Y is equal to the square root of X. Function y = square root of x, its properties and graph. Lesson and presentation on the topic: "Graph of the square root function. Domain of definition and construction of the graph"
Lesson and presentation on the topic: "Graph of the square root function. Domain of definition and construction of the graph"
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Educational aids and simulators in the Integral online store for grade 8
Electronic textbook for the textbook by Mordkovich A.G.
Electronic algebra workbook for 8th grade
Graph of the square root function
Guys, we have already met with constructing graphs of functions, and more than once. We constructed many linear functions and parabolas. In general, it is convenient to write any function as $y=f(x)$. This is an equation with two variables - for each value of x we get y. Having performed some given operation f, we map the set of all possible x to the set y. We can write almost any mathematical operation as a function f.Usually, when plotting function graphs, we use a table in which we record the values of x and y. For example, for the function $y=5x^2$ it is convenient to use the following table: Mark the resulting points on the Cartesian coordinate system and carefully connect them with a smooth curve. Our function is not limited. Only with these points can we substitute absolutely any value x from the given domain of definition, that is, those x for which the expression makes sense.
In one of the previous lessons, we learned a new operation for extracting the square root. The question arises: can we, using this operation, define some function and build its graph? Let's use the general form of the function $y=f(x)$. Let's leave y and x in their place, and instead of f we introduce the square root operation: $y=\sqrt(x)$.
Knowing the mathematical operation, we were able to define the function.
Graphing the Square Root Function
Let's graph this function. Based on the definition of the square root, we can calculate it only from non-negative numbers, that is, $x≥0$.Let's make a table:
Let's mark our points on the coordinate plane.
All we have to do is carefully connect the resulting dots.
Guys, pay attention: if the graph of our function is turned on its side, we get the left branch of a parabola. In fact, if the lines in the table of values are swapped (the top line with the bottom), then we get values just for the parabola.
Domain of the function $y=\sqrt(x)$
Using the graph of a function, it is quite easy to describe the properties.1. Scope of definition: $$.
b) $$.
Solution.
We can solve our example in two ways. In each letter we will describe different methods.
A) Let's return to the graph of the function constructed above and mark the required points of the segment. It is clearly seen that for $x=9$ the function is greater than all other values. This means that it reaches its greatest value at this point. When $x=4$ the value of the function is lower than all other points, which means that this is the smallest value.
$y_(most)=\sqrt(9)=3$, $y_(most)=\sqrt(4)=2$.
B) We know that our function is increasing. This means that each larger argument value corresponds to a larger function value. The highest and lowest values are achieved at the ends of the segment:
$y_(most)=\sqrt(11)$, $y_(most)=\sqrt(2)$.
Example 2.
Solve the equation:
$\sqrt(x)=12-x$.
Solution.
The easiest way is to construct two graphs of a function and find their point of intersection.
The intersection point with coordinates $(9;3)$ is clearly visible on the graph. This means that $x=9$ is the solution to our equation.
Answer: $x=9$.
Guys, can we be sure that this example has no more solutions? One of the functions increases, the other decreases. In general, they either do not have common points or intersect only at one.
Example 3.
Construct and read the graph of the function:
$\begin (cases) -x, x 9. \end (cases)$
We need to construct three partial graphs of the function, each on its own interval.
Let's describe the properties of our function:
1. Domain of definition: $(-∞;+∞)$.
2. $y=0$ for $x=0$ and $x=12$; $у>0$ for $хϵ(-∞;12)$; $y 3. The function decreases on the intervals $(-∞;0)U(9;+∞)$. The function is increasing on the interval $(0;9)$.
4. The function is continuous over the entire domain of definition.
5. There is no maximum or minimum value.
6. Range of values: $(-∞;+∞)$.
Problems to solve independently
1. Find the largest and smallest value of the square root function on the segment:a) $$;
b) $$.
2. Solve the equation: $\sqrt(x)=30-x$.
3. Construct and read the graph of the function: $\begin (cases) 2-x, x 4. \end (cases)$
4. Construct and read the graph of the function: $y=\sqrt(-x)$.
Did you look for x root of x equals? . A detailed solution with description and explanations will help you deal with even the most complex problem, and x is the root of y, no exception. We will help you prepare for homework, tests, olympiads, as well as for entering a university.
And no matter what example, no matter what math query you enter, we already have a solution.
For example, “x is the root of x equals.”
The use of various mathematical problems, calculators, equations and functions is widespread in our lives. They are used in many calculations, construction of structures and even sports. Man has used mathematics since ancient times and since then their use has only increased. However, now science does not stand still and we can enjoy the fruits of its activity, such as an online calculator that can solve problems such as x root of x equals, x root of y, root of x, root of x is equal to x, root of x is equal to x, root of x is equal to x, function y is root of minus x, function y minus root of x, x is root of y, x is root of x is equal to. On this page you will find a calculator that will help you solve any question, including x root of x equals. (for example, root of x).
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You can solve the problem x root of x equals on our website. The free online solver will allow you to solve an online problem of any complexity in a matter of seconds. All you need to do is simply enter your data into the solver. You can also watch the video instructions and learn how to correctly enter your task on our website. And if you still have questions, you can ask them in the chat at the bottom left of the calculator page.
Basic goals:
1) form an idea of the feasibility of a generalized study of the dependencies of real quantities using the example of quantities related by the relation y=
2) to develop the ability to construct a graph y= and its properties;
3) repeat and consolidate the techniques of oral and written calculations, squaring, extracting square roots.
Equipment, demonstration material: handouts.
1. Algorithm:
2. Sample for completing the task in groups:
3. Sample for self-test of independent work:
4. Card for the reflection stage:
1) I understood how to graph the function y=.
2) I can list its properties using a graph.
3) I did not make mistakes in independent work.
1. Self-determination for educational activities
Purpose of the stage:
1) include students in educational activities;
2) determine the content of the lesson: we continue to work with real numbers.
Organization of the educational process at stage 1:
– What did we study in the last lesson? (We studied the set of real numbers, operations with them, built an algorithm to describe the properties of a function, repeated the functions studied in 7th grade).
– Today we will continue to work with a set of real numbers, a function.
2. Updating knowledge and recording difficulties in activities
Purpose of the stage:
1) update educational content that is necessary and sufficient for the perception of new material: function, independent variable, dependent variable, graphs
y = kx + m, y = kx, y =c, y =x 2, y = - x 2,
2) update mental operations necessary and sufficient for the perception of new material: comparison, analysis, generalization;
3) record all repeated concepts and algorithms in the form of diagrams and symbols;
4) record an individual difficulty in activity, demonstrating at a personally significant level the insufficiency of existing knowledge.
Organization of the educational process at stage 2:
1. Let's remember how you can set dependencies between quantities? (Using text, formula, table, graph)
2. What is a function called? (A relationship between two quantities, where each value of one variable corresponds to a single value of another variable y = f(x)).
What is the name of x? (Independent variable - argument)
What is the name of y? (Dependent variable).
3. In 7th grade did we study functions? (y = kx + m, y = kx, y =c, y =x 2, y = - x 2,).
Individual task:
What is the graph of the functions y = kx + m, y =x 2, y =?
3. Identifying the causes of difficulties and setting goals for activities
Purpose of the stage:
1) organize communicative interaction, during which the distinctive property of the task that caused difficulty in learning activities is identified and recorded;
2) agree on the purpose and topic of the lesson.
Organization of the educational process at stage 3:
-What's special about this task? (The dependence is given by the formula y = which we have not yet encountered.)
– What is the purpose of the lesson? (Get acquainted with the function y =, its properties and graph. Use the function in the table to determine the type of dependence, build a formula and graph.)
– Can you formulate the topic of the lesson? (Function y=, its properties and graph).
– Write the topic in your notebook.
4. Construction of a project for getting out of a difficulty
Purpose of the stage:
1) organize communicative interaction to build a new method of action that eliminates the cause of the identified difficulty;
2) fix a new method of action in a symbolic, verbal form and with the help of a standard.
Organization of the educational process at stage 4:
Work at this stage can be organized in groups, asking the groups to construct a graph y =, then analyze the results. Groups can also be asked to describe the properties of a given function using an algorithm.
5. Primary consolidation in external speech
The purpose of the stage: to record the studied educational content in external speech.
Organization of the educational process at stage 5:
Construct a graph of y= - and describe its properties.
Properties y= - .
1.Domain of definition of a function.
2. Range of values of the function.
3. y = 0, y> 0, y<0.
y =0 if x = 0.
y<0, если х(0;+)
4.Increasing, decreasing functions.
The function decreases as x.
Let's build a graph of y=.
Let's select its part on the segment. Note that we have = 1 for x = 1, and y max. =3 at x = 9.
Answer: at our name. = 1, y max. =3
6. Independent work with self-test according to the standard
The purpose of the stage: to test your ability to apply new educational content in standard conditions based on comparing your solution with a standard for self-test.
Organization of the educational process at stage 6:
Students complete the task independently, conduct a self-test against the standard, analyze, and correct errors.
Let's build a graph of y=.
Using a graph, find the smallest and largest values of the function on the segment.
7. Inclusion in the knowledge system and repetition
The purpose of the stage: to train the skills of using new content together with previously studied: 2) repeat the educational content that will be required in the following lessons.
Organization of the educational process at stage 7:
Solve the equation graphically: = x – 6.
One student is at the blackboard, the rest are in notebooks.
8. Reflection of activity
Purpose of the stage:
1) record new content learned in the lesson;
2) evaluate your own activities in the lesson;
3) thank classmates who helped get the lesson result;
4) record unresolved difficulties as directions for future educational activities;
5) discuss and write down your homework.
Organization of the educational process at stage 8:
- Guys, what was our goal today? (Study the function y=, its properties and graph).
– What knowledge helped us achieve our goal? (Ability to look for patterns, ability to read graphs.)
– Analyze your activities in class. (Cards with reflection)
Homework
paragraph 13 (before example 2) № 13.3, 13.4
Solve the equation graphically.
The basic properties of the power function are given, including formulas and properties of the roots. The derivative, integral, power series expansion, and complex number representation of a power function are presented.
ContentA power function, y = x p, with exponent p has the following properties:
(1.1)
defined and continuous on the set
at ,
at ;
(1.2)
has many meanings
at ,
at ;
(1.3)
strictly increases with ,
strictly decreases at ;
(1.4)
at ;
at ;
(1.5)
;
(1.5*)
;
(1.6)
;
(1.7)
;
(1.7*)
;
(1.8)
;
(1.9)
.
Proof of properties is given on the page “Power function (proof of continuity and properties)”
Roots - definition, formulas, properties
A root of a number x of power n is a number that when raised to the power n gives x:.
Here n = 2, 3, 4, ... - a natural number greater than one.
You can also say that the root of a number x of degree n is the root (i.e. solution) of the equation
.
Note that the function is the inverse of the function.
The square root of x is the 2 root: .
The cube root of x is the 3rd root: .
Even degree
For even powers n = 2 m, the root is defined for x ≥ 0
.
.
A formula that is often used is valid for both positive and negative x:
.
For square root:
The order in which the operations are performed is important here - that is, first the square is performed, resulting in a non-negative number, and then the root is taken from it (the square root can be taken from a non-negative number). If we changed the order: , then for negative x the root would be undefined, and with it the entire expression would be undefined.
Odd degree
;
.
For odd powers, the root is defined for all x:
Properties and formulas of roots
.
The root of x is a power function: 0
When x ≥
;
;
,
;
.
the following formulas apply:
These formulas can also be applied for negative values of variables.
You just need to make sure that the radical expression of even powers is not negative.
Private values
The root of 0 is 0: .
Root 1 is equal to 1: .
The square root of 0 is 0: .
The square root of 1 is 1: .
.
Example. Root of roots
.
Let's look at an example of a square root of roots:
.
Let's transform the inner square root using the formulas above:
.
Now let's transform the original root:
So,
y = x p for different values of the exponent p.
The inverse of a power function with exponent p is a power function with exponent 1/p.
If, then.
Derivative of a power function
Derivative of nth order:
;
Deriving formulas > > >
Integral of a power function
P ≠ - 1
;
.
Power series expansion
At - 1
< x < 1
the following decomposition takes place:
Expressions using complex numbers
Consider the function of the complex variable z:
f (z) = z t.
Let us express the complex variable z in terms of the modulus r and the argument φ (r = |z|):
z = r e i φ .
We represent the complex number t in the form of real and imaginary parts:
t = p + i q .
We have:
Next, we take into account that the argument φ is not uniquely defined:
,
Let's consider the case when q = 0
, that is, the exponent is a real number, t = p.
.
Then
.
If p is an integer, then kp is an integer. Then, due to the periodicity of trigonometric functions:
That is, the exponential function with an integer exponent, for a given z, has only one value and is therefore unambiguous. If p is irrational, then the products kp for any k do not produce an integer. Since k runs through an infinite series of values k = 0, 1, 2, 3, ... , then the function z p has infinitely many values. Whenever the argument z is incremented 2π
(one turn), we move to a new branch of the function.
If p is rational, then it can be represented as: , Where m, n
.
- integers that do not contain common divisors. Then First n values, with k = k 0 = 0, 1, 2, ... n-1
.
, give n different values of kp: However, subsequent values give values that differ from the previous ones by an integer. For example, when k = k 0+n
.
we have: Trigonometric functions whose arguments differ by multiples of 2π First n values, with k = k.
, have equal values. Therefore, with a further increase in k, we obtain the same values of z p as for k = k Trigonometric functions whose arguments differ by multiples of Thus, an exponential function with a rational exponent is multivalued and has n values (branches). Whenever the argument z is incremented
(one turn), we move to a new branch of the function. After n such revolutions we return to the first branch from which the countdown began. In particular, a root of degree n has n values. As an example, consider the nth root of a real positive number z = x.,
.
.
In this case φ 2
,
.
0 = 0 , z = r = |z| = x So, for a square root, n = For even k, (- 1 ) k = 1.
.
For odd k,
I.N. Bronstein, K.A. Semendyaev, Handbook of mathematics for engineers and college students, “Lan”, 2009.
Where can you solve any problem in mathematics, as well as x root of x equals Online?
You can solve the problem x root of x equals on our website. The free online solver will allow you to solve an online problem of any complexity in a matter of seconds. All you need to do is simply enter your data into the solver. You can also watch the video instructions and learn how to correctly enter your task on our website. And if you still have questions, you can ask them in the chat at the bottom left of the calculator page.
Basic goals:
1) form an idea of the feasibility of a generalized study of the dependencies of real quantities using the example of quantities related by the relation y=
2) to develop the ability to construct a graph y= and its properties;
3) repeat and consolidate the techniques of oral and written calculations, squaring, extracting square roots.
Equipment, demonstration material: handouts.
1. Algorithm:
2. Sample for completing the task in groups:
3. Sample for self-test of independent work:
4. Card for the reflection stage:
1) I understood how to graph the function y=.
2) I can list its properties using a graph.
3) I did not make mistakes in independent work.
1. Self-determination for educational activities
Purpose of the stage:
1) include students in educational activities;
2) determine the content of the lesson: we continue to work with real numbers.
Organization of the educational process at stage 1:
– What did we study in the last lesson? (We studied the set of real numbers, operations with them, built an algorithm to describe the properties of a function, repeated the functions studied in 7th grade).
– Today we will continue to work with a set of real numbers, a function.
2. Updating knowledge and recording difficulties in activities
Purpose of the stage:
1) update educational content that is necessary and sufficient for the perception of new material: function, independent variable, dependent variable, graphs
y = kx + m, y = kx, y =c, y =x 2, y = - x 2,
2) update mental operations necessary and sufficient for the perception of new material: comparison, analysis, generalization;
3) record all repeated concepts and algorithms in the form of diagrams and symbols;
4) record an individual difficulty in activity, demonstrating at a personally significant level the insufficiency of existing knowledge.
Organization of the educational process at stage 2:
1. Let's remember how you can set dependencies between quantities? (Using text, formula, table, graph)
2. What is a function called? (A relationship between two quantities, where each value of one variable corresponds to a single value of another variable y = f(x)).
What is the name of x? (Independent variable - argument)
What is the name of y? (Dependent variable).
3. In 7th grade did we study functions? (y = kx + m, y = kx, y =c, y =x 2, y = - x 2,).
Individual task:
What is the graph of the functions y = kx + m, y =x 2, y =?
3. Identifying the causes of difficulties and setting goals for activities
Purpose of the stage:
1) organize communicative interaction, during which the distinctive property of the task that caused difficulty in learning activities is identified and recorded;
2) agree on the purpose and topic of the lesson.
Organization of the educational process at stage 3:
-What's special about this task? (The dependence is given by the formula y = which we have not yet encountered.)
– What is the purpose of the lesson? (Get acquainted with the function y =, its properties and graph. Use the function in the table to determine the type of dependence, build a formula and graph.)
– Can you formulate the topic of the lesson? (Function y=, its properties and graph).
– Write the topic in your notebook.
4. Construction of a project for getting out of a difficulty
Purpose of the stage:
1) organize communicative interaction to build a new method of action that eliminates the cause of the identified difficulty;
2) fix a new method of action in a symbolic, verbal form and with the help of a standard.
Organization of the educational process at stage 4:
Work at this stage can be organized in groups, asking the groups to construct a graph y =, then analyze the results. Groups can also be asked to describe the properties of a given function using an algorithm.
5. Primary consolidation in external speech
The purpose of the stage: to record the studied educational content in external speech.
Organization of the educational process at stage 5:
Construct a graph of y= - and describe its properties.
Properties y= - .
1.Domain of definition of a function.
2. Range of values of the function.
3. y = 0, y> 0, y<0.
y =0 if x = 0.
y<0, если х(0;+)
4.Increasing, decreasing functions.
The function decreases as x.
Let's build a graph of y=.
Let's select its part on the segment. Note that we have = 1 for x = 1, and y max. =3 at x = 9.
Answer: at our name. = 1, y max. =3
6. Independent work with self-test according to the standard
The purpose of the stage: to test your ability to apply new educational content in standard conditions based on comparing your solution with a standard for self-test.
Organization of the educational process at stage 6:
Students complete the task independently, conduct a self-test against the standard, analyze, and correct errors.
Let's build a graph of y=.
Using a graph, find the smallest and largest values of the function on the segment.
7. Inclusion in the knowledge system and repetition
The purpose of the stage: to train the skills of using new content together with previously studied: 2) repeat the educational content that will be required in the following lessons.
Organization of the educational process at stage 7:
Solve the equation graphically: = x – 6.
One student is at the blackboard, the rest are in notebooks.
8. Reflection of activity
Purpose of the stage:
1) record new content learned in the lesson;
2) evaluate your own activities in the lesson;
3) thank classmates who helped get the lesson result;
4) record unresolved difficulties as directions for future educational activities;
5) discuss and write down your homework.
Organization of the educational process at stage 8:
- Guys, what was our goal today? (Study the function y=, its properties and graph).
– What knowledge helped us achieve our goal? (Ability to look for patterns, ability to read graphs.)
– Analyze your activities in class. (Cards with reflection)
Homework
paragraph 13 (before example 2) № 13.3, 13.4
Solve the equation graphically.
