The module sign, perhaps, is one of the most famous phenomena in mathematics. At zv'yazku z tsim at rich schoolchildren post nutrition, as buduvat schedules of functions, scho to avenge the module. Let's report on the food chain.

1. Pobudova graphic functions, what to replace the module

example 1.

Induce the graph of the function y = x 2 - 8 | x | + 12.

Solution.

The parity of the function is significant. The values ​​for y(-x) are generated from the values ​​for y(x), which is given a pair function. Todi її schedule symmetrical shdo osі Oy. It will be the graph of the function y = x 2 - 8x + 12 for x ≥ 0, and the graph of Oy will be symmetrically displayed for negative x (Fig. 1).

butt 2.

Upcoming schedule mind y = | x 2 - 8x + 12 |.

– What is the scope of the proponated function? (y ≥ 0).

- How is the schedule redrawing? (Above the top of the abscissa or sticking out її).

Tse means that the graph of the function has the following order: the graph of the function y \u003d x 2 - 8x + 12 will fill out the part of the graph, which lies above the line Ox, without changing, and the part of the graph, which lies under the line of the abscissa, symmetrically shows that to axis Ox (Fig. 2).

Example 3.

To encourage the graph of the function y = | x 2 - 8 | x | + 12 | carry out a combination of transformations:

y = x 2 - 8x + 12 → y = x 2 - 8 | + 12 → y = | x 2 - 8 | x | + 12 |.

Suggestion: Figure 3.

Look at the transformation of fair to all types of functions. Let's make a table:

2. Pobudova graphs of functions, like in the formula "insert modules"

We have already learned about the butts of the quadratic function, how to avenge the module, as well as the basic rules for making graphs of functions of the form y = f(|x|), y = |f(x)| and y = |f(|x|)|. Qi transformation will help us for an hour to look at the offensive butt.

Example 4.

Let's look at a function of type y = |2 – |1 – |x|||. Viraz, who sets the function, remove the module insertion.

Solution.

Speeding up by the method of geometric transformations.

Let's write down the lanterns of the last changes and zrobimo in the middle of the armchair (Fig. 4):

y=x → y=| x | → y = -|x| → y = -|x| + 1 → y = |-|x| + 1|→ y = -|-|x| + 1|→ y = -|-|x| + 1 | + 2 → y = | 2 - | 1 - | x | | |.

Let's look at the vipadki, if the transformation of symmetry and parallel transference is the main technique for encouraging schedules.

Example 5.

Induce the graph of the function to the form y \u003d (x 2 - 4) / √ (x + 2) 2.

Solution.

The first time will be the schedule, we will remake the formula, which is the given function, that is taken away, otherwise the analytical function is given (Fig. 5).

y = (x 2 - 4) / √ (x + 2) 2 = (x - 2) (x + 2) / | x + 2 |.

Rozkriёmo at the banner of the module:

For x > -2, y = x - 2, and for x< -2, y = -(x – 2).

Destination area D(y) = (-∞; -2)ᴗ(-2; +∞).

Value area E(y) = (-4; +∞).

Points, in which the graph changes along the coordinate axes: (0; -2) and (2; 0).

The function changes over all x intervals (-∞; -2), grows at x out -2 to +∞.

Here we had a chance to decipher the sign of the module and to develop a graph of the function for the skin rash.

Example 6.

Let's look at the function y = | x + 1 | - | x - 2 |.

Solution.

Exploring the sign of the module, it is necessary to look at the different combination of signs of the submodule verses.

Possibly some vipadki:

(x + 1 - x + 2 = 3, with x ≥ -1 and x ≥ 2;

(-x - 1 + x - 2 = -3, with x< -1 и x < 2;

(x + 1 + x - 2 = 2x - 1, for x ≥ -1 i x< 2;

(-x - 1 - x + 2 = -2x + 1, with x< -1 и x ≥ 2 – пустое множество.

Same look function matime look:

(3, for x ≥ 2;

y = (-3, at x< -1;

(2x – 1, with -1 ≤ x< 2.

We took away the lump-set function, the graph of which is depicted as a little 6.

3. Algorithm for inducing graphs of functions in the form

y = a 1 | x - x 1 | + a 2 | x - x 2 | + … + a n | x - x n | + ax + b.

At the front butt, it was easy to open the signs of the module. If the sum of modules is larger, then it is problematic to look at all the combinations of signs of submodules. How can I induce the schedule of the function to whom?

It is important that the graph has a laman, with vertices at points, that the abscissas can be -1 and 2. At x = -1 and x = 2, submodules are equal to zero. In a practical way, we came close to the rule of encouraging such schedules:

Graph of the function in the form y = a 1 | x - x 1 | + a 2 | x - x 2 | + … + a n | x - x n | + ax + b є laman with inexhaustible extreme lanks. In order to induce such a laman, it is enough to know all її vertices (absciss of vertices є zero pіdmodulnyh virazіv) and one control point on the left and right of the non-skinned lankas.

Manager.

Induce the graph of the function y = | x | + | x - 1 | + | x + 1 | and to know is the least important.

Solution:

Zero submodular viruses: 0; -1; 1. Tops of lamanoi (0; 2); (-13); (13). Control point right-handed (2; 6), evil (-2; 6). There will be a schedule (Fig. 7). min f(x) = 2.

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Lesson 5

Target: master the basic skills of converting graphics from modules.

I. Informed by those that mark the lesson

II . Repetition and consolidation of the material covered

1. Vіdpovіdі on zapovіdnja schodo zavdannya (analysis of unviolated tasks).

2. Control assimilated to the material (letter review).

Option 1

f (x), induce the graph of the function y = f(-x) + 2?

2. Check out the function schedule:

Option 2

1. Yak, knowing the graph of the function y = f (x), induce the graph of the function y \u003d - f(x) - 1?

2. Check out the function schedule:

III. Introduction of new material

From the material of the previous lesson, it is clear that you should make a transformation of the graphics in an oversized way when you wake up. Therefore, let's take a look at the main ways of converting graphics to replace modules. Qi ways are universal and accessories for any functions. For simplicity, consider the schmatkovo-linear function f (x) with the area of ​​​​appointment D(f ), a schedule for the performances of the little one. Let's take a look at three standard transformations of graphics with modules.

1) Pobud's graph of the function y = | f(x) |

f /(x), yaksho Dx)>0,

For the purpose of the module, we take:Tse means that the graph of the function y = | f(x )| need to save part of the graph of the function y = f(x ), for which y ≥ 0. That part of the graph of the function y = f (x), for which y< 0, надо симметрично отразить вверх относительно оси абсцисс.

2) Pobudova graph of the function y = f(|x|)

G / O), yakscho Dx)\u003e 0,

Recognize the module and take:Therefore, to encourage the graph of the function y \u003d f(|x |) need to save part of the graph of the function y = f (x), for which x ≥ 0. In addition, the first part needs to be displayed symmetrically to the left along the y-axis.

3) Pobudov's schedule equalization |y| = f(x)

For the purpose of the module, it is possible that for f (x) ≥ 0 requires the graphs of two functions: y = f(x) and y = - f (X). Tse means, scho s pobudovi graphics equalization | = f (x) need to save part of the graph of the function y = f (x), for ikoї ≥ 0. In addition, this part needs to be shown symmetrically down along the x-axis.

Dearly, zalezhnist |y| = f (x) does not specify a function, then for x∈ (-2.6; 1.4) skin x value is given two y values. To that little one of the presentations itself, the schedule is equal | y | = f(x).

Vikoristujom looked at ways of converting graphics with modules to encourage graphics folding functions and alignment.

butt 1

Please let us schedule the function

We see that this function has a whole partSuch a graph will appear when the graph of the function y \u003d -1 / x 2 units to the right and 1 unit down. Graph tsієї functions є hyperbole.

butt 2

Please let us schedule the function

Prior to method 1, we save part of the graph from butt 1, for which y ≥ 0. That part of the graph, for which y< 0, симметрично отразим вверх относительно оси абсцисс.

butt 3

Please let us schedule the function

Vikoristovuyuchi method 2, we save part of the graph butt 1, for x ≥ 0. I save a part, moreover, it is mirrored to the left along the y-axis. Take away the graph of the function, symmetrical along the y-axis.

butt 4

Let's have a schedule

Vіdpovіdno to method 3 save part of the graph from the butt 1, for which ≥ 0. In addition, the part to be saved is symmetrically presumably down along the abscissa axis. We take away the schedule of thogo equal.

Obviously, the reviewed methods and the transformation of graphics can be scored at once.

butt 5

Please let us schedule the function

Vykoristovuemo schedule functioninducement at the butt 3. To encourage the dates of the schedule, save parts of the schedule 3, for those at ≥ 0. Those parts of the schedule 3, for those at< 0, симметрично отразим вверх относительно оси абсцисс.

In quiet situations, if the modules lie in a lower rank (lower in methods 1-3), it is necessary to open the modules.

butt 6

Please let us schedule the function

Virazi x - 1 ta x + 2, to enter under the signs of modules, change their signs at the points x = 1 i x = -2 is reasonable. Significantly qi points on the coordinate line. The stench breaks її into three intervals. Vykoristovuyuchi designation of the module, rozkriёmo modules in the skin area.

We take:

1. When

2. When

3. When

Let's make the graphs of these functions, the rotation intervals for the change x, de-route the signs of the module. Take laman straight.

To finish often with the help of schedules, equalize with modules for the future opening of the vicorous coordinate plane. Let's explain it with an example.

butt 7

Let's have a schedule

Viraz y - x changes its sign to a straight line y \u003d x. Let's call the straight line - the bisector of the first and third coordinate cuts. Tsya straight splits the points of the plane into two areas: 1 - points, spread over the straight line y - x; 2 - points, roztashovanі under tsієyu straight line. Rozkriёmo module in such areas. In area 1, take, for example, a control point (0; 5). Bachimo, scho for tsієї point viraz y - x\u003e 0. Curve the module, we take: y - x + y + x \u003d 4 or y = 2. It will be right at the borders of the first region. Obviously, in the region 2 virase y - x< 0. Раскрывая модуль, имеем: -(у - х) + у + х = 4 или х = 2. Строим эту прямую в пределах области 2. Получаем график данного уравнения.

3. Look at the graph of the shot-linear function and alignment:

4. Look at the function schedule, alignment, unevenness:

VIII. P_dbitya p_dbag_v lesson

The module sign, perhaps, is one of the most famous phenomena in mathematics. At zv'yazku z tsim at rich schoolchildren post nutrition, as buduvat schedules of functions, scho to avenge the module. Let's report on the food chain.

1. Pobudova graphic functions, what to replace the module

example 1.

Induce the graph of the function y = x 2 - 8 | x | + 12.

Solution.

The parity of the function is significant. The values ​​for y(-x) are generated from the values ​​for y(x), which is given a pair function. Todi її schedule symmetrical shdo osі Oy. It will be the graph of the function y = x 2 - 8x + 12 for x ≥ 0, and the graph of Oy will be symmetrically displayed for negative x (Fig. 1).

butt 2.

Upcoming schedule mind y = | x 2 - 8x + 12 |.

– What is the scope of the proponated function? (y ≥ 0).

- How is the schedule redrawing? (Above the top of the abscissa or sticking out її).

Tse means that the graph of the function has the following order: the graph of the function y \u003d x 2 - 8x + 12 will fill out the part of the graph, which lies above the line Ox, without changing, and the part of the graph, which lies under the line of the abscissa, symmetrically shows that to axis Ox (Fig. 2).

Example 3.

To encourage the graph of the function y = | x 2 - 8 | x | + 12 | carry out a combination of transformations:

y = x 2 - 8x + 12 → y = x 2 - 8 | + 12 → y = | x 2 - 8 | x | + 12 |.

Suggestion: Figure 3.

Look at the transformation of fair to all types of functions. Let's make a table:

2. Pobudova graphs of functions, like in the formula "insert modules"

We have already learned about the butts of the quadratic function, how to avenge the module, as well as the basic rules for making graphs of functions of the form y = f(|x|), y = |f(x)| and y = |f(|x|)|. Qi transformation will help us for an hour to look at the offensive butt.

Example 4.

Let's look at a function of type y = |2 – |1 – |x|||. Viraz, who sets the function, remove the module insertion.

Solution.

Speeding up by the method of geometric transformations.

Let's write down the lanterns of the last changes and zrobimo in the middle of the armchair (Fig. 4):

y=x → y=| x | → y = -|x| → y = -|x| + 1 → y = |-|x| + 1|→ y = -|-|x| + 1|→ y = -|-|x| + 1 | + 2 → y = | 2 - | 1 - | x | | |.

Let's look at the vipadki, if the transformation of symmetry and parallel transference is the main technique for encouraging schedules.

Example 5.

Induce the graph of the function to the form y \u003d (x 2 - 4) / √ (x + 2) 2.

Solution.

The first time will be the schedule, we will remake the formula, which is the given function, that is taken away, otherwise the analytical function is given (Fig. 5).

y = (x 2 - 4) / √ (x + 2) 2 = (x - 2) (x + 2) / | x + 2 |.

Rozkriёmo at the banner of the module:

For x > -2, y = x - 2, and for x< -2, y = -(x – 2).

Destination area D(y) = (-∞; -2)ᴗ(-2; +∞).

Value area E(y) = (-4; +∞).

Points, in which the graph changes along the coordinate axes: (0; -2) and (2; 0).

The function changes over all x intervals (-∞; -2), grows at x out -2 to +∞.

Here we had a chance to decipher the sign of the module and to develop a graph of the function for the skin rash.

Example 6.

Let's look at the function y = | x + 1 | - | x - 2 |.

Solution.

Exploring the sign of the module, it is necessary to look at the different combination of signs of the submodule verses.

Possibly some vipadki:

(x + 1 - x + 2 = 3, with x ≥ -1 and x ≥ 2;

(-x - 1 + x - 2 = -3, with x< -1 и x < 2;

(x + 1 + x - 2 = 2x - 1, for x ≥ -1 i x< 2;

(-x - 1 - x + 2 = -2x + 1, with x< -1 и x ≥ 2 – пустое множество.

Same look function matime look:

(3, for x ≥ 2;

y = (-3, at x< -1;

(2x – 1, with -1 ≤ x< 2.

We took away the lump-set function, the graph of which is depicted as a little 6.

3. Algorithm for inducing graphs of functions in the form

y = a 1 | x - x 1 | + a 2 | x - x 2 | + … + a n | x - x n | + ax + b.

At the front butt, it was easy to open the signs of the module. If the sum of modules is larger, then it is problematic to look at all the combinations of signs of submodules. How can I induce the schedule of the function to whom?

It is important that the graph has a laman, with vertices at points, that the abscissas can be -1 and 2. At x = -1 and x = 2, submodules are equal to zero. In a practical way, we came close to the rule of encouraging such schedules:

Graph of the function in the form y = a 1 | x - x 1 | + a 2 | x - x 2 | + … + a n | x - x n | + ax + b є laman with inexhaustible extreme lanks. In order to induce such a laman, it is enough to know all її vertices (absciss of vertices є zero pіdmodulnyh virazіv) and one control point on the left and right of the non-skinned lankas.

Manager.

Induce the graph of the function y = | x | + | x - 1 | + | x + 1 | and to know is the least important.

Solution:

Zero submodular viruses: 0; -1; 1. Tops of lamanoi (0; 2); (-13); (13). Control point right-handed (2; 6), evil (-2; 6). There will be a schedule (Fig. 7). min f(x) = 2.

Run out of food? Don't know how to schedule a function with a module?
I’ll help you to help the tutor -.

blog.website, with a new or private copy of the material sent on the original binding.

Pobudova graphic functions, scho to replace the sign of the module.

I agree, you respectfully read paragraph 23 and understand how the function looks like the function. Now let's take a look at some more applications, which are the responsibility of helping you with your schedules.

Example 1. Induce the schedule of the function

May the function of the mind, de.

1. Let's start the schedule of the submodular function, then the function. For whom we see a whole part of the shot. I’m wondering what you can do in two ways: by dividing the number on the banner “at the stump” or by writing the number in such a way that a viraz appears at the new one, a multiple of the banner. We can see the whole part in a different way.

Otzhe, submodular function can be seen . So, її schedule є hyperbole mind, shifted by 1 unit to the right and 3 units uphill.

Let's take a look at your schedule.

2. In order to capture the graph of the current function, it is necessary to remove a part of the generated function graph, which lies higher than the Ox axis, without changing, and the part of the graph, which lies below the Ox axis, is shown symmetrically near the upper plane. Vikonaemo qi transformation.

Wake up schedule.

The abscissa of the point of the line of the graph from the line

y = 0, then. Let's take it away.

Now, behind the graph, you can determine all the power of the function, find the least and most important function for the interval, and solve tasks with the parameter.

For example, you can vodpovisti on such a supply. “For any values ​​of the parameter A can i make one decision?

Conducted directly y=a with different values ​​of the parameter A. (Thin red lines straight on the stepping little one)

It can be seen that a<0 , then the graph of the induced function and there are no direct hot points, hence, there is no equal solution.

Yakscho 0< a<3 or a>3, then straight y=a and prompting the schedule to make two bright points, so that the equal can be two rozv'azki.

Yakshcho a = 0 or a = 3, then there is only one solution, that is, up to. at these values A that graph of the function is straight, it can exactly one point.

butt 2. Induce the function schedule

Solution

Let's start the graph of the function with negative values ​​of x. So , even then our function is visible to the eye , but the function is needed -- the function of the mind .

The graph of the function - the head of the parabola is "straight" to the left, shifted to 4 units right-hander. (T. do. we can show ).

Let's have a schedule of functions

And we will only look at that part of the yogo, as it is ripped to the right for the Oy axis. Reshta trioma.

Catch the respect that we have ignored the ordinate values ​​of the graph point, which lies on the y-axis. For which it is sufficient to calculate the value of the function at x = 0. In our opinion x = 0 took y=2.

Now we will prompt the schedule of the function when X< 0 . For whom we will create a line, symmetrical, which we have already created, like axis Oy.

In this manner, we prompted the schedule of the shukano function.

Example 3. Induce the schedule of the function

The task is no longer easy. Bachimo, what's wrong with seeing the functions with the module: i , i . Let's go in order:

Let's start by adding the function schedule without the required modules: Let's add the skin argument module. We take away the function of the mind, that is. To encourage such a graph, it is necessary to fix the symmetry of the axis Oy. Adding a new old module. Take away, nareshti, I will need a function. Since this function is removed from the front of the call module, we can see the function, and therefore, it is necessary to block the symmetry of the Oh.

Now report.

This is a shot-linear function, for stimulating the graphic, it is necessary to see the whole part, the lower one is taken up.

So, the graph of the functions is a hyperbole, shifted by 2 to the right and 4 down.

Let's calculate the coordinates and the point of the crossbar with the coordinate axes.

y = 0 at x = 0, the graph will also pass through the cob of coordinates.

2. Now let's call the function schedule.

For that, at the end of the chart, the back of the head zіtremo that yogo part, as if roztashovuєtsya levoruch vіd osі Oy:

, and then it can be imagined її symmetrically around the Oy axis. Respect, the asymptotes are also symmetrically displayed!

Now let's see the remaining schedule of the function: . For which part of the front graph, which lie higher than the Ox axis, it is left without change, and those that are located below the Ox axis are symmetrically visible near the upper plane. Again, do not forget that the asymptotes appear at once from the graph!

Wake up schedule.

Example 4

Schos is absolutely twisted and folded! A bunch of modules! And the square x has no module! It's impossible to wake up!

So chi is approximately so able to demystify the average statistical student of the 8th class, unknown from the technique of prompting graphs.

Ale, not me! To that we know the different ways of transforming the schedule of functions and we also know the different power of the module.

So let's do it in order.

The first problem is the presence of the module in x in the square. Not bad. We know what. Dobre. Also, our function can be written as . Tse is already more beautiful, that is similar to.

Dali. The function may be the most important module, which, it seems, will have to follow the rules of the schedule of the function. Let's look at that, which is a submodular viraz. See function . Yakby is not -2, then the function will reset the current module and know how to induce the graph of the function for help symmetry. Aha! Ale, if mi yogo will be instructed, then, having replaced yogo by 2 units down, we will take away the shukane!

Otzhe, schos begin to vimalovuvatsya. Let's try to put together an algorithm to encourage graphics.

1.

5. I for example, . All those that lie below the Ox axis seem to be symmetrical near the upper plane.

Hooray! Schedule ready!

Good luck to you at the difficult task of getting a schedule!