Basic elementary functions, their properties and graphs. Functions and graphs Irrational functions plotting
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Basic elementary functions are: constant function (constant), root n-th degree, power function, exponential, logarithmic function, trigonometric and inverse trigonometric functions.
Permanent function.
A constant function is given on the set of all real numbers by the formula , where C– some real number. A constant function assigns each actual value of the independent variable x same value of the dependent variable y- meaning WITH. A constant function is also called a constant.
The graph of a constant function is a straight line parallel to the x-axis and passing through the point with coordinates (0,C). For example, let's show graphs of constant functions y=5,y=-2 and , which in the figure below correspond to the black, red and blue lines, respectively.
Properties of a constant function.
Domain: the entire set of real numbers.
The constant function is even.
Range of values: set consisting of a singular number WITH.
A constant function is non-increasing and non-decreasing (that’s why it’s constant).
It makes no sense to talk about convexity and concavity of a constant.
There are no asymptotes.
The function passes through the point (0,C) coordinate plane.
Root of the nth degree.
Let's consider the basic elementary function, which is given by the formula, where n– a natural number greater than one.
The nth root, n is an even number.
Let's start with the root function n-th power for even values of the root exponent n.
As an example, here is a picture with images of function graphs 
and , they correspond to black, red and blue lines.
The graphs of even-degree root functions have a similar appearance for other values of the exponent.
Properties of the root functionn -th power for evenn .
The nth root, n is an odd number.
Root function n-th power with an odd root exponent n is defined on the entire set of real numbers. For example, here are the function graphs 
and , they correspond to black, red and blue curves.
Lesson topic:Graphing functions containing modules. Introduction to IF and FunctionsABS.
Mathematics and computer science teacher, Secondary School No. 2, Novobelokatay village, Belokataysky district, Yulia Rafailovna Galiullina.
Textbook “Algebra and the beginnings of mathematical analysis. 10-11 grade" ed. Kolmogorova, Ugrinovich N.D. "Informatics and ICT 10th grade."
Lesson type: training lesson using information technology.
The purpose of the lesson: test knowledge, skills and abilities on this topic.
Lesson objectives:
Educational
systematization and generalization of knowledge on this topic;
teach to determine the most convenient solution method;
teach how to graph a function using a spreadsheet.
Developmental
development of self-control ability;
activation of students’ mental activity;
Educational
nurturing learning motives and a conscientious attitude to work.
Teaching methods: partially search, research, individual.
Form of organization of educational activities: individual, frontal, cards.
Means of education: multimedia projector, screen, cards
During the classes
I. Organizing time
Greetings, checking those present. Explanation of the lesson
II. Repetition
Consolidating knowledge of plotting graphs in a spreadsheet processor.
Frontal survey.
-How to insert a graph into Excel?
- What types of graphs exist in Excel?
Consolidating knowledge on the topic chart with modules.
- What is the meaning of a function with a module?
Example analysis: y = | x | – 2.
There are two cases to consider when x=0. If x=0, then the function will look like y = x – 2. Construct a graph of this function in your notebooks.
Now let’s build a graph of the function using the MS Excel spreadsheet processor. This function can be graphed in two ways:
Method 1: Using the IF function
In order to build a graph, we first need to fill out a table of X and Y values.
We call cell A2-X, cell B2-U. Therefore, column A will contain the value of the variable, and column B will contain the value of the function.
In column A we enter a variable in the range from -5 to 5 in increments of 0.5. To do this, enter -5 in cell A3, and the formula =A4+0.5 in cell A4, copy the formula to subsequent cells, since here there is relative addressing, the formula will change when copied.
After filling in the X values, move on to the second column, to fill which you need to enter a formula. In cell B4, enter a formula in which we use the IF function.
Function " If" in MS Excel spreadsheets (Category - Boolean) analyzes the result of an expression or the contents of a specified cell and places one of two possible values or expressions in the specified cell.
Syntax of the "IF" function.
=IF (Boolean expression; Value_if_true; Value_if_false). A Boolean expression or condition that can evaluate to TRUE or FALSE. Value_if_true – the value that the logical expression takes if it is executed. Value_if_false is the value that the Boolean expression takes if it fails."
Logical expressions or conditions are constructed using comparison operators (, =, =) and logical operations (AND, OR, NOT).
Fig.22 IF function
The IF function is a logical function.
Let’s remember the meaning of a function with a modulus: if x=0, then the function will look like y = x – 2.
This wording must be entered into cell B4 in a clear table form. The value of X is in column A, therefore if A4
A4-2, otherwise = A4-2.
Fig.23 Arguments of the IF function
The formula looks like: =IF(A5A5-2,A5-2)
After filling out the table of values. Building a graph of a function
Menu item Insert-Diagrams-Scatter. Select one of the layouts. An empty chart field appears on the worksheet. In the context menu of this field, select Select data. The Select Data dialog box appears.
In this dialog box, select the name of the series in cell A1, or you can also enter the name from the keyboard.
In the X value field, select the column in which we entered the variable value.
In the Y value field, select the column in which we found the value of the function using the conditional IF operator.
Rice. 24. Graph of the function y = | x | – 2.
Method 2: Using a functionABS
You can also use the ABS function to build a graph with a module.
Let's plot the function y = | x | – 2 using the ABS function.
In example 2 the values of the variable X are given.
In cell B4, enter a formula using the ABC function
Fig.25. Entering the ABS function using the function wizard
The formula will look like: =ABS(A4)-2.
IV. Doing practical work
After analyzing two examples, students are given a practical task.
In these tasks you are given several functions with modules. You must choose which function is more appropriate to use in each example.
Practical work
Students consider the linear function y = x – 2 and graph it.
Task 1. Graph the function y = | x – 2 |
Task 2. Graph the function y = | x | – 2
Task 3. Graph the equation | y | = x – 2
Students consider the quadratic function y = x 2 – 2x – 3 and build a graph.
Task 1. Graph the function y = | x 2 – 2x – 3 |
Task 2. Graph the function y = | x 2 | – 2 | x | - 3
Task 3. Graph the equation | y | = x 2 – 2x - 3
V. Information about homework.
VI.Summing up the lesson, reflection. Students and the teacher summarize the lesson and analyze the implementation of the assigned tasks.
