Display function - presentation before the lesson from algebra (grade 10) on the topic. Mathematics presentation on the topic "Show function, її power and graph" Gra "Smart in class"
The presentation “Show Function, Power and Graph” presents the initial material on these topics. In the course of the presentation, the authoritativeness of the display function, the behavior of the coordinate system are examined, the applications of the distribution of tasks from the various authorities of the function, the leveling of those irregularities, and important theorems on the topic are discussed. For an additional presentation, the teacher can improve the effectiveness of the mathematics lesson. Yaskrave the appearance of the material helps to increase the respect of the scientists to the education of those, the animation effects help to demonstrate the understanding of the task. For a quick memory to understand, the power and peculiarities of the decision are victorious when seen in color.
The demonstration is based on the application of the display function y = 3 x with different indicators - positive and negative integers, and decimal fractions. Before the skin indicator, the value of the function is calculated. There will be a schedule for this function. On slide 2, a table was created, filled with coordinates of a point that should lie on the graph of the function y \u003d 3 x. Behind these points on the coordinate plane there will be a second line graph. In order of the graph, there will be similar graphs y \u003d 2x, y \u003d 5x and y \u003d 7x. Skin function is seen in different colors. Such colors have vikonan graphics and functions. It is obvious that the step of the display function of the graph becomes steeper and more close to the y-axis. Which slide describes the power of the show function. It is assigned that the assigned area is a numerical line (-∞; +∞), the Function is not paired or unpaired, in all areas the assigned function grows and does not have the largest or smallest value. The display function is bordered from below, but not bordered by the beast, without interruption of the designated area and bulged down. The range of the function value lies between (0;+∞).
Slide 4 shows the following function y = (1/3) x. There will be a function schedule. This is why the coordinates of the point, which lie on the graph of the function, the table, are filled in. Behind these points there will be a graph on a rectangular coordinate system. The instructions describe the power of the function. It is assigned that the entire numerical value is assigned to the area. This function is not unpaired, but paired, which changes in the entire area of \u200b\u200bapplication, does not have the highest, least value. The function y \u003d (1/3) x is fringed from the bottom and unfenced to the beast, on the distance it is uninterrupted, it can bulge down. The area of value is positive pіvvіs (0;+∞).
On the suggested application of the function y \u003d (1/3) x, one can see the power of the display function with a positive basis, less than one can clarify the statement about the її graphics. On the slide there are 5 views of such a function y = (1/a) x de 0 On slide 6, the graphs of functions y \u003d (1/3) x i y \u003d 3 x are arranged. It can be seen that the graphs are symmetrical along the ordinate axis. In order to improve the accuracy, the graphs were shaped in colors, with which the formulas of functions were seen. Next, a designated display function is given. On slide 7, the frame shows a designation, in which it is designated that the function of the form y \u003d a x, which is more positive than a, not equal to 1, is called display. Further, for the help of the table, the display function with a basis greater than 1, and a positive lesser 1 is given. Obviously, in practice, all power functions are similar, only a function with a basis, greater a, growing, and with a basis, less 1, less. In the distance, we look at the rozv'yazannya of butts. For butt 1, it is necessary to tie 3 x \u003d 9. The alignment is changed in a graphical way - the graph of the function y \u003d 3 x the graph of the function y \u003d 9 will be. The break point of these graphs is M (2; 9). Vidpovidno, rozv'azkom equal є value x=2. Slide 10 describes the solution of 5 x = 1/25. Similarly to the front butt, the solution is shown graphically. Demonstrated prompt graphs of functions y=5 x i y=1/25. The line point of these graphs is the point E (-2; 1/25), later, the alignment of x \u003d -2. Let's take a look at the solutions to the nervousness 3 x<27. Решение выполняется графически - определяется точка пересечения графиков у=3 х и у=27. Затем на плоскости координат хорошо видно, при каких значениях аргумента значения функции у=3 х будут меньшими 27 - это промежуток (-∞;3). Аналогично выполняется решение задания, в котором нужно найти множество решений неравенства (1/4) х <16. На координатной плоскости строятся графики функций, соответствующих правой и левой части неравенства и сравниваются значения. Очевидно, что решением неравенства является промежуток (-2;+∞). On the next slides, important theorems are presented that increase the power of the show function. Theorem 1 asserts that for positive equality a m = a n is true only if m = n. Theorem 2 presents the assertion that, with a positive value of the function y=a x, it will be greater than 1 for positive x, and less than 1 for negative x. The confirmation is confirmed by the image of the display function graph, which shows the behavior of the function in different intervals of the designated area. Theorem 3 says that for 0 p align="justify"> Further, for mastering the material, the scientists look at the applications of perfection of the twisted theoretical material. For example 5, it is necessary to induce a graph of the function y \u003d 2 2 x +3. The principle of inducing a graph of a function is demonstrated by transforming the back of the її y into the form y \u003d a x + a + b. Carried out in parallel with the transfer of the coordinate system y to the point (-1; 3) and the next cob of coordinates will be the graph of the function y \u003d 2 x. On slide 18, a graphic solution of 7 x \u003d 8 x is seen. It will be straight y \u003d 8 x and graph of the function y \u003d 7 x. The abscissa of the point of the line of the graph x=1 is equal to the solutions. The rest of the butt describes the breakdown of unevenness (1/4) x \u003d x + 5. Budyuyuyutsya graphs of both parts of the nerіvnostі and vіdnaєєєєєєєєєєєєєєє, yоogo solutions є value (-1; + ∞), for any value of the function y = (1/4) x zavzhda less value y = x +5. The presentation “Display function, power and schedule” is recommended to improve the effectiveness of the school mathematics lesson. The accuracy of the material in the presentation will help to reach the goals of learning for an hour of a distance lesson. The presentation can be proponated for independent work by students, as they did not master the topic well enough at the lesson.
Power of the function is analyzed for the schematic: it is anal for the schematic: 1. The area of Voznoi functions 1. The area of Voznoi function 2. Multiple knowledge of function 2. Bezlіch. 6. monotonicity of a function 6. monotonicity of a function 7. largest and smallest value 7. largest and smallest value 8. periodicity of a function 8. periodicity of a function 9. exchange function.
0 at x R. 5) Function n_ pair, n_ "title=" Display function, її graph and power y x 1 o 1) Designation area - the absence of all actual numbers (D(y)=R). 2) Anonymous value - the absence of all positive numbers (E(y) = R +). 3) There are no zeros. 4) y>0 at x R. 5) Function ni pair, ni" class="link_thumb">
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!}!} Display function, її graph and density y x 1 o 1) Designation area - the absence of all real numbers (D (y) \u003d R). 2) Anonymous value - the absence of all positive numbers (E(y) = R +). 3) There are no zeros. 4) y>0 for x R. 5) The function is neither paired nor unpaired. 6) The function is monotonic: it grows by R at a>1 and changes by R at 0 0 at x R. 5) Function ni pair, ni "> 0 at x R. 5) Function ni pair, ni unpair. 6) The function is monotonic: increases by R at a> 1 and changes to R at 0" x R. 5) Function of no pair, no "title="Display function, її graph and authority y x 1 o 1) Designation area - impersonal of all real numbers (D(y)=R). 2) Anonymous value - the absence of all positive numbers (E(y) = R +). 3) There are no zeros. 4) y>0 at x R. 5) Function ni pair, ni">
title="Display function, її graph and density y x 1 o 1) Designation area - the absence of all real numbers (D (y) \u003d R). 2) Anonymous value - the absence of all positive numbers (E(y) = R +). 3) There are no zeros. 4) y>0 at x R. 5) Function ni pair, ni">
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The growth of the village is subject to the law, de: A-change in the number of villages per hour; A 0 - Pochatkova village; t-hour, before, a- day fast. The growth of the village is subject to the law, de: A-change in the number of villages per hour; A 0 - Pochatkova village; t-hour, before, a- day fast. t 0 t0t0 t1t1 t2t2 t3t3 tntn A A0A0 A1A1 A2A2 A3A3 AnAn
The temperature of the kettle is changed according to the law, de: T-change of the temperature of the kettle by the hour; T 0 - water boiling point; t-hour, before, a- day fast. The temperature of the kettle is changed according to the law, de: T-change of the temperature of the kettle by the hour; T 0 - water boiling point; t-hour, before, a- day fast. t 0 t0t0 t1t1 t2t2 t3t3 tntn T T0T0 T1T1 T2T2 T3T3
Radioactive decay is subject to the law, de: Radioactive decay is subject to the law, de: N is the number of atoms that did not decay at some point in the hour t; N 0 - Pochatkov number of atoms (at the moment t = 0); t-hour; N is the number of atoms that did not fall apart, at some point in the hour t; N 0 - Pochatkov number of atoms (at the moment t = 0); t-hour; T-period is reversed. T-period is reversed. t 0 t 1 t 2 N N3N3 N4N4 t4t4 N0N0 t3t3 N2N2 N1N1
The essence of the power of the processes of organic change of values is due to the fact that for equal intervals of time the value of the value changes in the very same growth of the village Change of the temperature of the kettle Change of the vise of repetition Before the processes of organic change of the values are seen:
Match the numbers 1.3 34 and 1.3 40. Example 1. Match the numbers 1.3 34 and 1.3 40. 1. Reveal the numbers at the same level with the same basis (as it is necessary) 1.3 34 and 1, Z'yasuvati, increasing or decreasing - showing function a = 1.3; a>1, the display function is also growing. a=1.3; a>1, the display function is also growing. 3. Align step indicators (or function arguments) 34 1, the growth function is also shown. a=1.3; a>1, the display function is also growing. 3. Align step indicators (or function arguments) 34"> Untie graphically equalize 3 x = 4 x. Butt 2. Draw graphically equal 3 x = 4 x. Solution. Vikoristovuєmo functional-graphical method of rozv'yazannya rіvnyan: let's use one coordinate system of graphics functions y=3x and y=4-x. graphs of functions y = 3x and y = 4x. Respectfully, they stink one big point (1; 3). Otzhe, equal may be the same root x = 1. Match: 1 Match: 1 y=4-x
4th. Example 3. Expand graphically unevenness 3 х > 4 х. Solution. y=4 Vykoristovuy functional-graphical method of decoupling of irregularities:'яжіть графічно нерівність 3 х >4-х. Приклад 3. Розв'яжіть графічно нерівність 3 х >4-х Вирішення у=4-х Використовуємо функціонально-графічний метод розв'язання нерівностей: 1. Побудуємо в одній системі 1. Побудуємо в одній системі координат графіки функцій" class="link_thumb">
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!}!} Decompose graphically unevenness 3 х > 4 х. Example 3. Expand graphically unevenness 3 х > 4 х. Solution. y \u003d 4-x Vykoristovuєmo functional-graphical method of decoupling irregularities: 1. Let's stay in one system 1. Let's stay in one coordinate system graphics function coordinates graphics functions y = 3x and y = 4x. 2. We can see a part of the graph of the function y = 3x, but it is more detailed (because the sign >) the graph of the function y = 4x. 3. Significantly on the x-axis that part, yak confirms the sighting of a part of the graph (also: it is projected to see a part of the graph on the whole x). 4. Let's write the interval for the interval: The interval: (1;). Suggestion: (1;). 4th. Example 3. Expand graphically unevenness 3 х > 4 х. Solution. y \u003d 4-x Vicorist functional-graphical method of decomposing irregularities: 1. We will be in one system 1. We will be in one system of coordinates graphics of functions "\u003e 4-x. Example 3. Graphically decompose irregularities 3 x\u003e 4-x .=4 Vykoristovuy functional-graphical method of derivation of irregularities: 1. Let's stay in one system 1. Let's stay in one system of coordinates graphs of functions of coordinates graphs of functions y=3 x and y=4-x 2. We can see part of the graph of function y \u003d 3 x, expanded more (because the > sign) graph of the function y \u003d 4. 3. Significantly on the x axis that part, as you see the part of the graph on the whole x) 4. Write down the part of the graph look at the interval: Width: (1;). Width: (1;)."\u003e 4-x. Example 3. Expand graphically unevenness 3 х > 4 х. Solution. y=4 Vykoristovuy functional-graphical method of decoupling of irregularities:'яжіть графічно нерівність 3 х >4-х. Приклад 3. Розв'яжіть графічно нерівність 3 х >4-х Вирішення у=4-х Використовуємо функціонально-графічний метод розв'язання нерівностей: 1. Побудуємо в одній системі 1. Побудуємо в одній системі координат графіки функцій">
title="Rozv'яжіть графічно нерівність 3 х >4-х. Приклад 3. Розв'яжіть графічно нерівність 3 х >4-х. Рішення. у = 4-х Використовуємо функціонально-графічний метод розв'язання нерівностей: 1. Побудуємо в одній системі 1. Побудуємо в одній системі координат графіки функцій">
!}!} Decompose graphically irregularities: 1) 2 х >1; 2) 2 x 1; 2) 2 x "> 1; 2) 2 x "> 1; 2) 2 x "title="Design'яжіть графічно нерівності: 1) 2 х >1; 2) 2 х">
title="Rozv'яжіть графічно нерівності: 1) 2 х >1; 2) 2 х">
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Independent robot (test) 1. Enter the display function: 1. Enter the display function: 1) y = x 3; 2) y \u003d x 5/3; 3) y \u003d 3 x + 1; 4) y = 3 x +1. 1) y \u003d x 3; 2) y \u003d x 5/3; 3) y \u003d 3 x + 1; 4) y = 3 x +1. 1) y \u003d x 2; 2) y \u003d x -1; 3) y \u003d -4 + 2 x; 4) y \u003d 0.32 x. 1) y \u003d x 2; 2) y \u003d x -1; 3) y \u003d -4 + 2 x; 4) y \u003d 0.32 x. 2. Specify a function that grows on the entire target area: 2. Specify a function that grows on the entire target area: 1) y = (2/3) -x; 2) y \u003d 2; 3) y = (4/5) x; 4) y \u003d 0.9 x. 1) y \u003d (2/3) -x; 2) y \u003d 2; 3) y = (4/5) x; 4) y \u003d 0.9 x. 1) y \u003d (2/3) x; 2) y = 7.5 x; 3) y = (3/5) x; 4) y \u003d 0.1 x. 1) y \u003d (2/3) x; 2) y = 7.5 x; 3) y = (3/5) x; 4) y \u003d 0.1 x. 3. Specify a function that changes in the entire scope: 3. Specify a function that changes in the entire scope: 1) y = (3/11) -x; 2) y = 0.4 x; 3) y \u003d (10/7) x; 4) y \u003d 1.5 x. 1) y \u003d (2/17) -x; 2) y = 5.4 x; 3) y = 0.7 x; 4) y \u003d 3 x. 4. Enter the multiplier value of the function y=3 -2 x -8: 4. Enter the multiplier value of the function y=2 x+1 +16: 5. Enter the least of these numbers: 5. Enter the least of these numbers: 1) 3 - 1/3; 2) 27 -1/3; 3) (1/3) -1/3; 4) 1-1/3. 1) 3 -1/3; 2) 27 -1/3; 3) (1/3) -1/3; 4) 1-1/3. 5. Enter the largest of these numbers: 1) 5 -1/2; 2) 25 -1/2; 3) (1/5) -1/2; 4) 1-1/2. 1) 5 -1/2; 2) 25 -1/2; 3) (1/5) -1/2; 4) 1-1/2. 6. Explain graphically, how many roots may equal 2 x = x -1/3 (1/3) x = x 1/2 6. Explain graphically, how many roots may equal 2 x = x -1/3 ( 1/3) x \u003d x 1/2 1) 1 root; 2) 2 roots; 3) 3 roots; 4) 4 roots. 1. Specify the display function: 1) y = x 3; 2) y \u003d x 5/3; 3) y=3 x+1; 4) y = 3 x +1. 1) y \u003d x 3; 2) y \u003d x 5/3; 3) y=3 x+1; 4) y=3 x Indicate the function that grows in the entire target area: 2. Indicate the function that grows in the entire target area: 1) y = (2/3)-x; 2) y \u003d 2; 3) y = (4/5) x; 4) y \u003d 0.9 x. 1) y \u003d (2/3) -x; 2) y \u003d 2; 3) y = (4/5) x; 4) y \u003d 0.9 x. 3. Specify a function that changes in the entire scope: 3. Specify a function that changes in the entire scope: 1) y = (3/11)-x; 2) y = 0.4 x; 3) y \u003d (10/7) x; 4) y \u003d 1.5 x. 1) y \u003d (3/11) -x; 2) y = 0.4 x; 3) y \u003d (10/7) x; 4) y \u003d 1.5 x. 4. Enter the multiplier of the function value y=3-2 x-8: 4. Enter the multiplier of the function value y=3-2 x-8: 5. Enter the least of these numbers: 5. Enter the least of these numbers: 1) 3- 1/3; 2) 27-1/3; 3) (1/3)-1/3; 4) 1-1/3. 1) 3-1/3; 2) 27-1/3; 3) (1/3)-1/3; 4) 1-1/3. 6. Write graphically, how many roots may equal 2 x=x- 1/3 6. Write graphically, how many roots may equal 2 x=x- 1/3 1) 1 root; 2) 2 roots; 3) 3 roots; 4) 4 roots. 1) 1 root; 2) 2 roots; 3) 3 roots; 4) 4 roots. Reversal of the robot Select display functions, such as: Select display functions, such as: Option I - change in the area of appointment; Option I - changing the area of appointment; II option - increase the areas of appointment. II option - increase the areas of appointment.
Mathematics lesson on the topic “Display function” grade 10 (assistant “Algebra and the beginning of mathematical analysis grade 10” S.M. Nikolsky, M.K. Potapov and others.) is divided with additional computer technologies. At the lesson, the function is looked at, the authority of the function and the schedule are looked at. The values of power will be victorious at a distance, when the powers of the logarithmic function are brought, with the difference of showy equalities and irregularities. Lesson type: combinations of computer and interactive whiteboard. Computer technologies create great opportunities for the activation of primary activity. Widespread use of ICT for more subjects gives the opportunity to implement the principle of “recovery from hoarding”, and even if any subject has a greater chance of becoming loved by children. The first lesson for the topic: the first lesson for the topic. Method: combinations (verbal-study-practical). Meta lesson: formulate a statement about the display function, power and graphics. Lesson task: The material for the lesson is chosen in such a rank that it transfers to the work from students of various categories - from weak to strong students. Hid lesson I. Organizational moment (Slide 1-4). Presentation
II. Introduction of new material (Slide 5-6) Designated display function; The power of the display function; Show function graph. III. Usno -
consolidation of new knowledge (slide 7-16) 1) Z'yasuvati, chi є growing function (changing) 2) Repair: . 3) Pair with one: 4) The little one shows the graphics of the display functions. Spivvіdnesіt graph of the function from the formula. IV. Dynamic pause V. Consolidation and systematization of new knowledge (Slide 16-20) 1) Induce the graph of the function: y=(1/3) x; 2) Razvyazati graphic equalization: 3) Stopping the display function until the completion of application tasks: “The period for the decay of plutonium is about 140 dB. How much plutonium will be lost in 10 years, how much is 8 g of cob mass? VI. Test robot (slide 21) The skin learns the card from the tasks - test (Addendum 1) and the table for entering the recommendations (Addendum 2). Verify and evaluate (slide 22) VII. Homework (Slide 23-24) No. 4.55 (a, c, c) No. 4.59, No. 4.60 (a, g); No. 4.61 (d, h) Zavdannya (for the quiet, who squawk with mathematics): Atmospheric pressure deposits (in centimeters of mercury column) in altitude, which is expressed in kilometers. h above the level of the sea are expressed by the formula Calculate what will be the atmospheric pressure on the top of Elbrus, the height is 5.6 km? VIII. Pіdbitya pіdbagіv Literature
This presentation was recognized for repetition by the “Show Function” topic in the 10th grade. Won to avenge as theoretical vіdomosti z tsієї those, and rіznоіvnеі practical tasks. The distribution is made up of three blocks: The presentation shows different ways of untying the showy equalities and irregularities. Tsyu rozrobku can vykoristovuvat not only with the explanation of okremikh topics, but the first hour of preparation before sleep. To speed up the presentation ahead of time, create your own Google Post and see before: https://accounts.google.com “Show function” Mathematics teacher of the Moscow Autonomous Educational Institution Lyceum No. 3 of the Kropotkin district of the Krasnodar Territory Zozulya Olena Oleksiivna The display function is the function of the mind, where x is changed, - the given number, >0, 1. Apply: The power of the display function Designation area: current numbers Indefinite value: positive numbers When > 1, the function is growing; at 0 Display function graph , then the graph of any show function will pass through the point (0; 1) 1 1 x x y 0 0 Show rivnyannia Appointment Simplest rivnyannia Appointed Rivnyannya, who has a change of place at the stage show, is called showy. Apply: The simplest show is equal - the goal is equal to the mind. Methods for rozvyazannya foldable showy rіvnyan. Blame for the temples of the step with a smaller oscillator Blame for the temples of a step with a smaller showman 2) coefficients before changing however For example: Replacing the Change With which method of displaying, the alignment will be reduced to a square one. The way to replace the change of vikoristovuyut, as an indication of one of the steps in 2 times more, lower in the other. For example: 3 2 x - 4 3 x - 45 \u003d 0 coefficient in front of the replacement bed. For example: 2 2 - x - 2 x - 1 \u003d 1 b) a) the bases of the steps are the same; Submitted to the show function a) in equal form a x \u003d b x is divisible by b x For example: 2 x \u003d 5 x | : 5 x b) y equal A a 2 x + B (ab) x + C b 2 x = 0 divisible by b 2x. For example: 3 25 x - 8 15 x + 5 9 x = 0 | : 9 x Showing unevenness Pokazovі nerіvnostі - tse nerіvnostі, for some it is impossible to avenge at the showman's step. Apply: The simplest display of unevenness is the value of the unevenness of the mind: de a > 0, a 1, b – be a number. With the exception of the simplest inequalities, the victorious power grows and the ostentatious function changes. For razv'yazanny folded ostentatious inconsistencies vikoristovuyutsya themselves ways, like and pіd hour vyrіshennya ostentatious rivnyan. Display function Pobudova graph Pairing of numbers with different power levels of the display function Pairing of numbers 1 a) analytical method; b) graphic method. Task 1 Schedule the function y = 2 x x y -1 8 7 6 5 4 3 2 1 - 3 - 2 -1 0 1 2 3 x y 3 8 2 4 1 2 0 1 Task 2 Task 3 Match the number of 1. Solution -5 Task 4 C to increase the number p z 1 p = 2 > 1, then the function y = 2 t is growing. 0 1. Indication: > 1 p = Rezvyazannya pozovyh rivnya The simplest pozovy ryvnyannya Decision that hangs over the arches of the steps with a smaller oscillator Decision that breaks the replacement of the zminnoy vpadok 1; vypadok 2. Rivnyannia, yakі vyrishyuyutsya rozpodilom on the show function vypadok 1; Vipadok 2. The simplest impressions are equal Vidpovid: - 5.5. Response: 0; 3. Blame for the temples of a step with a smaller indicator Vidpovid: 5 x + 1 - (x - 2) = = x + 1 - x + 2 = 3 Replacement of the change (1) of the base of the steps is the same, the indicator of one of the steps is 2 times greater, lower in the other. 3 2 x - 4 3 x - 45 \u003d 0 t \u003d 3 x (t\u003e 0) t 2 - 4 t - 45 \u003d 0 t 1 + t 2 \u003d 4 t 1 \u003d 9; t 2 \u003d - 5 - not satisfied with the mind 3 x \u003d 9; 3 x = 3 2; x = 2. Response: 2 Replacement of change (2) The bases of the steps are the same, the coefficients before the change of the protege. According to vієta: - Not satisfied with the mind Vidpovid: 1 Approved for showing function Response: 0 Approved for the display function Validation: 0; 1. The simplest display of unevenness Under the folds of the unevenness The simplest display of nervousness Underlying irregularities Vidpovid: (-4; -1). 3 > 1 , then Elimination of ostentatious irregularities 3 > 1, then the sign of unevenness is overwritten by itself: 10 Elimination of ostentatious irregularities Method: Replacement of change Response: x 1, then Vikoristovuvana literature. A.G. Mordkovich: Algebra and the cob of mathematical analysis (professional study), 10th grade, 2011. O.M. Kolmogorov: Algebra and the beginning of mathematical analysis, 2008. Internet
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