later.

The main properties of the function:

Fundamentally, the graphs of functions look the same, etc.

I must say that the second case is less common in practice, but it does occur, so I felt it necessary to include it in this article.

Graph of a logarithmic function

Consider a function with natural logarithm .
Let's do a line drawing:

If you forgot what a logarithm is, please refer to school textbooks.

The main properties of the function:

Domain:

Range of values: .

The function is not limited from above: , albeit slowly, but the branch of the logarithm goes up to infinity.
Let us examine the behavior of the function near zero on the right: . So the axis is vertical asymptote for the graph of the function with "x" tending to zero on the right.

Be sure to know and remember the typical value of the logarithm: .

Fundamentally, the plot of the logarithm at the base looks the same: , , (decimal logarithm to base 10), etc. At the same time, the larger the base, the flatter the chart will be.

We will not consider the case, something I don’t remember when the last time I built a graph with such a basis. Yes, and the logarithm seems to be a very rare guest in problems of higher mathematics.

In conclusion of the paragraph, I will say one more fact: Exponential Function and Logarithmic Functionare two mutually inverse functions. If you look closely at the graph of the logarithm, you can see that this is the same exponent, just it is located a little differently.

Graphs of trigonometric functions

How does trigonometric torment begin at school? Correctly. From the sine

Let's plot the function

This line is called sinusoid.

I remind you that “pi” is an irrational number:, and in trigonometry it dazzles in the eyes.

The main properties of the function:

This function is periodical with a period. What does it mean? Let's look at the cut. To the left and to the right of it, exactly the same piece of the graph repeats endlessly.

Domain: , that is, for any value of "x" there is a sine value.

Range of values: . The function is limited: , that is, all the “games” sit strictly in the segment .
This does not happen: or, more precisely, it happens, but these equations do not have a solution.

How to build a parabola? There are several ways to graph a quadratic function. Each of them has its pros and cons. Let's consider two ways.

Let's start by plotting a quadratic function like y=x²+bx+c and y= -x²+bx+c.

Example.

Plot the function y=x²+2x-3.

Solution:

y=x²+2x-3 is a quadratic function. The graph is a parabola with branches up. Parabola vertex coordinates

From the vertex (-1;-4) we build a graph of the parabola y=x² (as from the origin. Instead of (0;0) - the vertex (-1;-4). From (-1;-4) we go to the right by 1 unit and up by 1, then left by 1 and up by 1, then: 2 - right, 4 - up, 2 - left, 4 - up, 3 - right, 9 - up, 3 - left, 9 - up. these 7 points are not enough, then - 4 to the right, 16 - up, etc.).

The graph of the quadratic function y= -x²+bx+c is a parabola whose branches are directed downwards. To build a graph, we are looking for the coordinates of the vertex and from it we build a parabola y= -x².

Example.

Plot the function y= -x²+2x+8.

Solution:

y= -x²+2x+8 is a quadratic function. The graph is a parabola with branches down. Parabola vertex coordinates

From the top we build a parabola y = -x² (1 - right, 1 - down; 1 - left, 1 - down; 2 - right, 4 - down; 2 - left, 4 - down, etc.):

This method allows you to build a parabola quickly and does not cause difficulties if you know how to plot the functions y=x² and y= -x². Disadvantage: if the vertex coordinates are fractional numbers, plotting is not very convenient. If you want to know the exact values ​​of the intersection points of the graph with the x-axis, you will have to additionally solve the equation x² + bx + c = 0 (or -x² + bx + c = 0), even if these points can be directly determined from the figure.

Another way to build a parabola is by points, that is, you can find several points on the graph and draw a parabola through them (taking into account that the line x=xₒ is its axis of symmetry). Usually, for this, they take the top of the parabola, the intersection points of the graph with the coordinate axes, and 1-2 additional points.

Plot the function y=x²+5x+4.

Solution:

y=x²+5x+4 is a quadratic function. The graph is a parabola with branches up. Parabola vertex coordinates

that is, the top of the parabola is the point (-2.5; -2.25).

Are looking for . At the point of intersection with the Ox axis y=0: x²+5x+4=0. Roots quadratic equation x1=-1, x2=-4, that is, we got two points on the graph (-1; 0) and (-4; 0).

At the intersection point of the graph with the Oy axis x=0: y=0²+5∙0+4=4. Got a point (0; 4).

To refine the graph, you can find an additional point. Let's take x=1, then y=1²+5∙1+4=10, that is, one more point of the graph - (1; 10). We mark these points on the coordinate plane. Taking into account the symmetry of the parabola with respect to the straight line passing through its vertex, we mark two more points: (-5; 6) and (-6; 10) and draw a parabola through them:

Plot the function y= -x²-3x.

Solution:

y= -x²-3x is a quadratic function. The graph is a parabola with branches down. Parabola vertex coordinates

The top (-1.5; 2.25) is the first point of the parabola.

At the points of intersection of the graph with the x-axis y=0, that is, we solve the equation -x²-3x=0. Its roots are x=0 and x=-3, that is, (0; 0) and (-3; 0) are two more points on the graph. The point (o; 0) is also the point of intersection of the parabola with the y-axis.

At x=1 y=-1²-3∙1=-4, i.e. (1; -4) is an additional point for plotting.

Building a parabola from points is a more time-consuming method compared to the first one. If the parabola does not intersect the Ox axis, more additional points will be required.

Before continuing the construction of graphs of quadratic functions of the form y=ax²+bx+c, consider the construction of graphs of functions using geometric transformations. Graphs of functions of the form y=x²+c are also most convenient to build using one of these transformations - parallel translation.

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