Presentation "power functions, their properties and graphs." Presentation of functions and their graphs Properties of power functions
Introduction. Mathematics, having long ago become the language of science and technology, is now increasingly penetrating everyday life and everyday language, and is increasingly being introduced into areas traditionally distant from it. As the great Galileo Galilei (1564 - 1642) figuratively noted, the book of nature is written in mathematical language, and its letters are mathematical signs and geometric figures, without them it is impossible to understand its words, without them wandering in an endless labyrinth is in vain. And it is the function that is the means of mathematical language that allows us to describe the processes of movement and changes inherent in nature. While studying the quadratic function in 9th grade, we performed transformations on the graph of this function. As a result of these transformations, plotting the graph was easy and simple. And I thought: “Is it possible to perform similar transformations with graphs of other functions, for example, a linear function, inverse proportionality, power function?” Therefore, I chose the topic of my work “Class of elementary functions and their graphs”, setting myself the goal: to understand and study the ways of forming elementary functions and transforming their graphs.
From the history of the development of the function. For the first time, a function entered mathematics under the name “variable quantity” in the famous work of the French mathematician and philosopher R. Descartes “Geometry”, and its appearance, according to F. Engels, served as a turning point in mathematics, thanks to which movement and dialectics were included in it. Without variables, I. Newton would not be able to express the laws of dynamics that describe the processes of mechanical movement of bodies - celestial and completely terrestrial, and modern scientists would not be able to calculate the trajectories of spacecraft and solve the endless number of technical problems of our era.
From the history of the development of the function. With the development of science, the concept of function was refined and generalized. Now it has become so general that it coincides with the concept of correspondence. Thus, a function in the general sense is any law (rule) according to which each object from a certain class, the domain of definition of a function, is associated with some object from another (or the same) class, the domain of possible values of the function. But we do not consider the concept of a function in such a general sense, but believe that both independent and dependent variables are quantities. Thus, a function is a dependence that connects with each value of one variable quantity (argument) from a certain area of its change a certain value of another quantity (function). If the argument is denoted by x, the value of the function by y, and the dependence itself - the function - by the symbol f, then the relationship between the values of the function and the argument is as follows: y=f(x).
Methods for specifying functions. There are three main ways of expressing dependencies between quantities: tabular, graphical and analytical (“formula”). The tabular method is important because it is the main one for detecting real dependencies and may also turn out to be the only means of specifying them (it is not always possible to choose a formula, and sometimes there is no need for it). Functions are often switched to the tabular specification when performing practical calculations, with it related: for example, the use of tables of square roots is convenient when carrying out calculations in which such roots are involved. From a mathematical point of view, the tabular assignment of continuous dependencies is always incomplete and provides only information about the values of the function at individual points.
Methods for specifying functions The graphical method of representing dependencies is also one of the means of recording them when studying real phenomena. This allows you to make various “self-recording” instruments, such as a seismograph, electrocardiograph, oscilloscope, etc., which display information about changes in measured quantities in the form of graphs. But if there is a graph, then the corresponding function is also defined. In such cases, we talk about graphically specifying the function. However, the graphical method of specifying a function is inconvenient for calculations; Moreover, like the tabular one, it is approximate and incomplete. The analytical (formular) function assignment is distinguished by its compactness, is easy to remember and contains complete information about the dependence. The function can be specified using a formula, for example: y=2x+5, S=at2/2, S=vt. These formulas can be derived using geometric or physical reasoning. Sometimes formulas are obtained as a result of processing an experiment; such formulas are called empirical.
Class of elementary functions Elementary functions include almost all functions found in a school textbook. First of all, there is a fairly representative set of well-known and well-studied functions, which are called basic elementary functions. These are functions: y=C, called constant, y= xа - power function (for a = 1, the function y=x is obtained, called identical). Graphs of these functions are attached. (Appendix 1-7) Having the basic elementary functions at your disposal, you can introduce a number of operations that allow you to combine them with each other as parts to obtain more complex and varied designs. Valid arithmetic operations on functions. [+] – addition, [-] – subtraction, [*] – multiplication, [:] – division. All those functions that can be obtained from basic elements using arithmetic operations are called elementary functions and constitute the class of elementary functions.
Formation of a class of elementary functions Having a certain set of basic functions f1, f2,f3,...fk and admissible operations F1, F2, ... Fs on them (they can be applied any number of times), we can obtain other functions, similar to how from the parts of a designer, using certain rules for connecting them, you can get different models. The class of all functions obtained in this way is denoted as follows:< f1,f2,...fk; F1,F2,...Fs>. In particular, if we take all basic elementary functions as basic and allow only arithmetic operations, we obtain a class of elementary functions. Taking as basis some of the basic elementary functions and allowing, perhaps, only some of the indicated operations, we obtain some subclasses of the class of elementary functions, some families of functions generated by this basis and these operations. Here are some examples of such families of functions, where (a) is understood as the operation of multiplication by any constant:
Construction of graphs To construct a graph of the function y = 3x2, you need to multiply the graph of the function y = x2 by 3. As a result, the graph of the function y = x2 will stretch 3 times along the ordinate axis, and if y = 0.3 x2, then the graph will be compressed to 0, 3 times along the Oy axis. (Appendix 8, 9).
Construction of graphs The graph of the function y=3(x -4)2 can be obtained by performing the following steps: - add the graphs of the identical function y=x and the constant y=-4, we obtain a graph of the function y=x-4; - multiply the graphs of the functions y=x-4 and y=x-4, we get the graph of the function y= (x -4)2; - multiply y= (x -4)2 by 3, we get a graph of the function y=3(x -4)2. Or simply shift the graph of the function y=3x2 along the Ox axis by 4 unit segments (Appendix 10).
Transformations of the original graph of the function y= f(x). From the above, we can draw the following conclusion that by performing various actions with graphs of elementary functions, we perform transformations of these graphs, namely: parallel translation, symmetry with respect to the line Ox and line Oy.
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“Functions and graphs” Presentation for a lesson GBOU NPO Professional Lyceum No. 80 Mathematics teacher Galina Ivanovna Savitskaya
“Functions and Graphs” 1. What is a function? Definition 2. Graphs of elementary functions 3. Properties of a function 5. Transformation of graphs of functions Exercises: Specify the properties of a function 4. How to construct a graph using given properties of a function
Let there be sets X and Y. If each element x from the set X, according to some rule, is associated with a single element y from the set Y, then we say that the function y = f(x) is given. DEFINITION OF X Y Y X 1 y 1 X 2 y 2 X 3 y 3 X 4 y 4 X f (law)
They say that y is a function of x y=f(x) In this case: X = – domain of definition of the function OOF or D(y) y – set of values of the function MZF or E(y) X – independent variable or argument Y – dependent variable or function
1) Formula x 1 2 3 4 5 y 1 8 15 20 22 Methods of specifying the function y = x 2 + 2x – 4 y = 3x f(x) = log 2 (3x+4) f(x) = COS 2x 2) Table
Y= f (x) Y X 0 ordinate axis abscissa axis origin of coordinates Methods of specifying a function 3) Graph 1 2 3 -1 -2 -3 -1 -2 -3 1 2 3
Y= f (x) Y X 0 1 2 3 -1 -2 -3 -1 -2 -3 1 2 3 A(-2;1) B(1;-2) M(x; Y) Graph of the function Y = f (x) is the set of points of the coordinate plane having coordinates (x; f (x)) or (x; Y)
1. Linear function Graphs of elementary functions y x Y = x y = 2x y = - x y = k x + in k – slope 0 y = x k=1 y = 2 x k=2 y = - x k=- 1 y = ½ x k = ½ 1 1 2 -1 y = ½ x
1. Linear function: Graphs of elementary functions y x y = k x + in k – slope 0 y = x +2 y = x -2 1 1 2 -1 y = x-2 y = x+2 y = x - 2
1. Linear function: Graphs of elementary functions y x y = k x + in k – slope 0 y = x y = 2 x = 3 1 1 1 2 -1 -2 3 2 3 y = 2 X = 3
2. Quadratic function y=ax 2 + b x + c Graphs of elementary functions 0 y x x 0 y 0 parabola Coordinates of the vertex of the parabola: x 0 = - b 2a y 0 = a (x 0) 2 + b x 0 + c if a > 0 The branches of the parabola are directed upward if a 0 a
Cubic function: y=ax 3 + b x 2 + cx + d Graphs of elementary functions cubic parabola y x 0 y=x 3 1 1 -1 -1 y=x 3
4. Inversely proportional function: Y= Graphs of elementary functions hyperbola k x y x 0 1 -1 1 -1 y x 0 1 -1 1 -1 y = 1 x y = - 1 x
5. Modular function: y = | x | Graphs of elementary functions y x 0 1 1 -1
PROPERTIES OF FUNCTIONS Y = f (x) Y x 0 a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 8 a 9 a 1 a 2 a 3 a 4
PROPERTIES OF FUNCTIONS y = f (x) Y x 0 a 1 a 9 1 . The domain of definition of a function is the set of values of the argument X for which the OOF function exists: X є [ a 1 ; a 9 ]
PROPERTIES OF FUNCTIONS Y = f (x) Y x 0 in 1 in 4 2. The set of function values is the set of all numbers that the MZF can take: y є [ in 4 ; in 1 ]
PROPERTIES OF FUNCTIONS Y = f (x) Y x 0 a 2 a 4 a 6 a 8 3. The roots (or zeros) of a function are those values of x at which the function is equal to zero (y = 0) f (x) = 0 at X = a 2; a 4; a 6; a 8
PROPERTIES OF FUNCTIONS y = f (x) Y x 0 a 1 a 2 a 4 a 6 a 8 a 9 4 . Areas of constant sign of a function are those values of x for which the function is greater or less than zero (i.e. y > 0 or y 0 for X є (a 1 ; a 2); (a 4 ; a 6); (a 8 ; a 9)
PROPERTIES OF FUNCTIONS y= f (x) Y x 0 a 2 a 4 a 6 a 8 4. Areas of constant sign of a function are those values of x at which the function is greater or less than zero (i.e. y > 0 or y
PROPERTIES OF FUNCTIONS y= f (x) Y x 0 a 3 a 5 a 7 a 9 5. The monotonicity of a function is the areas of increasing and decreasing function. The function increases as X є [ a 3 ; a 5 ] ; [a 7; a 9 ] a 1 The function decreases as X є [ a 1 ; a 3 ] ; [a 5; a 7 ]
PROPERTIES OF FUNCTIONS y = f (x) Y x 0 a 3 a 5 a 7 in 2 in 3 in 4 Extrema of the function F max (x) F min (x) F min (x) F max (x) = in 2 at the point extremum x = a 5 F min (x) = in 3 at the extremum point x = a 3 F min (x) = in 4 at the extremum point x = a 7
PROPERTIES OF FUNCTIONS y = f (x) y x 0 a 7 a 9 in 1 in 4 7. The highest and lowest values of a function (these are the highest and lowest points on the function graph) the highest value F (x) = in 1 at point x = a 9 smallest value F (x) = b 4 at point x = a 7
y x F(x) = x 2 y x F(x) = cos x x 0 0 X -X PROPERTIES OF FUNCTIONS Even and odd functions A function is called even if for any X from its domain of definition the rule f(x) = f is satisfied (- x) The graph of an even function is symmetrical about the Y axis f(x) X -X f(x)
PROPERTIES OF FUNCTIONS Even and odd functions A function is called odd if for any X from its domain of definition the rule f(x) = - f(x) is satisfied. The graph of an odd function is symmetrical with respect to the origin y x 0 y=x 3 x f(x) - f(x) - x y x 0 y = 1 x 1 -1 1 -1
2 2 4 6 8 10 x -2 -4 -6 -8 -10 0 4 6 y -2 -4 y= f (x) T = 4 Periodicity of functions If the pattern of the graph of a function is repeated, then such a function is called periodic, and the length segment along the X axis is called the period of the function (T) A periodic function obeys the rule f(x) = f(x+T) PROPERTIES OF FUNCTIONS
2 2 4 6 x -2 -4 -6 0 4 6 y -2 -4 -6 y= f (x) Т = 6 PROPERTIES OF FUNCTIONS Function y=f(x) is periodic with period Т = 6
1 1 2 3 4 5 x -1 -2 -3 -4 -5 0 2 3 4 y -1 -2 -3 -4 Indicate the properties of the function 1) OOF 2) MZF 3) Zeros of the function 4) Positive function Negative function 5 ) The function increases The function decreases 6) Extrema of the function F max (x) F min (x) 7) The largest value of the function The smallest value of the function y = f (x)
1 1 2 3 4 5 x -1 -2 -3 -4 -5 0 2 3 4 y -1 -2 -3 -4 Indicate the properties of the function y = f (x)
2 2 4 6 8 10 x -2 -4 -6 -8 -10 0 4 6 8 y -2 -4 -6 -8 Indicate the properties of the function y = f (x)
2 2 x -2 0 y -2 Indicate the properties of the function y = f (x)
3 3 x -1 0 y -1 -4 -5 Construct a graph of the function Given: a) The domain of definition is the interval [-4;3] b) The values of the function make up the interval [- 5;3] c) The function decreases on the intervals [ -4; 1 ] and [ 2 ;3] increases on the interval [- 1 ; 2 ] d) Zeros of the function: -2 and 2
TRANSFORMATION OF FUNCTION GRAPHES Knowing the graph of an elementary function, for example f(x) = x 2, you can construct a graph of a “complex” function, for example f(x) = 3(x +2) 2 - 16 using graph transformation rules
Rules for converting graphs 1 rule: Displacement along the X axis If you add or subtract a number to the argument X, the graph will shift to the left or right along the X axis f(x) f(x ± a) convert to 0 y x 0 y x 4 -4 F (x) = x 2 F(x) = (x+4) 2 F(x) = (x-4) 2
If you add or subtract a number to the function Y, the graph will shift up or down along the Y axis f(x) f(x) = X ± a convert to Rules for converting graphs 2 rule: displacement along the Y axis y x 4 - 4 0 y x F(x) = x 2 F(x) = x 2 + 4 F(x) = x 2 - 4
If the argument X is multiplied or divided by the number K, then the graph will be compressed or stretched K times along the X axis f(x) f(k · x) converted to Rules for converting graphs 3 rule: compression (stretching) of the graph along the X axis y x F (x) = sin x F(x) = sin 2x
If you add or subtract a number to the function Y, the graph will move up or down along the Y axis f(x) f(x) ± a convert to y x F(x) = sin x F(x) = sin x 2 Rules for converting graphs Rule 3: C compressing (stretching) the graph along the X axis
If the function is multiplied or divided by the number K, then the graph will be stretched or compressed K times along the Y axis f(x) k · f(x) converted to Rules for converting graphs 4th rule: compression (stretching) of the graph along the Y axis y x F( x) = cos x F(x) = cos x 1 2
If the function is multiplied or divided by the number K, then the graph will be stretched or compressed K times along the Y axis f(x) k · f(x) converted to Rules for converting graphs 4th rule: compression (stretching) of the graph along the Y axis y x F( x) = cos x F(x) = 2cos x
If you change the sign to the opposite one before the function, then the graph will be symmetrically flipped relative to the X axis f(x) - f(x) converted to Rules for converting graphs 5 rule: flipping the graph relative to the X axis y x F(x) = x 2 F(x) = - x 2
Presentation “Power functions, their properties and graphs” is a visual aid for conducting a school lesson on this topic. Having studied the features and properties of a power with a rational exponent, it is possible to make a complete analysis of the properties of a power function and its behavior on the coordinate plane. During this presentation, the concept of a power function, its various types, the behavior of the graph on the coordinate plane of a function with a negative, positive, even, odd exponent are considered, an analysis of the properties of the graph is made, and examples of solving problems using the studied theoretical material are described.
Using this presentation, the teacher has the opportunity to increase the effectiveness of the lesson. The slide clearly shows the construction of the graph; with the help of color highlighting and animation, the features of the function’s behavior are highlighted, forming a deep understanding of the material. A bright, clear and consistent presentation of the material ensures better memorization of it.
The demonstration begins with the property of a degree with a rational exponent, learned in previous lessons. It is noted that it transforms into the root a p/q = q √a p for non-negative a and unequal to one q. It is recalled how this is done using the example 1.3 3/7 = 7 √1.3 3 . The following is a definition of the power function y=x k, in which k is a rational fractional exponent. The definition is boxed for memorization.
Slide 3 demonstrates the behavior of the function y=x 1 on the coordinate plane. This is a function of the form y=x, and the graph is a straight line passing through the origin of coordinates and located in the first and third quarters of the coordinate system. The figure shows an image of the graph of the function, highlighted in red.
Next, we consider the degree of the 2-power function. Slide 4 shows an image of the graph of the function y=x 2 . Schoolchildren are already familiar with this function and its graph - a parabola. Slide 5 looks at a cubic parabola - a graph of the function y=x 3 . Its behavior has also already been studied, so students can recall the properties of the graph. The graph of the function y=x 6 is also considered. It also represents a parabola - its image is attached to the description of the function. Slide 7 shows a graph of the function y=x 7 . This is also a cubic parabola.
Then the properties of functions with negative exponents are described. Slide 8 describes the type of power function with a negative integer exponent y=x -n =1/x n. An example of a graph of such a function is the graph y=1/x 2. It has a discontinuity at the point x=0, consists of two parts located in the first and second quarters of the coordinate system, each of which, as it tends to infinity, is pressed against the abscissa axis. It is noted that this behavior of the function is typical for even n.
On slide 10, a graph of the function y = 1/x 3 is constructed, parts of which lie in the first and third quarters. The graph also breaks at the point x=0 and has asymptotes y=0 and x=0. It is noted that this behavior of the graph is typical for a function in which the degree is an odd number.
Slide 11 describes the behavior of the graph of the function y=x0. This is the straight line y=1. It is also demonstrated on a rectangular coordinate plane.
Next, the difference between the location of the branch of the function y=x n is analyzed with increasing exponent n. For visual demonstration, functional dependencies are marked in the same color as the graphs. As a result, it is clear that with an increase in the function index, the graph branch is pressed more closely to the ordinate axis, and the graph becomes steeper. In this case, the graph of the function y=x 2.3 occupies a middle position between y=x 2 and y=x 3.
On slide 13, the considered behavior of the power function is generalized into a pattern. It is noted that at 0<х<1 при увеличении показателя степени, уменьшается значение выражения х 5 < х 4 < х 3 , следовательно и √х 5 < √х 4 < √х 3 . Для х, большего 1, верно обратное утверждение - при увеличении показателя степени значение степенной функции увеличивается, то есть х 5 >x 4 > x 3, therefore, √x 5 > √x 4 > √x 3.
What follows is a detailed consideration of the behavior on the coordinate plane of the power function y=x k, in which the exponent is the improper fraction m/n, where m>n. In the figure, the description of this function is accompanied by a constructed graph in the first quarter of the coordinate system, which represents a branch of the parabola y=x 7/2. The properties of the function for m/n>1 are described on slide 15 using the example of the graph y=x 7/2. It is noted that it has a domain of definition - ray. 6.The function increases from 0 to + as x)
