We choose a rectangular coordinate system on the plane and add a value to the argument on the abscissa axis X, but in the y-axis - the value of the function y = f(x).

Schedule function y = f(x) all points are called impersonal points, in which abscissas lie in the area of ​​assigned function, and ordinates are equal to the corresponding values ​​of the function.

In other words, the graph of the function y \u003d f (x) is the nameless point of the plane, X, at some of them are satisfied with the y = f(x).

On fig. 45 and 46 pointed function graphs y = 2x + 1і y \u003d x 2 - 2x.

Strictly seeming, following the difference between the graph of the function (more precisely, the mathematical designation of what was given more) and crossed the curve, as a rule, I give more or less precise sketch of the graph (that and those, as a rule, are not less than a graph, but less than one part, ruffled in the kіtsev and part of the plane). Nadali, however, we sound like a "graphic" and not "a sketch of the graph."

For additional graphics, you can know the value of the function at the point. Same as a point x = a belong to the area of ​​assigned function y = f(x), then the value of the number f(a)(so the value of the function at the point x = a) next write like this. Useful through a dot with an abscissa x = a draw a straight line parallel to the y-axis; Directly transfer the function schedule y = f(x) at one point; ordinate tsієї point i bude, z vyznachennya graphics, dorivnyuє f(a)(Fig. 47).

For example, for the function f(x) = x 2 - 2x From the help chart (Fig. 46) we know f(-1) = 3, f(0) = 0, f(1) = -l, f(2) = 0 just fine.

The graph of the function clearly illustrates the behavior and power of the function. For example, looking at Fig. 46 clear what the function is y \u003d x 2 - 2x takes on a positive value when X< 0 i at x > 2, Negative - at 0< x < 2; наименьшее значение функция y \u003d x 2 - 2x accept for x = 1.

To encourage graphics functions f(x) it is necessary to know all points of the plane, coordinates X,at those who are satisfied with the jealousy y = f(x). Most of the time, it is impossible to grow, the shards of such points are infinitely rich. Therefore, the graph of the function is depicted approximately with greater or lesser accuracy. The simplest is the method of inducing a graph for a number of dots. Win the argument X set the final number of values ​​- say, x 1, x 2, x 3, ..., x k and set up a table, to which the selected value of the function is included.

The table looks like this:

Having added such a table, we can name a few points of the graph of the function y = f(x). Let's add a smooth line to the points, we will take an approximate view of the graph of the function y = f(x).

It should be noted that the method of inducing a graph for a number of dots is no longer appropriate. In fact, the behavior of the graph between the designated points and the behavior of the yoga posture in the extreme between the extreme points taken are filled with the unknown.

butt 1. To encourage graphics functions y = f(x) xtos having added a table to the value of the argument of that function:

Vіdpovіdnі five points is shown in fig. 48.

On the support, the rotting of the points of the vines has been made, so the graph of the function is a straight line (shown in Fig. 48 by a dotted line). Chi can vvazhat tsey vysnovok over it? As there are no additional mirkuvans that confirm this whiskers, it is unlikely that one can be taken into account by them. above.

For priming your firmness, let's look at the function

.

The calculation shows that the values ​​of the function at points -2, -1, 0, 1, 2 are described by the above table. However, the graph of the function is not a straight line (indications in Fig. 49). Another butt can be a function y = x + l + sinx;її The values ​​can also be described by the above table.

Use it to show how the “pure” method looks like a graphic behind a kilkom with dots. Therefore, for prompting the schedule of a given function, as a rule, such a method is needed. At the same time, the power of function is raised, with the help of which one can induce a sketch of a schedule. Then, counting the values ​​of the function at a number of points (the choice of which to lie in the established powers of the function), we know the most important points of the graph. І, nareshti, draw a curve through the prompted points, vicorist to the power of the function.

Deyakі (the most simple and the most victorious) of the power of functions, zastosovuvani perebuvannya eskіzu graphics, peacefully pіznіshe, now razberemo deyakі often zastosovuvanі methodi pobudovi graphіv.

Graph of the function y = | f(x)|.

Frequently communicated to the schedule of the function y=| f(x)|, de f(x) - function is set. Guessing how to fight. For the appointment of the absolute value of the number, you can write

Ze means that the graph of the function y=| f(x) | you can select graphics, functions y = f(x) in the coming order: all points of the graph of the function y = f(x), if the ordinates can be non-negative, the next is left without change; far, change point of the graph of the function y = f(x), which can generate negative coordinates, next induce the corresponding points of the graph of the function y = -f(x)(this is part of the graph of the function
y = f(x), which lies below the axis X, next symmetrically to the axis X).

butt 2. Induce the function schedule y = | x |.

Beremo schedule function y = x(Fig. 50, a) that part of the graph at X< 0 (what to lie under the sky X) symmetrically in line with the axis X. As a result, we take the function schedule y = | x |(Fig. 50, b).

butt 3. Induce the function schedule y=| x 2 - 2x |.

At a glance, we will call the schedule of the function y \u003d x 2 - 2x. The graph of the function is a parabola, the needles are straight uphill, the vertex of the parabola has coordinates (1; -1), the graph is redrawn all the abscissas at the points 0 and 2. On the interval (0; 2) the function gains negative values, that same part of the graph as symmetrically imaginable along the abscissa axis. On baby 51, a schedule of functions was prompted y = | x 2 -2x |, appearing from the graph of the function y = x 2 - 2x

Graph of the function y = f(x) + g(x)

Let's look at the task and the graph of the function y = f(x) + g(x). How to set function schedules y = f(x)і y = g(x).

Respectfully, the scope of the function y = |f(x) + g(х)| є impersonal usіh quiet value of x, for any assigned functions y = f(x) і y = g(x), so that the area of ​​assignment is the overlap of the areas of assignment, functions f(x) and g(x).

Come on specks (x 0, y 1) that (x 0, y 2) is likely to lie with the graphs of functions y = f(x)і y = g(x), i.e. y 1 = f(x0), y2=g(x0). Same point (x0;. y1 + y2) lie on the graph of the function y = f(x) + g(x)(more f(x 0) + g(x 0) = y 1+y2),. moreover, be it a point of the graph of a function y = f(x) + g(x) can be taken in this way. Otzhe, graph of the function y = f(x) + g(x) can be removed from the graphs of functions y = f(x). і y = g(x) replacement of the skin point ( x n, y 1) function schedule y = f(x) point (x n, y 1 + y 2), de y 2 = g(x n), then by the sound of the skin point ( x n, y 1) function graphics y = f(x) vzdovzh osi at by the amount y 1 \u003d g (x n). With whom, such points are seen less X n for which offensive functions are assigned y = f(x)і y = g(x).

This method prompts the graph of the function y = f(x) + g(x) is called adding graphs of functions y = f(x)і y = g(x)

butt 4. On the baby by the method of folding the graphs, a graph of the function was induced
y = x + sinx.

When prompted schedule functions y = x + sinx we thought that f(x) = x, A g(x) = sinx. To encourage the graph of the function, select the specks behind the abcises -1.5π, -, -0.5, 0, 0.5,, 1.5, 2. Value f(x) = x, g(x) = sinx, y = x + sinx calculable at selected points and the results are placed at the table.

"The transformation of functions" - Goydalkami. Zsuv along the axis of the angle. Increasing the fullness - increase a (amplitude) kolivan again. Zsuv on the axis x levoruch. The task of the lesson. 3 balls. Music. Look at the graph of the function and assign D(f), E(f) and T: Squeezing along the x axis. Zsuv along the axis uni. Add a red color to the palette - change the k (frequency) of the electromagnetic colors.

"Functions of a few changes" - similar to higher orders. The function of two variables can be represented graphically. Differential and integral calculations. Internal and boundary points. Designation of inter-functions of 2-x change. Course of mathematical analysis. Berman. Between the functions of 2 shifts. Function chart. Theorem. Fenced area.

"Understanding the function" - ways to encourage graphs of quadratic functions. The development of various methods of managing a function is an important methodical technique. Peculiarities of turning a quadratic function. Genetic interpretation of the concept of "function". Functions and graphics in the school course of mathematics. The notice about the linear function is seen when the schedule of the current linear function is prompted.

"Theme Function" - Analysis. It is necessary to tell not those who do not know the scientists, but those who know the wine. Laying the foundations for a successful building EDI and joining the VNZ. Synthesis. If the learners practice in a different way, then the teacher can practice with them in a different way. Analogy. Uzagalnennya. Rozpodіl zavdan ЄDI z main blocks zmіstu school course of mathematics.

"Change of graphs of functions" - Repeat and see the transformation of graphs. Improve skin function. symmetry. The purpose of the lesson: Pobudova graphic folding functions. Applied a changeling, explanatory leather look of a turning. Reorganization of schedules of functions. Stretching. Close the graphs of functions with additional transformation of graphs of elementary functions.

"Function graphs" - Function of the mind. The function value area is all values ​​of the fallow exchange rate. The graph of the function is a parabola. The graph of the function is a cubic parabola. The graph of the function is a hyperbole. The scope of the function is the scope of the value of the function. Skin direct spіvvіdnesіt z її equals: The area of ​​​​appointed function - the value of the independent change.

Lesson on the topic: "The graph of the power of the function $y=x^3$. Apply a graph"

Additive materials
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Electronic manual for the 7th grade "Algebra for 10 credits"
Educational complex 1C "Algebra, 7-9 class"

The power of the function $y=x^3$

Let's describe the characteristics of this function:

1. x - independent change, y - fallow change.

2. Destination area: it is obvious that, given any value of the argument (x), the value of the function (y) can be assigned. Apparently, the scope of the assigned function is the entire numerical straight line.

3. Scope of meaning: you can but be-yakim. Obviously, the area of ​​​​value is also a numerical straight line.

4. If x=0, then y=0.

Graph of the function $y=x^3$

1. Compiling a table of values:

2. For positive x values, the graph of the function $ y = x ^ 3 $ is already similar to a parabola, the pins are more "squeezed" to the OY axis.

3. If the negative values ​​of the x function $y=x^3$ can have an opposite value, then the graph of the function is symmetrical to the cob of coordinates.

Now we can see the points on the coordinate plane and the graph will appear (div. Fig. 1).

This curve is called a cubic parabola.

Apply

I. The small ship ran out of fresh water. It is necessary to bring enough water from the city. Water is brought in late and paid for a new cube, so that it can be poured a little less. How many cubes do you need to close, so as not to overpay for the occupied cube and refill the tank? It seems that the cistern may have the same length, width and height, as if it were 1.5 m.

Solution:

1. Let's call the graph of the function $ y = x ^ 3 $.
2. We know point A, coordinate x, which is 1.5. It is important that the coordinate of the function is between the values ​​3 and 4 (div. small 2). You also need to remember 4 cubes.

Induce function

We respect your service for the construction of graphics functions online, all rights to any belong to the company Desmos. For the introduction of functions, speed up the left column. You can enter manually or for the help of the virtual keyboard at the bottom of the window. To increase the window with the schedule, you can attach it as the left column, and the virtual keyboard.

Advance schedules online

• Visual display of functions to be introduced
• Pobudov more folding graphics
• Pobudova graphs, assignments implicitly (for example, elіps x^2/9+y^2/16=1)
• Possibility to save graphics and apply them to them, as it becomes available to everyone on the Internet
• Scale control, color of lines
• Possibility of encouraging graphs for points, using constants
• Pobudova one hour a few schedules of functions
• Pobudov's graphs in the polar coordinate system (select r and θ(\theta))

With us, it is easy to create graphics of various folds online. Pobudov to get through the mittevo. Request service for defining the break point of functions, displaying graphs for further moving to a Word document as an illustration for the execution of a task, for analyzing the behavioral features of graphs of functions. The optimal browser for working with graphics on this side is Google Chrome. For other browsers, the correctness of the work is not guaranteed.

Pobudov's schedule of functions, how to solve the modules, call out the chimali difficulties for schoolchildren. Prote is not so bad. To finish the memory of some algorithms in the execution of such tasks, and you can easily induce the schedule to create your own seemingly folding functions. Let's take a look at what algorithms are.

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

It is important that the value of the functions y = | f(x) | : y > 0

Pobud's graph of the function y = | f(x) | folded from the next few simple steps.

1) Be careful and respectful of the graph of the function y = f(x).

2) Leave without changing all the points of the graph, if they are more off axis 0x or on it.

3) Part of the graph, which lies below the 0x axis, is shown symmetrically along the 0x axis.

Example 1. Draw the graph of the function y = | x 2 - 4x + 3 |

1) We will be the graph of the function y \u003d x 2 - 4x + 3. Obviously, the graph of the function is a parabola. We know the coordinates of all points of the parabola's crossbar with the coordinate axes and the coordinates of the vertex of the parabola.

x 2 - 4x + 3 = 0.

x1=3, x2=1.

Also, the parabola goes over 0x at the points (3, 0) and (1, 0).

y = 0 2 - 4 0 + 3 = 3.

Also, the parabola changes all 0y at the point (0, 3).

Parabolic vertex coordinates:

x in \u003d - (-4/2) \u003d 2, y in \u003d 2 2 - 4 2 + 3 \u003d -1.

Again, the point (2, -1) is the vertex of the given parabola.

Small parabola, victorious otrimani data (Fig. 1)

2) Part of the graph, which lies below the 0x axis, is supposed to be symmetrical to the 0x axis.

3) We take the schedule of the output function ( Rice. 2, shown as a dotted line).

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

Respectfully, the functions of the form y = f(|x|) are the guys:

y(-x) = f(|-x|) = f(|x|) = y(x). So the graphs of such functions are symmetrical about the axis 0y.

Pobudov's graph of the function y = f(|x|) is composed of an offensive clumsy procession.

1) Induce the graph of the function y = f (x).

2) Omit that part of the graph, for which x ≥ 0, so that part of the graph is ripped off at the right side of the plane.

3) Shown in paragraph (2) part of the graph is symmetrical to the axis 0y.

4) As a residual graph, you can see the aggregation of the curves taken from paragraphs (2) and (3).

Example 2. Draw a graph of the function y = x 2 - 4 · | + 3

Shards x 2 = |x| 2 , then the resulting function can be rewritten to look like this: y = | x | 2 - 4 · | x | + 3. And now we can zastosovuvaty zastosovuvati more algorithm.

1) Be careful and respectfully, the graph of the function y \u003d x 2 - 4 x + 3 (div. also Rice. 1).

2) We leave that part of the graph, for which x ≥ 0, then the part of the graph is ripped at the right side of the plane.

3) View the right part of the graph symmetrically up to the 0y axis.

(Fig. 3).

Example 3. Draw a graph of the function y = log 2 | x |

Zastosovuєmo scheme, given more.

1) We will be the graph of the function y = log 2 x (Fig. 4).

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

It is important that the functions mean y = | f(|x|)| tezh є guys. True, y(-x) = y = |f(|-x|)| = y = | f(|x|)| = y(x), and to that their graphs are symmetrical to the axis 0y. Anonymous value of such functions: y ≥ 0. Also, the graphs of such functions are expanded on the upper surface.

To induce the graph of the function y = |f(|x|)|, it is necessary:

1) Gently induce the graph of the function y = f(|x|).

2) Remove without changing that part of the graph, as it is known more for axis 0x or on it.

3) A part of the graph, expanded below the 0x axis, is displayed symmetrically along the 0x axis.

4) As a residual graph, you can see the aggregation of the curves taken from paragraphs (2) and (3).

Example 4. Draw the graph of the function y = | -x 2 + 2 | x | - 1 |.

1) Respectfully, that x 2 = | 2. Mean the replacement of the output function y = -x 2 + 2|x| - 1

you can twist the function y=-|x| 2+2|x| - 1, because these graphics are avoided.

Future schedule y = - | x | 2+2|x| - 1. For which zastosovuєmo algorithm 2.

a) We will be the graph of the function y \u003d -x 2 + 2x - 1 (Fig. 6).

b) We are leaving that part of the schedule, as it is stashed at the right side of the plane.

c) It is possible to remove a part of the graph symmetrically up to the 0y axis.

d) Removing the image graph for the baby with a dotted line (Mal. 7).

2) There is no more point on the 0x axis, the points on the 0x axis can be left without changing.

3) A part of the graph, expanded below the 0x axis, is supposed to be symmetrically around 0x.

4) Removing the graph is shown on the small dotted line (Fig. 8).

Example 5. Induce the graph of the function y = | (2 | x | - 4) / ( | x | + 3) |

1) It is necessary to induce the heart of the graph of the function y = (2 | x | - 4) / ( | x | + 3). For which we turn to algorithm 2.

a) Carefully plot the function y = (2x - 4) / (x + 3) (Fig. 9).

Respectfully, that the given function is a shot-linear and її graph є hyperbole. To induce a crooked spine, it is necessary to determine the asymptotics of the graph. Horizontal - y \u003d 2/1 (addition of coefficients at x y to the number and banner of the fraction), vertical - x \u003d -3.

2) That part of the graph, which is more common than axis 0x or on it, is left without change.

3) A part of the graph, expanded below the 0x axis, seems to be symmetrically about 0x.

4) The rest of the graph is shown a little (Fig. 11).

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