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Download functions and their graphs ppt. Functions, their properties and graphs

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Objectives of the lesson: To become familiar with the concept of “function”, consolidate it with examples To learn new terms To learn methods for studying functions To consolidate knowledge on the topic when solving problems To learn how to build graphs of functions Kolomina N.N.

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A little history The word “function” (from the Latin functio - accomplishment, execution) was first used in 1673 by the German mathematician Leibniz. The definition of a function “A function of a variable quantity is an analytical expression composed in some way from this quantity and numbers or constant quantities” was made in 1748 by the German and Russian mathematician Leonhard Euler N.N. Colomina.

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Definition. “The dependence of the variable y on the variable x, in which each value of the variable x corresponds to a single value of the variable y, is called a function.” Symbolically, the functional relationship between the variable y (function) and the variable x (argument) is written using the equality Methods for specifying functions: tabular (table), graphical (graph), analytical (formula). Kolomina N.N.

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General scheme for studying a function 1. Domain of definition of a function. 2.Investigation of the range of values of the function. 3. Study of the function for parity. 4. Study of intervals of increasing and decreasing function. 5. Study of a function for monotonicity. 5. Study of a function for an extremum. 6. Study of the function for periodicity. 7. Determination of intervals of constancy of sign. 8. Determination of the points of intersection of the graph of a function with the coordinate axes. 9. Graphing a function. Kolomina N.N.

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Domain of definition of a function The domain of definition (existence) of a function is the set of all real values of the argument for which it can have a real value. For example, for the function y=x the domain of definition is the set of all real values of the numbers R; for the function y=1/x the domain of definition is the set R except x=0. Kolomina N.N.

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[-3;5] 0 x y 7 -5 [-5;7) [-5;7] (-3;5] Find the domain of definition of the function whose graph is shown in the figure. 5 -3 Domain of definition of the function - values, which is taken by the independent variable x. Kolomina N.N.

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Set of function values. The set of values of a function is the set of all real values of the function y that it can take. For example, the set of values of the function y= x+1 is the set R, the set of values of the function is the set of real numbers greater than or equal to 1. y= X2 +1 Kolomina N.N.

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Find the set of values of the function whose graph is shown in the figure. y x 0 -6 -4 6 6 (-4;6) [-6;6] (-6;6) [-4;6] The set of function values is the values that the dependent variable y takes. Kolomina N.N.

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Study of the function for parity. A function is called even if, for all values of x in the domain of definition of this function, when the sign of the argument is changed to the opposite, the value of the function does not change, i.e. . For example, the parabola y = X2 is an even function, because (-X2)= X2. Schedule even function symmetrical about the y axis. Kolomina N.N.

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One of the following figures shows the graph of an even function. Provide this schedule. x x x x y y y The graph is symmetrical about the Oy axis 0 0 0 0 Kolomina N.N.

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A function is called odd if, for all values of x in the domain of definition of this function, when the sign of the argument changes to the opposite, the function changes only in sign, i.e. . For example, the function y = X3 is odd, because (-X)3 = -X3. The graph of an odd function is symmetrical about the origin. Not every function has the property of even or odd. For example, the function is neither even nor odd: X2+ X3 (-X)2+ (-X)3 = X2 – X3; X2 + X3 X2 – X3; = / Kolomina N.N.

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x x x x y y One of the following figures shows the graph of an odd function. Provide this schedule. The graph is symmetrical with respect to point O. O O O O Kolomina N.N.

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Among the many functions, there are functions whose values only increase or decrease as the argument increases. Such functions are called increasing or decreasing. A function is called increasing in the interval a x b if for any X1 and belonging to this interval, at X1 X2 the inequality holds. Definition of intervals of increasing and decreasing /\ /\ X2 /\ /\ 1 2 The function is said to be decreasing in the interval a x b, if for any X1 and X2 belonging to this interval, for X1 X2 the inequality /\ /\ /\ 2 1 > N.N. Kolomina holds.

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[-6;7] [-5;-3] U [-3;7] [-3;2] x 0 2 6 -5 7 -3 -6 -2 3 The figure shows the graph of the function y = f(x ), specified on the interval (-5;6). Indicate the intervals where the function increases. at Kolomin N.N.

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y x 1 2 4 0 The zero of the function is the value of x at which y = 0. In the figure, these are the points of intersection of the graph with the Ox axis. The figure shows a graph of the function y = f(x). Specify the number of zeros of the function. 0 Kolomina N.N.

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Study of a function for monotonicity. Both increasing and decreasing functions are called monotonic, and the intervals in which the function increases or decreases are called monotonic intervals. For example, the function y = X2 at x 0 increases monotonically. The function y = X3 monotonically increases on the entire numerical axis, and the function y = -X3 monotonically decreases on the entire numerical axis. /\ /\ Kolomina N.N.

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Examine the function for monotonicity Function y=x2 Function y=x2 at x<0 монотонно убывает, при х>0 monotonically increases x -2 -1 0 1 2 y 4 1 0 1 4 Kolomina N.N.

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Inverse function If a function takes each of its values only for a single value of x, then such a function is called invertible. For example, the function y=3x+5 is invertible, because each value of y is accepted with a single value of the argument x. On the contrary, the function y = 3X2 is not invertible, since, for example, it takes the value y = 3 both for x = 1 and for x = -1. For any continuous function (one that has no discontinuity points) there is a monotone single-valued and continuous inverse function. Kolomina N.N.

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Dictation Find the range of values Explore the intervals of increasing and decreasing functions. No. Option-1 No. Option-2 Find the domain of definition of the function 1 1 2 2 Indicate the method of specifying the function 3 3 Examine the function for parity 4 4 5 5 x -2 -1 0 1 y 3 5 7 9 Kolomina N.N.

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Functions. 1. Linear function 2. Quadratic function 3. Power function 4. Exponential function 5. Dogarithmic function 6. Trigonometric function Kolomina N.N.

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Linear function y = kx + b k – angular coefficient b x y α 0 b – free coefficient k = tan α Kolomina N.N.

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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 elementary functions 3. Properties of a function 5. Transformation of graphs of functions Exercises: Indicate the properties of a function 4. How to construct a graph from given properties functions

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 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 at 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 greatest and smallest value function (these are the highest and lowest points on the graph of the function) the largest value of F (x) = 1 at the point x = a 9 the smallest value of F (x) = 4 at the 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 the segment along the X axis is called the period of the function (T) 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) Function positive Function negative 5) The function increases The function decreases 6) Extrema of the function F max (x) F min (x) 7) Highest value functions 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 GRAPHICS 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 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

One of the most important questions when learning algebra is function. Studying begins in 7th grade. However, students often perceive the material with great difficulty. And even in 11th grade the topic causes difficulties. This presentation is a summary of the material and I hope it will help alleviate the difficulties in studying this topic.

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Function. Properties and graphs of functions. Loginova N.V. mathematics teacher MBOU "Secondary School No. 16" Izhevsk 12/14/2014

Repetition on the topic: 1. What is a function? Definition. The dependence of the variable y on the variable x, in which each value of the variable x corresponds to a single value of the variable y, is called a function. Definition. The correspondence f between two sets X and Y, in which each element of the set X is associated with a single element of the set Y, is called a function. 0 1 1 12/14/2014 Loginova N.V. MBOU "Secondary School No. 16" 2 1 1

2. How can you define a function? The variable x is called the independent variable or argument. The variable y is called the dependent variable. The variable y is also said to be a function of the variable x. The values of the dependent variable are called the values of the function. Methods for specifying functions: tabular, graphical, analytical (using a formula), verbal. 12/14/2014 Loginova N.V. MBOU "Secondary School No. 16" 3

Recall that the graph of a function is the set of all points of the coordinate plane, the abscissas of which are equal to the values of the argument, and the ordinates are equal to the corresponding values of the function. All the values that the dependent variable takes form the range of the function. All values of the independent variable form the domain of definition of the function. 3. What is a graph? 4. What is the domain of definition and domain of value of a function? 12/14/2014 Loginova N.V. MBOU "Secondary School No. 16" 4

1. 2. 3. ; () 4. 5. 6. ; () ; () Find the domain of definition of the function 12/14/2014 Loginova N.V. MBOU "Secondary School No. 16" 5

Name the functions with the same domain of definition 1. 2. 3. 4. 5. 6. Specify a function with a formula with a domain of definition 7. 8. 12/14/2014 Loginova N.V. MBOU "Secondary School No. 16" 6

The figure shows a graph of the dependence of temperature T (º C) on the time of day t (hour) T ºc 4 2 t 0 2 4 6 8 10 12 14 16 18 h -2 -4 12/14/2014 Loginova N.V. MBOU "Secondary School No. 16" 7

Boundedness of a function A function y = f (x) is called bounded from below on the set X  D (f) if all values of the function on the set X are greater than a certain number. if there is a number m such that for any value x  X the inequality f (x) > m is satisfied. 12/14/2014 Loginova N.V. MBOU "Secondary School No. 16" 8

Continuity of function x y 0 x y 0 12/14/2014 Loginova N.V. MBOU "Secondary School No. 16" 9

The domain of definition of the function D (f) is a symmetric set; 2. For any x  X the equality holds: f (– x) = f (x) x y f (– x) = – f (x) x y Even and odd functions 12/14/2014 Loginova N.V. MBOU "Secondary School No. 16" 10

Convexity of the function x y 0 x y 0 12/14/2014 Loginova N.V. MBOU "Secondary School No. 16" 11

Periodicity of a function A periodic function is called a function that satisfies the condition: f(x+T)=f(x) for any x. The smallest value of T is called the period of the function 12/14/2014 Loginova N.V. MBOU "Secondary School No. 16" 12

General scheme for studying a function 1. Domain of definition of a function. 2. Study of the range of values of the function. 3. Determination of the intersection points of the function graph with the coordinate axes (function zeros). 4. Study of the function for monotonicity (increase, decrease of the function) 5. Determination of intervals of constancy of sign. 6. Study of a function for continuity. 7. Study of the function for parity. 8. The largest and smallest values of a function. 9. Limited function. 10. Convexity of function. 11. Periodicity of the function. 12. Graphing a function. 12/14/2014 Loginova N.V. MBOU "Secondary School No. 16" 13

x y 0 k > 0 increasing x y 0 k

Properties of the function D (f) = (- ;0)(0;+) E (f) = (- ;0)(0;+) Monotonicity k > 0 k

Properties of the function y = kx 2 1. D (f) = (-  ;+ ) k > 0 k

Graph of a constant function x y 0 y = C C 12/14/2014 Loginova N.V. MBOU "Secondary School No. 16" 17

Rice. 1 Fig. 2 Fig. 3 Fig. 6 Fig. 5 Fig. 4 Fig. 7 Fig. 8 12/14/2014 Loginova N.V. MBOU "Secondary School No. 16" 18

Reading Function Graphs Independent work Loginova N.V. MBOU "Secondary School No. 16" 19

1 2 3 4 Option 1 Specify the domain of definition of the function Option 2 Specify the set of values of the function 1 12/14/2014 Loginova N.V. MBOU "Secondary School No. 16" 20

1 2 3 4 2 1st option Indicate the number of the even function 2nd option Indicate the number of the odd function 12/14/2014 Loginova N.V. MBOU "Secondary School No. 16" 21

3 On which of the following sets is the function whose graph is shown in Figure 1 option 2 option increasing decreasing 1 2 3 4 1 2 3 4 12/14/2014 Loginova N.V. MBOU "Secondary School No. 16" 22

1 2 3 4 1 option Find all values for which the inequality holds. Option 2 Find all values for which the inequality holds. 4 12/14/2014 Loginova N.V. MBOU "Secondary School No. 16" 23

Federal Agency for Education. State educational institution Average vocational education. Dimitrovgrad Technical College. Project by Stanislav Vereshchuk. Topic: “Properties and graphs of elementary functions.” Head: teacher Kuzmina V.V. Dimitrovgrad 2007

1. Definition of a function. 2. Linear function: increasing; decreasing; special cases. 3. Quadratic function. Quadratic function. 4. Power function: Power function: with even natural exponent; with an odd natural exponent; with an integer negative exponent; with a real indicator. 5. List of used literature.

Definition of a function. The relationship between the elements of two sets X and Y, in which each element x of the first set corresponds to one element of the second set, is called a function and is written y = f(x). All the values that the independent variable x takes are called the domain of the function. All the values that the dependent variable y takes are called the value set of a function or the range of a function. The graph of a function is the set of all points of the coordinate plane, the abscissas of which are equal to the values of the argument, and the ordinates are equal to the corresponding values of the function.

0 and b 0): 1. The domain of definition of the function is the set of all real numbers D(f)=R. 2. The set of values of a linear function is the set of all real numbers E(f)=R. 3. When k>0 the function increases" title="Properties of a linear function (provided k > 0 and b 0): 1. The domain of definition of the function is the set of all real numbers D(f) = R. 2. The set values of a linear function - the set of all real numbers E(f) = R. 3. When k>0 the function increases" class="link_thumb"> 5 !} Properties of a linear function (provided k > 0 and b 0): 1. The domain of definition of the function is the set of all real numbers D(f)=R. 2. The set of values of a linear function is the set of all real numbers E(f)=R. 3. When k>0 the function increases. y=kx+b (k>0) 0 and b 0): 1. The domain of definition of the function is the set of all real numbers D(f)=R. 2. The set of values of a linear function is the set of all real numbers E(f)=R. 3. When k>0 the function increases "> 0 and b 0): 1. The domain of definition of the function is the set of all real numbers D(f)=R. 2. The set of values of a linear function is the set of all real numbers E(f)=R 3. When k>0 the function increases. y=kx+b (k>0)"> 0 and b 0): 1. The domain of definition of the function is the set of all real numbers D(f)=R. 2. The set of values of a linear function is the set of all real numbers E(f)=R. 3. When k>0 the function increases" title="Properties of a linear function (provided k > 0 and b 0): 1. The domain of definition of the function is the set of all real numbers D(f) = R. 2. The set values of a linear function - the set of all real numbers E(f) = R. 3. When k>0 the function increases"> title="Properties of a linear function (provided k > 0 and b 0): 1. The domain of definition of the function is the set of all real numbers D(f)=R. 2. The set of values of a linear function is the set of all real numbers E(f)=R. 3. When k>0 the function increases"> !}

Properties of a linear function (subject to k

Special cases of a linear function: 1.If b=0, then the linear function is given by the formula y=кx. This function is called direct proportionality. The graph of direct proportionality is a straight line passing through the origin. y=кx (k>0) y=кx (k 0) y=кx (k"> 0) y=кx (k"> 0) y=кx (k" title="Special cases of a linear function: 1.If b=0, then the linear function is given by the formula y=кx. Such a function is called direct proportionality. The graph of direct proportionality is a straight line passing through the origin. y=кx (k>0) y=кx (k"> title="Special cases of a linear function: 1.If b=0, then the linear function is given by the formula y=кx. This function is called direct proportionality. The graph of direct proportionality is a straight line passing through the origin. y=кx (k>0) y=кx (k"> !}

Special cases of a linear function: 2.If k=0, then the linear function is given by the formula y=b. Such a function is called constant. The graph of a constant function is a straight line parallel to the Ox axis. If k=0 u b=0, then the graph of the constant function coincides with the Ox axis.

Properties of a power function with an even natural exponent: 1. The domain of definition D(f)=R is the set of all real numbers. 2. The range of values E(f)=R + is the set of all non-negative numbers. 3.The function is even, i.e. f(-x)=f(x). 4.Zeros of the function: y=0 at x=0. 5. The function decreases from - to 0 as x (-,0]. 6. The function increases from 0 to + as x [-3;7]\nThat's right!\n\nу\n7\n\n3\n-5\n\n-3\n\n0\n-2\n\n4\n\n[-3;2 ]\n-6\n\nCheck (1)\n\nKolomina N.N..jpg","smallImageUrl":"\/\/pedsovet.su\/_load-files\/load\/48\/64 \/3\/f\/2-page-14_300.jpg"),("number":15,"text":"The figure shows the graph of the function y = f(x).\nIndicate the number\nof zeros of the function.\ ny\n\nThink about it!\n1\n\n1\n\n2\n\n2\n\n3\n\n4\n\n4\n\n0\n\nThink about it!\nThat’s right!\n \nx\n\nThink about it!\n\nCheck (1)\nKolomina N.N.\n\n0\n\nThe zero of the function is the value of x at which y = 0. In the\nfigure these are the points of intersection of the graph with axis Oh..jpg","smallImageUrl":"\/\/pedsovet.su\/_load-files\/load\/48\/64\/3\/f\/2-page-15_300.jpg") ,("number":16,"text":"Which of the functions are\nincreasing and which are decreasing?\n\n1) y 5\n\nx\n\nincreasing, because 5  1\n \n2) y 0.5\n\n3) y 10\n\nx\n\nx\n\ndecreasing, because 0  0.5  1\n\nincreasing, because 10  1\n\nth, because  1\n4) y  x increasing\nx\n\n 2\n5) y  \n 3\n\n6) y  49\nKolomina N.N.\n\nx\n\n2\ndescending, because 0   1\n3\n1\n1\ndescending, because..jpg","smallImageUrl":" \/\/pedsovet.su\/_load-files\/load\/48\/64\/3\/f\/2-page-16_300.jpg"),("number":17,"text": "The study of a function for monotonicity.\nBoth increasing and decreasing functions\nare called monotonic, and the intervals\nin which the function increases or decreases are called intervals of monotonicity.\n\/\\\n\nFor example, the function y = X2 for x 0 monotonically\nincreases.\nThe function y= X3 on the entire numerical axis monotonically\nincreases, and\nthe function y= -X3 on the entire numerical axis monotonically\ndecreases.\nKolomina N.N..jpg","smallImageUrl":"\/\ /pedsovet.su\/_load-files\/load\/48\/64\/3\/f\/2-page-17_300.jpg"),("number":18,"text":"Explore the function for monotonicity\nx\nу\n\nFunction y=x2\n\n-2 -1 0\n4 1 0\n\n1\n1\n\n2\n4\n\ny\n6\n5\n4\n3 \n2\n1\n\n-6\n4\n\n-5\n5\n\n-4\n6\n\n-3\n\n-2 - -1\n1\n2\n3\ n4\n5\n6\n\nKolomina N.N..jpg","smallImageUrl":"\/\/pedsovet.su\/_load-files\/load\/48\/64\/3\/f\ /2-page-18_300.jpg"),("number":19,"text":"Inverse function\nIf a function y  f (x) takes each of its\nvalues only for a single value x, then\nsuch a function is called reversible.\nFor example, the function y=3x+5 is reversible, because \neach value of y is taken with a single\nvalue of the argument x. On the contrary, the function y = 3X2 is not invertible, since, for example, it takes the value y = 3 both for x = 1 and for x = -1.\nFor any continuous function (one that does not have breakpoints) there is a monotonic\nunambiguous and continuous inverse function.\nKolomina N.N..jpg","smallImageUrl":"\/\/pedsovet.su\/_load-files\/load\/48\/64\/3\/ f\/2-page-19_300.jpg"),("number":20,"text":"Dictation\n№\n\n№\n\nOption-1\n\nOption-2\n\nFind domain of definition of the function\n1\n\nу  х2  1\n\n1\n\nу\n\nFind the range of values\n2\n\nу\n\n3\n\nх 1\nх2  2\ n\nх 1\n2\n2\nу\nх 2\nIndicate the method of specifying the function\n\nх\n\n-2\n\n-1\n\n0\n\n1\n\nу\ n\n3\n\n5\n\n7\n\n9\n\n3\n\nх2  1\n\n x  3, x   3;\nh x   2\n x  3, x  3.\n\nStudy the function for parity\n4\n\n4\nStudy the intervals of increasing and decreasing functions.\n\n5\nKolomina N.N..jpg","smallImageUrl":"\ /\/pedsovet.su\/_load-files\/load\/48\/64\/3\/f\/2-page-20_300.jpg"),("number":21,"text":" Functions.\n1.Linear function\n2.Quadratic function\n3.Power function\n4.Exponential function\n5.Dogarithmic function\n6. Trigonometric\nfunction\nKolomina N.N..jpg","smallImageUrl":"\/\/pedsovet.su\/_load-files\/load\/48\/64\/3\/f\/2-page -21_300.jpg"),("number":22,"text":"Linear function\n\ny = kx + b\ny\nb – free\ncoefficient\nk – angular\ncoefficient\n\nk = tan α \nKolomina N.N..jpg","smallImageUrl":"\/\/pedsovet.su\/_load-files\/load\/48\/64\/3\/f\/2-page-22_300. jpg"),("number":23,"text":"Quadratic function\n\ny = ax2 + bx + c, a ≠ 0\ny\n\n2\n\n b  b  4ac\nx1 ,2 \n2a\nb\nxв  \n2а\n4ac  b2\nyв \n4a\nKolomina N.N..jpg","smallImageUrl":"\/\/pedsovet.su\/_load-files\ /load\/48\/64\/3\/f\/2-page-23_300.jpg"),("number":24,"text":"Power function\n\ny = xn\n\ny \n\ny = xnn, where n = 2k, k  Z\n\ny = xnn, where n = 2k +1, k  Z\n\n1\n01\n\nKolomina N.N..jpg", "smallImageUrl":"\/\/pedsovet.su\/_load-files\/load\/48\/64\/3\/f\/2-page-24_300.jpg"),("number":25 ,"text":"Exponential function\nx\ny = a , a > 0, a ≠ 1\ny\n\ny=a\n01\n\nx\n\n1\nKolomina N.N..jpg", "smallImageUrl":"\/\/pedsovet.su\/_load-files\/load\/48\/64\/3\/f\/2-page-25_300.jpg"),("number":26 ,"text":"Logarithmic function\ny\n\ny = loga x , and >.jpg","smallImageUrl":"\/\/pedsovet.su\/_load-files\/load\/48\/64 \/3\/f\/2-page-26_300.jpg"),("number":27,"text":"Independent work\nBuild graphs of functions and find:\n1. D(y)-domain;\n2. E(y) is the set of its values;\n3. Check for parity (oddness);\n4. Find intervals of monotonicity and\nOption-1\nOption-2\nintervals\nof constancy of sign;\n1.\n5. Determine points 1.intersection with axes\n2.\n\n2.\n\n3.\n\n3.\n\n4.\n\n4.\n\n5.\n\n5.\n\nKolomina N.N..jpg" ,"smallImageUrl":"\/\/pedsovet.su\/_load-files\/load\/48\/64\/3\/f\/2-page-27_300.jpg"),("number": 28,"text":"Questions for review\n1. Formulate the definition of a function.\n2. What is called the domain of definition of a function?\n3. What is called the domain of change\nof a function?\n4. In what ways can a function be\nspecified?\n5. How is\nthe domain of definition of a function?\n6.What functions are called even and how are they examined for\nparity?\n7.What functions\nare called odd and how are they examined for oddity?\n8.Give examples\nof functions that are not even , nor odd.\n9.What functions are called\nincreasing? Give examples.\n10.What functions are called decreasing?\nGive examples.\n11.What functions are called inverse?\n12.How are the graphs of direct and\ninverse functions arranged?\n \nKolomina N.N..jpg","smallImageUrl":"\/\/pedsovet.su\/_load-files\/load\/48\/64\/3\/f\/2-page-28_300. jpg"),("number":29,"text":"Sources\nLinks to images:\nGraph:http:\/\/goldenbakes.com\/wordpress\/wpcontent\/uploads\/2013\/07\ /\nSectors_Investment_Funds.jpg\nChecked sheet: http:\/\/demeneva.ru\/rmk\/fon\/59.png\nTemplate author: Natalya Nikolaevna Kolomina, mathematics teacher\nMKOU "Khotkovskaya Secondary School" Duminichsky district, Kaluga region. \nPresentations:\nhttp:\/\/festival.1september.ru\/articles\/644838\/presentation\/pril.pptx Mukhina Galina\nGennadievna\nhttp:\/\/prezentacii.com\/matematike\/223- their graphics voystva-funkciy-i-ih-grafiki.html\nhttp:\/\/semenova-klass.moy.su\/_ld\/1\/122____.ppt Elena Yuryevna Semenova\nBogomolov N.V. Mathematics: textbook. for colleges\/ N.V. Bogomolov,\nP.I. Samoilenko.-3rd ed., stereotype.- M.: Bustard, 2005.-395 pp.\n\nKolomina N.N..jpg"," smallImageUrl":"\/\/pedsovet.su\/_load-files\/load\/48\/64\/3\/f\/2-page-29_300.jpg")]">

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Topic 1.4 Functions, their properties and graphs

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A little history The word “function” (from the Latin functio - accomplishment, execution) was first used in 1673 by the German mathematician Leibniz. In the main mathematical work “Geometry” (1637) Rene Descartes first introduced the concept of a variable quantity, created a method of coordinates, and introduced symbols for variable quantities (x, y, z, ...) Kolomin N.N. The definition of a function “A function of a variable quantity is an analytical expression composed in some way from this quantity and numbers or constant quantities” was made in 1748 by the German and Russian mathematician Leonhard Euler

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Definition. “The dependence of the variable y on the variable x, in which each value of the variable x corresponds to a single value of the variable y, is called a function.” y 6 5 4 3 2 1 x -6 -5 6 Symbolically, the functional relationship between the variable y (function) and the variable x (argument) is written using the equality y  f (x) -4 -3 -2 -1 -1 -2 -3 -4 -5 -6 Methods of specifying functions: tabular (table), graphical (graph), analytical (formula). Kolomina N.N. 0 1 2 3 4 5

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Find the domain of definition of the function whose graph is shown in the figure. 1 2 3 4 Think [-5;7) th! [-5;7]Think about it! (-3;5] Check (1) Kolomina N.N. y Think th! Correct! [-3;5] 5 -5 0 7 x -3 The domain of definition of the function is the values that the independent variable x takes.

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Set of function values. The set of values of a function is the set of all real values of the function y that it can take. For example, the set of values of the function y= x+1 is the set 2 R, y= X +1 the set of values of the function is the set of real numbers greater than or equal to 1. Kolomina N.N.

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Find the set of values of the function whose graph is shown in the figure. 1 2 Think about it! [-6;6] y 6 Think about it! [-4;6] That's right! -4 3 (-6;6) 4 Think about it! (-4;6) 0 6 x -6 Check (1) Kolomina N.N. The set of function values is the values that the dependent variable y takes.

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Study of the function for parity. A function y  f (x) is called even if, for all values of x in the domain of definition of this function, when the sign of the argument changes to the opposite, the value of the function does not change, i.e. . f ( x) parabola  f (x) y= X2 is an even For example, a function, because (-X2)= X2. The graph of an even function is symmetrical about the axis Kolomin N.N. OU.

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One of the following figures shows the graph of an even function. y y Specify this schedule. Think about it! Think about it! 1 0 x y 0 y x 2 Correct! Think about it! 3 Check (1) Kolomina N.N. 4 0 x 0 The graph is symmetrical about the Oy x axis

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A function y  f (x) is called odd if, for all values of x in the domain of definition of this function, when the sign of the argument changes to the opposite, the function changes only in sign, i.e. f ( x)  f (x) . For example, the function y = X3 is odd, because (-X)3 = -X3. The graph of an odd function is symmetrical about the origin. Not every function has the property of even or odd. For example, the function f (x)  X2+ X3 is neither even nor odd: f ( x)  (-X)2+ (-X)3 = X2 – X3; Kolomina N.N. X2 + X3 = / X2 – X3 ;

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One of the following figures shows the graph of an odd function. Please provide this schedule. y Right! Think about it! O 1 x y O Think! О Check (1) Kolomina N.N. 3 u Think! 2 x x O x 4 The graph is symmetrical about point O.

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Determination of intervals of increasing and decreasing 1 /\ /\ /\ /\ Among the many functions there are functions whose values only increase or only decrease with increasing argument. Such functions are called increasing or decreasing. A function is called increasing in the interval a x b if for any X1 and X2 belonging to this interval, for X1 X2 the inequality 2 /\ /\ /\ The function y  f (x) is said to be decreasing in the interval a x b if for any X1 and X2 belonging to this interval, for X1 X2 the inequality f (x1) > f (x2) takes place. Kolomin N.N.

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The figure shows a graph of the function y = f(x), specified on the interval (-5;6). Indicate the intervals where the function increases. Think 1 2 3rd! [-6;7] Think about it! [-5;-3] U Think! [-3;7] That's right! y 7 3 -5 -3 0 -2 4 [-3;2] -6 Check (1) Kolomina N.N. 2 6 x

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The figure shows a graph of the function y = f(x). Specify the number of zeros in the function. y Think! 1 1 2 2 3 4 4 0 Think about it! Right! x Think about it! Check (1) Kolomina N.N. 0 The zero of the function is the x value at which y = 0. In the figure, these are the points of intersection of the graph with the Ox axis.

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Which functions are increasing and which are decreasing? 1) y 5 x increasing, because 5  1 2) y 0.5 3) y 10 x x decreasing, because 0  0.5  1 increasing, because 10  1 aya, because  1 4) y  x increasing x  2 5) y    3 6) y 49 Kolomina N.N. x 2 decreasing, because 0   1 3 1 1 decreasing, because 49  and 0  1 49 49 1

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Study of a function for monotonicity. Both increasing and decreasing functions are called monotonic, and intervals in which the function increases or decreases are called monotonic intervals. /\ For example, the function y = X2 at x 0 increases monotonically. The function y = X3 monotonically increases on the entire numerical axis, and the function y = -X3 monotonically decreases on the entire numerical axis. Kolomina N.N.

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Examine the function for monotonicity of x y Function y=x2 -2 -1 0 4 1 0 1 1 2 4 y 6 5 4 3 2 1 -6 4 -5 5 -4 6 -3 -2 - -1 1 2 3 4 5 6 Kolomina N.N. 0 1 2 3 The function y=x2 x at x0 increases monotonically

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