Concentration of attention:
Definition. Function species called exponential function .
Comment. Exclusion from base values a numbers 0; 1 and negative values a is explained by the following circumstances:
The analytic expression itself a x in these cases it retains its meaning and can be encountered in solving problems. For example, for the expression x y dot x = 1; y = 1 is included in the range of valid values.
Build graphs of functions: and.
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| Exponential function graph | |
| y = a x, a> 1 | y = a x , 0< a < 1 |
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Exponential function properties
| Exponential function properties | y = a x, a> 1 | y = a x , 0< a < 1 |
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| 2. Range of values of the function | ||
| 3.Intervals of comparison with unit | at x> 0, a x > 1 | at x > 0, 0< a x < 1 |
| at x < 0, 0< a x < 1 | at x < 0, a x > 1 | |
| 4. Parity, oddness. | The function is neither even nor odd (general function). | |
| 5. Monotony. | increases monotonically by R | decreases monotonically by R |
| 6. Extremes. | The exponential function has no extrema. | |
| 7 asymptote | O axis x is the horizontal asymptote. | |
| 8. For any valid values x and y; |  |
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When the table is filled in, tasks are solved in parallel with the filling.
Task number 1. (To find the domain of definition of the function).
What argument values are valid for functions:
Task number 2. (To find the range of values of the function).
The figure shows the graph of the function. Specify the scope and range of values of the function:
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Task number 3. (To indicate the intervals of comparison with the unit).
Compare each of the following degrees with a unit:
Task number 4. (To study the function for monotony).
Compare the largest real numbers m and n if:
Task number 5. (To study the function for monotony).
Make a conclusion on the basis a, if:
y (x) = 10 x; f (x) = 6 x; z (x) - 4 x
How are the graphs of exponential functions relative to each other for x> 0, x = 0, x< 0?
The graphs of the functions are plotted in one coordinate plane:
y (x) = (0,1) x; f (x) = (0.5) x; z (x) = (0.8) x.
How are the graphs of exponential functions relative to each other for x> 0, x = 0, x< 0?
| Number
one of the most important constants in mathematics. By definition, it equals the sequence limit
with unlimited
increasing n
... Designation e introduced Leonard Euler
in 1736 He calculated the first 23 digits of this number in decimal notation, and the number itself was named in honor of Napier "neper number".
Number e plays a special role in mathematical analysis. Exponential function with the foundation e, called exponential and denoted y = e x. First signs the numbers e easy to remember: two, comma, seven, Leo Tolstoy's year of birth - two times, forty-five, ninety, forty-five. |
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Homework:
Kolmogorov p. 35; No. 445-447; 451; 453.
Repeat the algorithm for plotting graphs of functions containing a variable under the modulus sign.
To use the preview of presentations, create yourself a Google account (account) and log into it: https://accounts.google.com
Slide captions:
MAOU "Sladkovskaya Secondary School" Exponential function, its properties and graph Grade 10
A function of the form y = ax, where a is a given number, a> 0, and ≠ 1, x-variable, is called exponential.
The exponential function has the following properties: OOF: the set R of all real numbers; Mn.zn .: the set of all positive numbers; The exponential function y = ax is increasing on the set of all real numbers, if a> 1, and decreasing, if 0
Graphs of the function y = 2 x and y = (½) x 1. The graph of the function y = 2 x passes through the point (0; 1) and is located above the Ox axis. a> 1 D (y): x є R E (y): y> 0 Increases over the entire domain of definition. 2. The graph of the function y = also passes through the point (0; 1) and is located above the Ox axis. 0
Using the properties of increasing and decreasing exponential function, you can compare numbers and solve exponential inequalities. Compare: a) 5 3 and 5 5; b) 4 7 and 4 3; c) 0.2 2 and 0.2 6; d) 0.9 2 and 0.9. Solve: a) 2 x> 1; b) 13 x + 1 0.7; d) 0.04 x a b or a x 1, then x> b (x
Solve graphically the equations: 1) 3 x = 4-x, 2) 0.5 x = x + 3.
If you remove a boiling kettle from the fire, then at first it cools down quickly, and then it cools down much more slowly, this phenomenon is described by the formula T = (T 1 - T 0) e - kt + T 1 Application of the exponential function in life, science and technology
The growth of wood occurs according to the law: A - change in the amount of wood over time; A 0 - the initial amount of wood; t -time, k, a- some constants. Air pressure decreases with altitude according to the law: P - pressure at height h, P0 - pressure at sea level, and - some constant.
Population growth The change in the number of people in the country over a short period of time is described by the formula, where N 0 is the number of people at time t = 0, N is the number of people at time t, a is a constant.
The law of organic reproduction: under favorable conditions (absence of enemies, a large amount of food), living organisms would multiply according to the law of exponential function. For example: one housefly can produce 8 x 10 14 offspring over the summer. Their weight would be several million tons (and the weight of the offspring of a pair of flies would exceed the weight of our planet), they would occupy a huge space, and if you line them up in a chain, then its length will be greater than the distance from the Earth to the Sun. But since, besides flies, there are many other animals and plants, many of which are natural enemies of flies, their number does not reach the above values.
When a radioactive substance decays, its amount decreases, after a while half of the original substance remains. This period of time t 0 is called the half-life. The general formula for this process: m = m 0 (1/2) -t / t 0, where m 0 is the initial mass of the substance. The longer the half-life, the slower the substance disintegrates. This phenomenon is used to determine the age of archaeological finds. Radium, for example, decays according to the law: M = M 0 e -kt. Using this formula, scientists calculated the age of the Earth (radium decays in about a time equal to the age of the Earth).
On the subject: methodological developments, presentations and notes
The use of integration in the educational process as a way to develop analytical and creative abilities ...
The presentation "Exponential function, its properties and graph" graphically presents educational material on this topic. During the presentation, the properties of the exponential function, its behavior in the coordinate system are considered in detail, examples of solving problems using the properties of the function, equations and inequalities are considered, important theorems on the topic are studied. With the help of a presentation, the teacher can improve the effectiveness of the math lesson. A vivid presentation of the material helps to keep the attention of students on the study of the topic, animation effects help to more clearly demonstrate solutions to problems. For faster memorization of concepts, properties and features of the solution, color highlighting is used.
The demonstration begins with examples of the exponential function y = 3 x with various exponents - positive and negative integers, fractions and decimal. The function value is calculated for each indicator. Further, a graph is plotted for the same function. On slide 2, a table is built filled with the coordinates of the points belonging to the graph of the function y = 3 x. A corresponding graph is plotted using these points on the coordinate plane. Similar graphs are plotted next to the graph y = 2 x, y = 5 x and y = 7 x. Each function is highlighted in a different color. The graphs of these functions are made in the same colors. Obviously, with an increase in the base of the exponential function, the graph becomes steeper and is more pressed to the ordinate axis. The same slide describes the properties of the exponential function. It is noted that the domain of definition is the number line (-∞; + ∞). The function is not even or odd, the function increases in all domains of definition and does not have the greatest or least value. The exponential function is bounded from below, but not bounded from above, continuous on the domain of definition, and convex downward. The range of values of the function belongs to the interval (0; + ∞).
Slide 4 presents a study of the function y = (1/3) x. The function is plotted. For this, the table is filled with the coordinates of points belonging to the function graph. These points are used to plot a graph on a rectangular coordinate system. The properties of the function are described side by side. It is noted that the entire number axis is the scope. This function is not odd or even, decreasing over the entire domain of definition, does not have the largest, smallest values. The function y = (1/3) x is bounded from below and unbounded from above, is continuous on the domain of definition, and has convexity downward. The range of values is the positive semiaxis (0; + ∞).
Using the given example of the function y = (1/3) x, it is possible to single out the properties of the exponential function with a positive base less than one and to clarify the idea of its graph. Slide 5 shows a general view of such a function y = (1 / a) x, where 0 Slide 6 compares the graphs of the functions y = (1/3) x and y = 3 x. It can be seen that these graphs are symmetrical about the ordinate axis. To make the comparison more clear, the graphs are painted in colors that highlight the function formulas. The following is the definition of the exponential function. On slide 7, a definition is highlighted in the box, which indicates that a function of the form y = a x, where a positive a, not equal to 1, is called exponential. Next, using the table, the exponential function is compared with a base greater than 1 and positive less than 1. Obviously, almost all the properties of the function are similar, only a function with a base greater than a increases, and with a base less than 1, decreases. The following is a solution of examples. In example 1, you need to solve the equation 3 x = 9. The equation is solved graphically - the graph of the function y = 3 x and the graph of the function y = 9 are plotted. The point of intersection of these graphs is M (2; 9). Accordingly, the solution to the equation is x = 2. Slide 10 describes the solution to equation 5 x = 1/25. Similarly to the previous example, the solution to the equation is determined graphically. Demonstrated the construction of graphs of functions y = 5 x and y = 1/25. The point of intersection of these graphs is the point E (-2; 1/25), which means that the solution to the equation is x = -2. Further, it is proposed to consider the solution to the inequality 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;+∞). The next slides present important theorems that reflect the properties of the exponential function. Theorem 1 states that for positive a, the equality a m = a n is true if m = n. In Theorem 2, the statement is presented that for positive a, the value of the function y = ax will be greater than 1 for positive x, and less than 1 for negative x. The statement is confirmed by the image of the exponential function graph, which shows the behavior of the function at different intervals of the domain of definition. It is noted in Theorem 3 that for 0 Further, for the assimilation of the material by the students, examples of solving problems using the studied theoretical material are considered. In example 5, it is necessary to plot the function y = 2 2 x +3. The principle of constructing a graph of a function is demonstrated, first transforming it into the form y = a x + a + b. A parallel transfer of the coordinate system to the point (-1; 3) is performed and the graph of the function y = 2 x is plotted relative to this origin. Slide 18 shows a graphical solution to the equation 7 x = 8-x. The straight line y = 8-x and the graph of the function y = 7 x are constructed. The abscissa of the point of intersection of the graphs x = 1 is the solution to the equation. The last example describes the solution to the inequality (1/4) x = x + 5. Graphs of both sides of the inequality are plotted and it is noted that its solution is the values (-1; + ∞), at which the values of the function y = (1/4) x are always less than the values y = x + 5. The presentation "Exponential function, its properties and graph" is recommended to improve the effectiveness of a school mathematics lesson. The clarity of the material in the presentation will help to achieve the learning objectives during the distance lesson. The presentation can be offered for independent work to students who have not mastered the topic well enough in the lesson.
Properties of the function Let's analyze it according to the scheme: Analyze it according to the scheme: 1. domain of definition of a function 1. domain of definition of a function 2. set of values of a function 2. set of values of a function 3. zeros of a function 3. zeros of a function 4. intervals of constant sign of a function 4. intervals of constant sign of a function 5. even or odd function 5. even or odd function 6. monotonicity of a function 6. monotonicity of a function 7. largest and smallest values 7. largest and smallest values 8. periodicity of a function 8. periodicity of a function 9. boundedness of a function 9. boundedness of a function
0 for x R. 5) The function is neither even, nor "title =" (! LANG: Exponential function, its graph and properties y x 1 о 1) Domain of definition - the set of all real numbers (D (y) = R). 2) The set of values is the set of all positive numbers (E (y) = R +). 3) There are no zeros. 4) y> 0 for x R. 5) The function is neither even nor" class="link_thumb">
10
!} Exponential function, its graph and properties y x 1 о 1) Domain of definition - the set of all real numbers (D (y) = R). 2) The set of values is the set of all positive numbers (E (y) = R +). 3) There are no zeros. 4) y> 0 for x R. 5) The function is neither even nor odd. 6) The function is monotonic: it increases by R for a> 1 and decreases by R for 0 0 for x R. 5) The function is neither even, nor "> 0 for x R. 5) The function is neither even, nor odd. 6) The function is monotonic: it increases by R for a> 1 and decreases by R for 0"> 0 for x R. 5) The function is neither even, nor "title =" (! LANG: Exponential function, its graph and properties yx 1 о 1) The domain of definition is the set of all real numbers (D (y) = R). 2) The set of values is the set of all positive numbers (E (y) = R +). 3) There are no zeros. 4) y> 0 for x R. 5) The function is neither even nor">
title="Exponential function, its graph and properties y x 1 о 1) Domain of definition - the set of all real numbers (D (y) = R). 2) The set of values is the set of all positive numbers (E (y) = R +). 3) There are no zeros. 4) y> 0 for x R. 5) The function is neither even nor">
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The growth of wood occurs according to the law, where: A - the change in the amount of wood over time; A 0 - the initial amount of wood; t-time, k, a- some constants. The growth of wood occurs according to the law, where: A - the change in the amount of wood over time; A 0 - the initial amount of wood; t-time, k, a- some constants. t 0 t0t0 t1t1 t2t2 t3t3 tntn А A0A0 A1A1 A2A2 A3A3 AnAn
The temperature of the kettle changes according to the law, where: Т - the change in the temperature of the kettle with time; T 0 is the boiling point of water; t-time, k, a- some constants. The temperature of the kettle changes according to the law, where: Т - the change in the temperature of the kettle with time; T 0 is the boiling point of water; t-time, k, a- some constants. t 0 t0t0 t1t1 t2t2 t3t3 tntn T T0T0 T1T1 T2T2 T3T3
Radioactive decay occurs according to the law, where: Radioactive decay occurs according to the law, where: N is the number of non-decayed atoms at any time t; N 0 - the initial number of atoms (at time t = 0); t-time; N is the number of non-decayed atoms at any moment of time t; N 0 - the initial number of atoms (at time t = 0); t-time; T is the half-life. T is the half-life. t 0 t 1 t 2 N N3N3 N4N4 t4t4 N0N0 t3t3 N2N2 N1N1
An essential property of organic processes and changes in quantities is that over equal time intervals the value of a quantity changes in the same ratio Wood growth Change in teapot temperature Change in air pressure The processes of organic change in quantities include: Radioactive decay
Compare the numbers 1.3 34 and 1.3 40. Example 1. Compare the numbers 1.3 34 and 1.3 40. General solution method. 1. Present the numbers as a power with the same base (if necessary) 1,3 34 and 1, Find out whether the exponential function is increasing or decreasing a = 1.3; a> 1, consequently the exponential function increases. a = 1.3; a> 1, consequently the exponential function increases. 3. Compare exponents (or function arguments) 34 1, next the exponential function increases. a = 1.3; a> 1, consequently the exponential function increases. 3. Compare exponents (or function arguments) 34 "> Solve the equation 3x = 4x graphically. Example 2. Solve graphically the equation 3 x = 4-x. Solution. We use a functional-graphical method for solving equations: we will construct graphs of the functions y = 3 x and y = 4-x in one coordinate system. graphs of functions y = 3 x and y = 4-x. Note that they have one common point (1; 3). Hence, the equation has a single root x = 1. Answer: 1 Answer: 1 y = 4-x
4-x. Example 3. Solve graphically the inequality 3 x> 4 x. Solution. y = 4-x We use the functional-graphical method of solving inequalities: 1. Let's build in one system 1. Let us construct in one coordinate system the graphics of the functions "title =" (! LANG: Solve graphically the inequality 3 x> 4. Example 3. Solve graphically inequality 3 x> 4. Solution.y = 4 We use a functional-graphic method for solving inequalities: 1. Let's construct in one system 1. Let's construct graphs of functions in one coordinate system" class="link_thumb">
24
!} Solve the inequality 3x> 4x graphically. Example 3. Solve graphically the inequality 3 x> 4 x. Solution. y = 4-x We use the functional-graphical method for solving inequalities: 1. Let's build in one system 1. Let us construct in one coordinate system the graphs of the functions of the coordinates of the graphs of the functions y = 3 x and y = 4-x. 2. Select the part of the graph of the function y = 3 x located above (since the> sign) the graph of the function y = 4-x. 3. Mark on the x-axis the part that corresponds to the selected part of the graph (otherwise: project the selected part of the graph onto the x-axis). 4. Let's write the answer in the form of an interval: Answer: (1;). Answer: (1;). 4-x. Example 3. Solve graphically the inequality 3 x> 4 x. Solution. y = 4-x We use the functional-graphical method for solving inequalities: 1. Let's build in one system 1. Let's build in one coordinate system the graphs of the functions "> 4. Example 3. Solve graphically the inequality 3 x> 4. Solution. y = 4-x We use the functional-graphical method for solving the inequalities: 1. Let's construct in one system 1. Let's construct in one coordinate system the graphs of the coordinate functions of the graphs of the functions y = 3 x and y = 4. 2. Let's select a part of the graph of the function y = 3 x, located above (because the> sign) of the graph of the function y = 4. 3. Mark on the x-axis that part that corresponds to the selected part of the graph (otherwise: project the selected part of the graph on the x-axis) 4. Write down the answer as an interval: Answer: (1;). Answer: (1;). "> 4-x. Example 3. Solve graphically the inequality 3 x> 4 x. Solution. y = 4-x We use the functional-graphical method of solving inequalities: 1. Let's build in one system 1. Let us construct in one coordinate system the graphics of the functions "title =" (! LANG: Solve graphically the inequality 3 x> 4. Example 3. Solve graphically inequality 3 x> 4. Solution.y = 4 We use a functional-graphic method for solving inequalities: 1. Let's construct in one system 1. Let's construct graphs of functions in one coordinate system">
title="Solve the inequality 3x> 4x graphically. Example 3. Solve graphically the inequality 3 x> 4 x. Solution. y = 4-x We use the functional-graphical method for solving inequalities: 1. Let's build in one system 1. We'll construct graphs of functions in one coordinate system">
!} Solve graphically the inequalities: 1) 2 x> 1; 2) 2 x one; 2) 2 x "> 1; 2) 2 x"> 1; 2) 2 x "title =" (! LANG: Solve graphically the inequality: 1) 2 x> 1; 2) 2 x">
title="Solve graphically the inequalities: 1) 2 x> 1; 2) 2 x">
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Independent work (test) 1. Indicate the exponential function: 1. Indicate the exponential function: 1) y = x 3; 2) y = x 5/3; 3) y = 3 x + 1; 4) y = 3 x + 1. 1) y = x 3; 2) y = x 5/3; 3) y = 3 x + 1; 4) y = 3 x + 1. 1) y = x 2; 2) y = x -1; 3) y = -4 + 2 x; 4) y = 0.32 x. 1) y = x 2; 2) y = x -1; 3) y = -4 + 2 x; 4) y = 0.32 x. 2. Indicate a function that increases throughout the entire domain of definition: 2. Indicate a function that increases in the entire domain of definition: 1) y = (2/3) -x; 2) y = 2 -x; 3) y = (4/5) x; 4) y = 0.9 x. 1) y = (2/3) -x; 2) y = 2 -x; 3) y = (4/5) x; 4) y = 0.9 x. 1) y = (2/3) x; 2) y = 7.5 x; 3) y = (3/5) x; 4) y = 0.1 x. 1) y = (2/3) x; 2) y = 7.5 x; 3) y = (3/5) x; 4) y = 0.1 x. 3. Indicate a function that decreases over the entire domain of definition: 3. Indicate a function that decreases over the entire domain of definition: 1) y = (3/11) -x; 2) y = 0.4 x; 3) y = (10/7) x; 4) y = 1.5 x. 1) y = (2/17) -x; 2) y = 5.4 x; 3) y = 0.7 x; 4) y = 3 x. 4. Indicate the set of values of the function y = 3 -2 x -8: 4. Indicate the set of values of the function y = 2 x + 1 + 16: 5. Indicate the smallest of these numbers: 5. Indicate the smallest 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. Indicate the largest of the given 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. Find out graphically how many roots the equation has 2 x = x -1/3 (1/3) x = x 1/2 6. Find out graphically how many roots the equation has 2 x = x -1/3 (1/3) x = x 1/2 1) 1 root; 2) 2 roots; 3) 3 roots; 4) 4 roots. 1. Indicate the exponential function: 1) y = x 3; 2) y = x 5/3; 3) y = 3 x + 1; 4) y = 3 x + 1. 1) y = x 3; 2) y = x 5/3; 3) y = 3 x + 1; 4) y = 3 x Indicate a function that increases over the entire domain of definition: 2. Indicate a function that increases over the entire domain of definition: 1) y = (2/3) -x; 2) y = 2; 3) y = (4/5) x; 4) y = 0.9 x. 1) y = (2/3) -x; 2) y = 2; 3) y = (4/5) x; 4) y = 0.9 x. 3. Indicate a function that decreases over the entire domain of definition: 3. Indicate a function that decreases over the entire domain of definition: 1) y = (3/11) -x; 2) y = 0.4 x; 3) y = (10/7) x; 4) y = 1.5 x. 1) y = (3/11) -x; 2) y = 0.4 x; 3) y = (10/7) x; 4) y = 1.5 x. 4. Indicate the set of values of the function y = 3-2 x-8: 4. Indicate the set of values of the function y = 3-2 x-8: 5. Indicate the smallest of these numbers: 5. Indicate the smallest 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. Find out graphically how many roots the equation has 2 x = x- 1/3 6. Find out graphically how many roots the equation has 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. Verification work Select the exponential functions that: Select the exponential functions that: I option - decrease on the domain of definition; Option I - decrease on the domain of definition; Option II - increase by the area of definition. Option II - increase by the area of definition.
