Let's analyze the properties of the function according to the scheme: Let's analyze according to the scheme: 1. domain of definition of the function 1. domain of definition of the function 2. set of values of the function 2. set of values of the function 3. zeros of the function 3. zeros of the function 4. intervals of constant sign of the function 4. intervals of constant sign of the function 5. even or odd of a function 5. even or odd of a function 6. monotonicity of a function 6. monotonicity of a function 7. greatest and least values 7. greatest and least 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=" Exponential function, its graph and properties y x 1 o 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" class="link_thumb"> 10 !} Exponential function, its graph and properties y x 1 o 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 odd. 6) The function is monotonic: it increases by R when a>1 and decreases by R when 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 on R for a>1 and decreases for R for 0"> 0 for x R. 5) The function is neither even nor " title=" Exponential function, its graph and properties y x 1 o 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 o 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"> !}
Wood growth occurs according to the law, where: A - change in the amount of wood over time; A 0 - initial amount of wood; t-time, k, a- some constants. Wood growth occurs according to the law, where: A - change in the amount of wood over time; A 0 - initial amount of wood; t-time, k, a- some constants. t 0 t0t0 t1t1 t2t2 t3t3 tntn A A0A0 A1A1 A2A2 A3A3 AnAn
The temperature of the kettle changes according to the law, where: T is the change in the temperature of the kettle over time; T 0 - boiling point of water; t-time, k, a- some constants. The temperature of the kettle changes according to the law, where: T is the change in the temperature of the kettle over time; T 0 - 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 undecayed atoms at any time t; N 0 - initial number of atoms (at time t=0); t-time; N is the number of undecayed atoms at any time t; N 0 - initial number of atoms (at time t=0); t-time; T - half-life. T - half-life. t 0 t 1 t 2 N N3N3 N4N4 t4t4 N0N0 t3t3 N2N2 N1N1
C An essential property of organic processes and changes in quantities is that over equal periods of time the value of a quantity changes in the same ratio. Growth of wood Change in temperature of a kettle Change in air pressure Processes of organic changes 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 numbers as powers with the same base (if necessary) 1.3 34 and 1. Find out whether the exponential function a = 1.3 is increasing or decreasing; a>1, then the exponential function increases. a=1.3; a>1, then the exponential function increases. 3. Compare exponents (or function arguments) 34 1, then the exponential function increases. a=1.3; a>1, then the exponential function increases. 3. Compare exponents (or function arguments) 34"> 
Solve graphically the equation 3 x = 4-x. Example 2. Solve graphically the equation 3 x = 4-x. Solution. We use the functional-graphical method for solving equations: we will construct graphs of the functions y=3x and y=4x in one coordinate system. graphs of functions y=3x and y=4x. We notice that they have one common point (1;3). This means that the equation has a single root x=1. Answer: 1 Answer: 1 y=4's
4. 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 construct in one system 1. Let's construct in one coordinate system graphs of the functions " title="Solve graphically the inequality 3 x >4-x. Example 3. Solve graphically inequality 3 x > 4-x. Solution. y = 4-x We use the functional-graphical method for solving inequalities: 1. Construct graphs of functions in one coordinate system." class="link_thumb"> 24 !} Solve graphically the inequality 3 x > 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 construct in one coordinate system graphs of functions of coordinates graphs of functions y=3 x and y=4-x. 2. Select the part of the graph of the function y=3x, located above (since the > sign) of the graph of the function y=4x. 3. Mark on the x-axis the part that corresponds to the selected part of the graph (in other words: project the selected part of the graph onto the x-axis). 4. Let's write the answer as an interval: Answer: (1;). Answer: (1;). 4. 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 construct in one system 1. Let's construct graphs of functions "> 4-x in one coordinate system. 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 construct in one coordinate system graphs of functions of coordinates graphs of functions y=3 x and y=4-x 2. Select part of the graph of the function y=3. x, located above (since the > sign) of 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 (in other words: project the selected part of the graph onto the x-axis 4. Write down the answer). as an interval: Answer: (1;). Answer: (1;)."> 4. 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 construct in one system 1. Let's construct in one coordinate system graphs of the functions " title="Solve graphically the inequality 3 x >4-x. Example 3. Solve graphically inequality 3 x > 4-x. Solution. y = 4-x We use the functional-graphical method for solving inequalities: 1. Construct graphs of functions in one coordinate system."> title="Solve graphically the inequality 3 x > 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 construct graphs of functions in one coordinate system"> !}
Solve graphically the inequalities: 1) 2 x >1; 2) 2 x 1; 2) 2 x "> 1; 2) 2 x "> 1; 2) 2 x " title="Solve graphically the inequalities: 1) 2 x >1; 2) 2 x"> title="Solve graphically the inequalities: 1) 2 x >1; 2) 2 x"> !}
Independent work (test) 1. Specify the exponential function: 1. Specify 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=x2; 2) y=x -1; 3) y=-4+2 x; 4) y=0.32 x. 1) y=x2; 2) y=x -1; 3) y=-4+2 x; 4) y=0.32 x. 2. 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 -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. Specify the set of values of the function y=3 -2 x -8: 4. Specify the set of values of the function y=2 x+1 +16: 5. Specify the smallest of the given numbers: 5. Specify the smallest of the given 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. Specify 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. Find out graphically how many roots the equation 2 x = x -1/3 (1/3) x = x 1/2 has 6. Find out graphically how many roots the equation 2 x = x -1/3 (1/3) has x = x 1/2 1) 1 root; 2) 2 roots; 3) 3 roots; 4) 4 roots.
1. Specify 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-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. 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. Specify the set of values of the function y=3-2 x-8: 4. Specify the set of values of the function y=3-2 x-8: 5. Specify the smallest of the given numbers: 5. Specify the smallest of the given 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 2 x=x- 1/3 has 6. Find out graphically how many roots the equation 2 x=x- 1/3 has 1) 1 root; 2) 2 roots; 3) 3 roots; 4) 4 roots. 1) 1 root; 2) 2 roots; 3) 3 roots; 4) 4 roots. Test work Select exponential functions that: Select exponential functions that: I option – decrease on the domain of definition; Option I – decrease in the area of definition; Option II – increases in the area of definition. Option II – increases in the area of definition.
Concentration of attention:
Definition. Function species is called exponential function .
Comment. Exclusion from base values a numbers 0; 1 and negative values a is explained by the following circumstances:
The analytical expression itself a x in these cases, it retains its meaning and can be used in solving problems. For example, for the expression x y dot x = 1; y = 1 is within the range of acceptable values.
Construct graphs of functions: and.
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| Graph of an Exponential Function | |
| y = a x, a > 1 | y = a x , 0< a < 1 |
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Properties of the Exponential Function
| Properties of the Exponential Function | y = a x, a > 1 | y = a x , 0< a < 1 |
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| 2. Function range | ||
| 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. Even, odd. | The function is neither even nor odd (a function of general form). | |
| 5.Monotony. | monotonically increases by R | decreases monotonically by R |
| 6. Extremes. | The exponential function has no extrema. | |
| 7.Asymptote | O-axis x is a horizontal asymptote. | |
| 8. For any real values x And y; |  |
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When the table is filled out, tasks are solved in parallel with the filling.
Task No. 1. (To find the domain of definition of a function).
What argument values are valid for functions:
Task No. 2. (To find the range of values of a function).
The figure shows the graph of the function. Specify the domain of definition and range of values of the function:
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Task No. 3. (To indicate the intervals of comparison with one).
Compare each of the following powers with one:
Task No. 4. (To study the function for monotonicity).
Compare real numbers by size m And n If:
Task No. 5. (To study the function for monotonicity).
Draw a conclusion regarding the basis a, If:
y(x) = 10 x ; f(x) = 6 x ; z(x) - 4x
How are the graphs of exponential functions relative to each other for x > 0, x = 0, x< 0?
The following function graphs 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 equal to the limit of the sequence
with unlimited
increasing n
. Designation e entered 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 the “non-Pierre number.”
Number e plays a special role in mathematical analysis. Exponential function with base e, called exponent and is designated y = e x. First signs numbers e easy to remember: two, comma, seven, year of birth of Leo Tolstoy - two times, forty-five, ninety, forty-five. |
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Homework:
Kolmogorov paragraph 35; No. 445-447; 451; 453.
Repeat the algorithm for constructing graphs of functions containing a variable under the modulus sign.
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MAOU "Sladkovskaya Secondary School" Exponential function, its properties and graph, grade 10
A function of the form y = a x, where a is a given number, a > 0, a ≠ 1, the x-variable, is called exponential.
The exponential function has the following properties: O.O.F: the set R of all real numbers; Multivalent: the set of all positive numbers; The exponential function y=a x 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 throughout 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.
Using the increasing and decreasing properties of an 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 heat, it first cools down quickly, and then the cooling occurs 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
Wood growth occurs according to the law: A - change in the amount of wood over time; A 0 - initial amount of wood; t - time, k, a - some constants. Air pressure decreases with height according to the law: P is pressure at height h, P0 is pressure at sea level, and is some constant.
Population growth The change in the number of people in a 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.
Law of organic reproduction: under favorable conditions (absence of enemies, large amount of food), living organisms would reproduce according to the law of the 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 they were lined up in a chain, its length would be greater than the distance from the Earth to the Sun. But since, in addition to 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 quantity decreases, after some time half of the original substance remains. This period of time t 0 is called the half-life. The general formula for this process is: 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 decays. 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 approximately a time equal to the age of the Earth).
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” clearly presents educational material on this topic. During the presentation, the properties of the exponential function, its behavior in the coordinate system are discussed in detail, examples of solving problems using the properties of the function, equations and inequalities are considered, and important theorems on the topic are studied. With the help of a presentation, a teacher can improve the effectiveness of a mathematics lesson. Vivid presentation of the material helps to keep students' attention on studying the topic, and animation effects help demonstrate solutions to problems more clearly. 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 decimals. For each indicator, the value of the function is calculated. Next, a graph is built for the same function. On slide 2, a table is constructed filled with the coordinates of the points belonging to the graph of the function y = 3 x. Based on these points on the coordinate plane, a corresponding graph is constructed. Similar graphs y=2 x, y=5 x and y=7 x are constructed next to the graph. Each function is highlighted in different colors. The graphs of these functions are made in the same colors. Obviously, as the base of the exponential function increases, the graph becomes steeper and is closer 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, over all domains of definition the function increases and does not have the greatest or least value. The exponential function is bounded below but not bounded above, continuous on its 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. A graph of the function is constructed. To do this, the table is filled with the coordinates of the points belonging to the graph of the function. Using these points, a graph is constructed on a rectangular coordinate system. The properties of the function are described nearby. It is noted that the domain of definition is the entire numerical axis. This function is not odd or even, decreasing over the entire domain of definition, and does not have a maximum or minimum value. The function y=(1/3) x is bounded from below and unbounded from above, is continuous in its domain of definition, and has a downward convexity. The range of values is the positive semi-axis (0;+∞).
Using the given example of the function y = (1/3) x, we can highlight the properties of an exponential function with a positive base less than one and clarify the idea of its graph. Slide 5 shows the 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. To make the comparison more clear, the graphs are colored in the same colors as the function formulas. Next, the definition of an exponential function is presented. On slide 7, a definition is highlighted in the frame, which indicates that a function of the form y = a x, where positive a, not equal to 1, is called exponential. Next, using the table, we compare an exponential function with a base greater than 1 and a positive one less than 1. Obviously, almost all the properties of the function are similar, only a function with a base greater than a is increasing, and with a base less than 1, it is decreasing. The solution to the examples is discussed below. In example 1, it is necessary to solve the equation 3 x =9. The equation is solved graphically - a graph of the function y=3 x and a graph of the function y=9 are plotted. The intersection point of these graphs is M(2;9). Accordingly, the solution to the equation is the value x=2. Slide 10 describes the solution to the equation 5 x =1/25. Similar to the previous example, the solution to the equation is determined graphically. The construction of graphs of the functions y=5 x and y=1/25 is demonstrated. The intersection point of these graphs is point E(-2;1/25), which means the solution to the equation is x=-2. Next, 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 following 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 valid when m = n. Theorem 2 states that for positive a, the value of the function y=a x will be greater than 1 for positive x, and less than 1 for negative x. The statement is confirmed by the image of the graph of the exponential function, which shows the behavior of the function at various intervals of the domain of definition. Theorem 3 notes that for 0 Next, to help students master the material, they consider examples of solving problems using the studied theoretical material. In example 5, it is necessary to construct a graph of the function y=2·2 x +3. The principle of constructing a graph of a function is demonstrated by first transforming it into the form y = a x + a + b. A parallel transfer of the coordinate system to the point (-1;3) is carried out and a graph of the function y = 2 x is constructed relative to this origin. Slide 18 looks at the graphical solution to the equation 7 x = 8-x. A straight line y=8x and a graph of the function y=7x are constructed. The abscissa of the intersection point 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 increase the effectiveness of a school mathematics lesson. The clarity of the material in the presentation will help achieve learning goals during a distance lesson. The presentation can be offered for independent work to students who have not mastered the topic well enough in class.
