Function of the form , where is called quadratic function.

Graph of quadratic function − parabola.

Consider the cases:

CASE I, CLASSICAL PARABOLA

I.e , ,

To build, fill in the table by substituting x values ​​into the formula:

Mark points (0;0); (1;1); (-1;1) etc. on the coordinate plane (the smaller the step we take x values ​​(in this case, step 1), and the more x values ​​we take, the smoother the curve), we get a parabola:

It is easy to see that if we take the case , , , that is, then we get a parabola symmetric about the axis (ox). It is easy to verify this by filling out a similar table:

II CASE, "a" DIFFERENT FROM ONE

What will happen if we take , , ? How will the behavior of the parabola change? With title="(!LANG: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):!}

The first picture (see above) clearly shows 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 argue similarly in the cases of pictures 2 and 3.

And when the parabola "becomes wider" parabola:

Let's recap:

1)The sign of the coefficient is responsible for the direction of the branches. With title="(!LANG: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”, “compression” of the parabola. The larger , the narrower the parabola, the smaller |a|, the wider the parabola.

CASE III, "C" APPEARS

Now let's put into play (that is, we consider the case when ), we will consider parabolas of the form . It is easy to guess (you can always refer to the table) that the parabola will move up or down along the axis, depending on the sign:

IV CASE, "b" APPEARS

When will the parabola “tear off” from the axis and will finally “walk” along the entire coordinate plane? When it ceases to be equal.

Here, to construct a parabola, we need formula for calculating the vertex: , .

So at this point (as at the point (0; 0) new system coordinates) we will build a parabola, which is already within our power. If we are dealing with the case , then from the top we set aside 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 top we set aside one single segment to the right, two - up, 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 template, because in our case.

When constructing a parabola after finding the coordinates of the vertex is veryIt is convenient to consider the following points:

1) parabola must pass through the point . Indeed, substituting x=0 into the formula, we get that . That is, the ordinate of the point of intersection of the parabola with the axis (oy), this is. In our example (above), the parabola intersects the y-axis at , 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 a parabola symmetrical about the axis of symmetry, 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 (ox). To do this, we solve the equation. Depending on the discriminant, we will get one (, ), two ( title="(!LANG:Rendered by QuickLaTeX.com" height="14" width="54" style="vertical-align: 0px;">, ) или нИсколько () точек пересечения с осью (ох) !} . In the previous example, we have the root of the discriminant - not an integer, when building it, it doesn’t really make sense for us to find the roots, but we can clearly see that we will have two points of intersection with the (oh) axis (since title = "(!LANG: Rendered by QuickLaTeX.com" height="14" width="54" style="vertical-align: 0px;">), хотя, в общем, это видно и без дискриминанта.!}

So let's work 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) find the coordinates of the vertex of the parabola by the formula , .

3) we find the point of intersection of the parabola with the axis (oy) by the free term, we build a point symmetrical to the given one with respect to the axis of symmetry 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 top of the parabola (as at the point (0; 0) of the new coordinate system), we build a parabola. If title="(!LANG: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 themselves have not yet “surfaced”), solving the equation

Example 1

Example 2

Remark 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 build it, because we have already been given the coordinates of the vertex . Why?

Let's take a square trinomial and select a full square in it: Look, here we got that , . We previously called the top of the parabola, that is, now,.

For example, . We mark the top of the parabola on the plane, we understand that the branches are directed downwards, the parabola is expanded (relatively). That is, we perform steps 1; 3; 4; 5 from the algorithm for constructing a parabola (see above).

Remark 2. If the parabola is given in a form similar to this (that is, represented as a product of two linear factors), then we immediately see the points of intersection of the parabola with the (x) axis. In this case - (0;0) and (4;0). For the rest, we act according to the algorithm, opening the brackets.

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 everything possible types 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 use 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 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:

1. The point of intersection of the desired line with the ordinate vector will have a value equal to c.
2. 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:

1. a \u003d 1, therefore, the branches are directed upwards;
2. extremum coordinates: x = - (-5) / 2 = 5/2; y = (5/2) 2 - 5 * (5/2) + 4 = -15/4;
3. intersects with the y-axis at the value y = 4;
4. find the discriminant: D = 25 - 16 = 9;
5. looking for roots

• X 1 \u003d (5 + 3) / 2 \u003d 4; (4, 0);
• X 2 \u003d (5 - 3) / 2 \u003d 1; (10).

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:

1. a \u003d 3, therefore, the branches are directed upwards;
2. extremum coordinates: x = - (-2) / 2 * 3 = 1/3; y = 3 * (1/3) 2 - 2 * (1/3) - 1 = -4/3;
3. with the y-axis will intersect at the value y \u003d -1;
4. 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 considered 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.

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 issue - how to correctly and FAST build a graph. During the study higher mathematics without knowing the graphs of the basic elementary functions, it will be difficult, so it is very important to remember how the graphs of a parabola, hyperbola, sine, cosine, etc. look like, to remember some function values. 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 short recommendation by 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 from the point of view correct 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. In this case, it was highly 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 algorithm construction can be figuratively called a "shuttle" or the principle of "back and forth" 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 .

It will be a BIG mistake if, when drawing up a drawing, by negligence, you 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 irrational number: , this will be required when building a graph, which, in fact, I will build without ceremony. Three points is 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 mutual 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? Right. 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.

Important notes!
1. If instead of formulas you see abracadabra, clear your cache. How to do it in your browser is written here:
2. Before you start reading the article, pay attention to our navigator for the most useful resource for

To understand what will be written here, you need to know well what a quadratic function is and what it is eaten with. If you consider yourself a pro at quadratic functions, welcome. But if not, you should read the thread.

Let's start with a small checks:

1. What does a quadratic function look like in general form (formula)?
2. What is the name of the graph of a quadratic function?
3. How does the leading coefficient affect the graph of a quadratic function?

If you can answer these questions right off the bat, keep reading. If at least one question caused difficulties, go to.

So, you already know how to handle a quadratic function, analyze its graph and build a graph by points.

Well, here it is: .

Let's take a quick look at what they do. odds.

1. The senior coefficient is responsible for the “steepness” of the parabola, or, in other words, for its width: the larger, the narrower (steeper) the parabola, and the smaller, the wider (flatter) parabola.
2. The free term is the coordinate of the intersection of the parabola with the y-axis.
3. And the coefficient is somehow responsible for the displacement of the parabola from the center of coordinates. Here's more about that now.

Why do we always start building a parabola? What is her distinguishing point?

This vertex. And how to find the coordinates of the vertex, remember?

The abscissa is searched for by the following formula:

Like this: what more, topics to the left the top of the parabola moves.

The ordinate of a vertex can be found by substituting into the function:

Substitute yourself and count. What happened?

If you do everything right and simplify the resulting expression as much as possible, you get:

It turns out that the more modulo, topics above will vertex parabolas.

Finally, let's move on to plotting.
The easiest way is to build a parabola starting from the top.

Example:

Plot the function.

Solution:

First, let's define the coefficients: .

Now let's calculate the vertex coordinates:

And now remember: all parabolas with the same leading coefficient look the same. So, if we build a parabola and move its vertex to a point, we get the graph we need:

Simple, right?

There is only one question left: how to quickly draw a parabola? Even if we draw a parabola with a vertex at the origin, we still have to build it point by point, which is long and inconvenient. But all parabolas look the same, maybe there is a way to speed up their drawing?

When I was at school, my math teacher told everyone to cut out a parabola-shaped stencil out of cardboard so they could draw it quickly. But you won’t be able to walk everywhere with a stencil, and they won’t be allowed to take it to the exam. So, we will not use foreign objects, but we will look for a pattern.

Consider the simplest parabola. Let's build it by points:

The rule here is this. If we move from the top to the right (along the axis) to, and upwards (along the axis) to, then we will get to the point of the parabola. Further: if from this point we move to the right by and up by, we will again get to the point of the parabola. Next: right on and up on. What's next? Right on and up on. And so on: move to the right, and to the next odd number up. Then we do the same with the left branch (after all, the parabola is symmetrical, that is, its branches look the same):

Great, this will help build any parabola from the vertex with the highest coefficient equal to. For example, we have learned that the vertex of a parabola is at a point. Construct (on your own, on paper) this parabola.

Built?

It should turn out like this:

Now we connect the obtained points:

That's all.

OK, well, now build only parabolas with?

Of course not. Now let's figure out what to do with them, if.

Let's consider some typical cases.

Great, we learned how to draw a parabola, now let's practice on real functions.

So, draw graphs of such functions:

Answers:

3. Top: .

Do you remember what to do if the senior coefficient is less?

We look at the denominator of the fraction: it is equal. So we'll move like this:

• right - up
• right - up
• right - up

and also to the left:

4. Top: .

Oh, what to do with it? How to measure cells if the vertex is somewhere between the lines?..

And we cheat. First, let's draw a parabola, and only then move its vertex to a point. Not even, let's do it even more tricky: Let's draw a parabola, and then move axes:- on the down, a - on right:

This technique is very convenient in the case of any parabola, remember it.

Let me remind you that we can represent the function in this form:

For example: .

What does this give us?

The fact is that the number that is subtracted from in brackets () is the abscissa of the vertex of the parabola, and the term outside the brackets () is the ordinate of the vertex.

This means that, having built a parabola, you just need to move the axis to the left and the axis to down.

Example: let's plot a function graph.

Let's select a full square:

What number subtracted from in brackets? This (and not how you can decide without thinking).

So, we build a parabola:

Now we shift the axis down, that is, up:

And now - to the left, that is, to the right:

That's all. This is the same as moving a parabola with its vertex from the origin to a point, only the straight axis is much easier to move than a crooked parabola.

Now, as usual, myself:

And do not forget to erase the old axles with an eraser!

I am as answers for verification, I will write you the ordinates of the vertices of these parabolas:

Did everything fit?

If yes, then you are great! Knowing how to handle a parabola is very important and useful, and here we have found that it is not difficult at all.

GRAPHING A QUADRATIC FUNCTION. BRIEFLY ABOUT THE MAIN

quadratic function is a function of the form, where, and are any numbers (coefficients), is a free member.

The graph of a quadratic function is a parabola.

Top of the parabola:
, i.e. the larger \displaystyle b , the more left the top of the parabola moves.
Substitute in the function, and get:
, i.e. the greater \displaystyle b modulo , the higher the top of the parabola will be

The free term is the coordinate of the intersection of the parabola with the y-axis.

Well, the topic is over. If you are reading these lines, then you are very cool.

Because only 5% of people are able to master something on their own. And if you have read to the end, then you are in the 5%!

Now the most important thing.

You've figured out the theory on this topic. And, I repeat, it's ... it's just super! You are already better than the vast majority of your peers.

The problem is that this may not be enough ...

For what?

For successful passing the exam, for admission to the institute on the budget and, MOST IMPORTANTLY, for life.

I will not convince you of anything, I will just say one thing ...

People who received a good education, earn much more than those who did not receive it. This is statistics.

But this is not the main thing.

The main thing is that they are MORE HAPPY (there are such studies). Perhaps because much more opportunities open up before them and life becomes brighter? Do not know...

But think for yourself...

What does it take to be sure to be better than others on the exam and be ultimately ... happier?

FILL YOUR HAND, SOLVING PROBLEMS ON THIS TOPIC.

On the exam, you will not be asked theory.

You will need solve problems on time.

And, if you haven’t solved them (LOTS!), you will definitely make a stupid mistake somewhere or simply won’t make it in time.

It's like in sports - you need to repeat many times to win for sure.

Find a collection anywhere you want necessarily with solutions detailed analysis and decide, decide, decide!

You can use our tasks (not necessary) and we certainly recommend them.

In order to get a hand with the help of our tasks, you need to help extend the life of the YouClever textbook that you are currently reading.

How? There are two options:

1. Unlock access to all hidden tasks in this article -
2. Unlock access to all hidden tasks in all 99 articles of the tutorial - Buy a textbook - 499 rubles

Yes, we have 99 such articles in the textbook and access to all tasks and all hidden texts in them can be opened immediately.

Access to all hidden tasks is provided for the entire lifetime of the site.

In conclusion...

If you don't like our tasks, find others. Just don't stop with theory.

“Understood” and “I know how to solve” are completely different skills. You need both.

Find problems and solve!