What does the root of x function look like? Square root function graph, graph transformations
Basic goals:
1) to form an idea of the expediency of a generalized study of the dependences of real quantities on the example of quantities related by the relation y=
2) to form the ability to plot y= and its properties;
3) repeat and consolidate the methods of oral and written calculations, squaring, extracting the square root.
Equipment, demo material: Handout.
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 figured out how to graph the function y=.
2) I can list its properties according to the schedule.
3) I did not make mistakes in my independent work.
4) I made mistakes in independent work (list these mistakes and indicate their reason).
During the classes
1. Self-determination to learning activities
Purpose of the stage:
1) include students in learning 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, actions with them, built an algorithm for describing the properties of a function, repeated the functions studied in grade 7).
– Today we will continue to work with the set of real numbers, a function.
2. Updating knowledge and fixing difficulties in activities
Purpose of the stage:
1) update the educational content necessary and sufficient for the perception of new material: function, independent variable, dependent variable, graphs
y \u003d kx + m, y \u003d kx, y \u003d c, y \u003d x 2, y \u003d - x 2,
2) to update the mental operations necessary and sufficient for the perception of new material: comparison, analysis, generalization;
3) fix all repeated concepts and algorithms in the form of schemes and symbols;
4) to fix an individual difficulty in activity, demonstrating the insufficiency of existing knowledge at a personally significant level.
Organization of the educational process at stage 2:
1. Let's remember how you can set the dependencies between the quantities? (Via text, formula, table, graph)
2. What is called a function? (The relationship between two quantities, where each value of one variable corresponds to a single value of the other variable y = f(x)).
What is x called? (Independent variable - argument)
What is the name of u? (Dependent variable).
3. Did we learn functions in 7th grade? (y = kx + m, y = kx, y =c, y =x 2 , y = - x 2 , ).
Individual task:
What is the graph of functions y = kx + m, y =x 2 , y = ?
3. Identification of the causes of difficulties and setting the goal of the activity
Purpose of the stage:
1) organize communicative interaction, during which the distinctive property of the task that caused difficulty in educational activities is revealed and fixed;
2) agree on the purpose and topic of the lesson.
Organization of the educational process at stage 3:
What is special about this task? (The dependence is given by the formula y = which we have not met yet).
- What is the purpose of the lesson? (Get acquainted with the function y \u003d, its properties and graph. The function in the table determines the type of dependence, build a formula and graph.)
- Can you guess the topic of the lesson? (Function y=, its properties and graph).
- Write the topic in your notebook.
4. Building a project for getting out of a difficulty
Purpose of the stage:
1) organize communicative interaction to build a new mode of action that eliminates the cause of the identified difficulty;
2) fix new way actions in a sign, verbal form and with the help of a standard.
Organization of the educational process at stage 4:
The work at the stage can be organized into groups by inviting the groups to plot y = , then analyze the results. Also, groups can be offered to describe the properties of this function according to the algorithm.
5. Primary consolidation in external speech
The purpose of the stage: to fix the studied educational content in external speech.
Organization of the educational process at stage 5:
Build a graph y= - and describe its properties.
Properties y= - .
1.Scope of function definition.
2.Scope of function values.
3. y=0, y>0, y<0.
y=0 if x=0.
y<0, если х(0;+)
4.Increase, decrease function.
The function is decreasing at x.
Let's plot y=.
Let's select its part on the segment . Let us note that at Naim. = 1 for x = 1, and y max. \u003d 3 for x \u003d 9.
Answer: naim. = 1, at the max. =3
6. Independent work with self-test according to the standard
The purpose of the stage: to test your ability to apply the new learning content in typical conditions based on comparing your solution with a standard for self-testing.
Organization of the educational process at stage 6:
Students perform the task on their own, conduct a self-test according to the standard, analyze, correct errors.
Let's plot y=.
Using the 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 in conjunction with previously learned: 2) repeat the learning content that will be required in the following lessons.
Organization of the educational process at stage 7:
Solve graphically the equation: \u003d x - 6.
One student at the blackboard, the rest in notebooks.
8. Reflection of activity
Purpose of the stage:
1) fix the new content learned in the lesson;
2) evaluate their own activities in the lesson;
3) thank classmates who helped to get the result of the lesson;
4) fix unresolved difficulties as directions for future learning activities;
5) Discuss and write down homework.
Organization of the educational process at stage 8:
- Guys, what was the goal for us today? (Study the function y \u003d, its properties and graph).
- What knowledge helped us achieve the goal? (The ability to look for patterns, the ability to read graphs.)
- Review your activities in class. (Reflection cards)
Homework
item 13 (up to example 2) № 13.3, 13.4
Solve the equation graphically.
Basic goals:
1) to form an idea of the expediency of a generalized study of the dependences of real quantities on the example of quantities related by the relation y=
2) to form the ability to plot y= and its properties;
3) repeat and consolidate the methods of oral and written calculations, squaring, extracting the square root.
Equipment, demonstration material: handout.
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 figured out how to graph the function y=.
2) I can list its properties according to the schedule.
3) I did not make mistakes in my independent work.
4) I made mistakes in independent work (list these mistakes and indicate their reason).
During the classes
1. Self-determination to learning activities
Purpose of the stage:
1) include students in learning 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, actions with them, built an algorithm for describing the properties of a function, repeated the functions studied in grade 7).
– Today we will continue to work with the set of real numbers, a function.
2. Updating knowledge and fixing difficulties in activities
Purpose of the stage:
1) update the educational content necessary and sufficient for the perception of new material: function, independent variable, dependent variable, graphs
y \u003d kx + m, y \u003d kx, y \u003d c, y \u003d x 2, y \u003d - x 2,
2) to update the mental operations necessary and sufficient for the perception of new material: comparison, analysis, generalization;
3) fix all repeated concepts and algorithms in the form of schemes and symbols;
4) to fix an individual difficulty in activity, demonstrating the insufficiency of existing knowledge at a personally significant level.
Organization of the educational process at stage 2:
1. Let's remember how you can set the dependencies between the quantities? (Via text, formula, table, graph)
2. What is called a function? (The relationship between two quantities, where each value of one variable corresponds to a single value of the other variable y = f(x)).
What is x called? (Independent variable - argument)
What is the name of u? (Dependent variable).
3. Did we learn functions in 7th grade? (y = kx + m, y = kx, y =c, y =x 2 , y = - x 2 , ).
Individual task:
What is the graph of functions y = kx + m, y =x 2 , y = ?
3. Identification of the causes of difficulties and setting the goal of the activity
Purpose of the stage:
1) organize communicative interaction, during which the distinctive property of the task that caused difficulty in educational activities is revealed and fixed;
2) agree on the purpose and topic of the lesson.
Organization of the educational process at stage 3:
What is special about this task? (The dependence is given by the formula y = which we have not met yet).
- What is the purpose of the lesson? (Get acquainted with the function y \u003d, its properties and graph. The function in the table determines the type of dependence, build a formula and graph.)
- Can you guess the topic of the lesson? (Function y=, its properties and graph).
- Write the topic in your notebook.
4. Building a project for getting out of a difficulty
Purpose of the stage:
1) organize communicative interaction to build a new mode of action that eliminates the cause of the identified difficulty;
2) fix a new mode of action in a sign, verbal form and with the help of a standard.
Organization of the educational process at stage 4:
The work at the stage can be organized into groups by inviting the groups to plot y = , then analyze the results. Also, groups can be offered to describe the properties of this function according to the algorithm.
5. Primary consolidation in external speech
The purpose of the stage: to fix the studied educational content in external speech.
Organization of the educational process at stage 5:
Build a graph y= - and describe its properties.
Properties y= - .
1.Scope of function definition.
2.Scope of function values.
3. y=0, y>0, y<0.
y=0 if x=0.
y<0, если х(0;+)
4.Increase, decrease function.
The function is decreasing at x.
Let's plot y=.
Let's select its part on the segment . Let us note that at Naim. = 1 for x = 1, and y max. \u003d 3 for x \u003d 9.
Answer: naim. = 1, at the max. =3
6. Independent work with self-test according to the standard
The purpose of the stage: to test your ability to apply the new learning content in typical conditions based on comparing your solution with a standard for self-testing.
Organization of the educational process at stage 6:
Students perform the task on their own, conduct a self-test according to the standard, analyze, correct errors.
Let's plot y=.
Using the 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 in conjunction with previously learned: 2) repeat the learning content that will be required in the following lessons.
Organization of the educational process at stage 7:
Solve graphically the equation: \u003d x - 6.
One student at the blackboard, the rest in notebooks.
8. Reflection of activity
Purpose of the stage:
1) fix the new content learned in the lesson;
2) evaluate their own activities in the lesson;
3) thank classmates who helped to get the result of the lesson;
4) fix unresolved difficulties as directions for future learning activities;
5) Discuss and write down homework.
Organization of the educational process at stage 8:
- Guys, what was the goal for us today? (Study the function y \u003d, its properties and graph).
- What knowledge helped us achieve the goal? (The ability to look for patterns, the ability to read graphs.)
- Review your activities in class. (Reflection cards)
Homework
item 13 (up to example 2) № 13.3, 13.4
Solve the equation graphically.
Lesson and presentation on the topic: "Power functions. Cubic root. Properties of the cubic root"
Additional materials
Dear users, do not forget to leave your comments, feedback, suggestions! All materials are checked by an antivirus program.
Teaching aids and simulators in the online store "Integral" for grade 9
Educational complex 1C: "Algebraic problems with parameters, grades 9-11" Software environment "1C: Mathematical constructor 6.0"
Definition of a power function - cube root
Guys, we continue to study power functions. Today we are going to talk about the Cube Root of x function.What is a cube root?
A number y is called a cube root of x (third degree root) if $y^3=x$ is true.
They are denoted as $\sqrt(x)$, where x is the root number, 3 is the exponent.
$\sqrt(27)=3$; $3^3=27$.
$\sqrt((-8))=-2$; $(-2)^3=-8$.
As we can see, the cube root can also be extracted from negative numbers. It turns out that our root exists for all numbers.
The third root of a negative number is equal to a negative number. When raised to an odd power, the sign is preserved, the third power is odd.
Let's check the equality: $\sqrt((-x))$=-$\sqrt(x)$.
Let $\sqrt((-x))=a$ and $\sqrt(x)=b$. Let's raise both expressions to the third power. $–x=a^3$ and $x=b^3$. Then $a^3=-b^3$ or $a=-b$. In the notation of the roots, we obtain the desired identity.
Properties of cube roots
a) $\sqrt(a*b)=\sqrt(a)*\sqrt(6)$.b) $\sqrt(\frac(a)(b))=\frac(\sqrt(a))(\sqrt(b))$.
Let's prove the second property. $(\sqrt(\frac(a)(b)))^3=\frac(\sqrt(a)^3)(\sqrt(b)^3)=\frac(a)(b)$.
We found that the number $\sqrt(\frac(a)(b))$ in the cube is equal to $\frac(a)(b)$ and then it is equal to $\sqrt(\frac(a)(b))$, which and needed to be proven.
Guys, let's plot our function graph.
1) The domain of definition is the set of real numbers.
2) The function is odd because $\sqrt((-x))$=-$\sqrt(x)$. Next, consider our function for $x≥0$, then reflect the graph relative to the origin.
3) The function increases for $х≥0$. For our function, a larger value of the argument corresponds to a larger value of the function, which means increasing.
4) The function is not limited from above. In fact, from an arbitrarily large number, you can calculate the root of the third degree, and we can move up to infinity, finding ever larger values of the argument.
5) For $x≥0$, the smallest value is 0. This property is obvious.
Let's build a graph of the function by points for x≥0.
Let's build our graph of the function on the entire domain of definition. Remember that our function is odd. 
Function properties:
1) D(y)=(-∞;+∞).
2) Odd function.
3) Increases by (-∞;+∞).
4) Unlimited.
5) There is no minimum or maximum value.
7) E(y)= (-∞;+∞).
8) Convex downwards by (-∞;0), convex upwards by (0;+∞).
Examples of solving power functions
Examples1. Solve the equation $\sqrt(x)=x$.
Solution. Let's build two graphs on the same coordinate plane $y=\sqrt(x)$ and $y=x$.
As you can see, our graphs intersect at three points. Answer: (-1;-1), (0;0), (1;1).
2. Build a graph of the function. $y=\sqrt((x-2))-3$.
Solution. Our graph is obtained from the graph of the function $y=\sqrt(x)$, by parallel shifting two units to the right and three units down. 
3. Build a function graph and read it. $\begin(cases)y=\sqrt(x), x≥-1\\y=-x-2, x≤-1 \end(cases)$.
Solution. Let's build two graphs of functions on the same coordinate plane, taking into account our conditions. For $х≥-1$ we build a graph of a cubic root, for $х≤-1$ a graph of a linear function. 
1) D(y)=(-∞;+∞).
2) The function is neither even nor odd.
3) Decreases by (-∞;-1), increases by (-1;+∞).
4) Unlimited from above, limited from below.
5) There is no maximum value. The smallest value is minus one.
6) The function is continuous on the entire real line.
7) E(y)= (-1;+∞).
Tasks for independent solution
1. Solve the equation $\sqrt(x)=2-x$.2. Plot the function $y=\sqrt((x+1))+1$.
3. Build a graph of the function and read it. $\begin(cases)y=\sqrt(x), x≥1\\y=(x-1)^2+1, x≤1 \end(cases)$.
Lesson and presentation on the topic: "Graph of the square root function. Scope and plotting"
Additional materials
Dear users, do not forget to leave your comments, feedback, suggestions. All materials are checked by an antivirus program.
Teaching aids and simulators in the online store "Integral" for grade 8
Electronic textbook for the textbook Mordkovich A.G.
Algebra electronic workbook for grade 8
Graph of the square root function
Guys, we have already met with the construction of graphs of functions, and more than once. We have built sets of linear functions and parabolas. In general, it is convenient to write any function as $y=f(x)$. This is a two-variable equation - for each value of x, we get y. After performing some given operation f, we map the set of all possible x to the set y. As a function f, we can write almost any mathematical operation.Usually, when plotting functions, we use a table in which we write down the x and y values. For example, for the function $y=5x^2$, it is convenient to use the following table: Mark the obtained 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 of 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 of extracting the square root. The question arises, can we, using this operation, set some function and build its graph? Let's use the general form of the function $y=f(x)$. We 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.
Plotting the Square Root Function
Let's plot this function. Based on the definition of the square root, we can only calculate it from non-negative numbers, that is, $x≥0$.Let's make a table:
Let's mark our points on the coordinate plane.
It remains for us to carefully connect the obtained points.
Guys, pay attention: if the graph of our function is turned on its side, then we get the left branch of the parabola. In fact, if the lines in the table of values are interchanged (the top line with the bottom), then we get the values \u200b\u200bjust for the parabola.
Function domain $y=\sqrt(x)$
Using the graph of the function, the properties are quite easy to describe.1. Domain of definition: $$.
b) $$.
Solution.
We can solve our example in two ways. Each letter describes a different way.
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. Hence, it reaches its maximum value at this point. For $х=4$ the value of the function is lower than all other points, which means that here 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 value of the argument corresponds to a larger value of the function. The largest and smallest values are reached at the ends of the segment:
$y_(naib)=\sqrt(11)$, $y_(naim)=\sqrt(2)$.
Example 2
Solve the equation:
$\sqrt(x)=12-x$.
Solution.
The easiest way is to plot two function graphs and find their intersection point.
The graph clearly shows the point of intersection with the coordinates $(9;3)$. So, $x=9$ 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 is increasing, the other is decreasing. In the general case, they either do not have common points, or intersect only in one.
Example 3
Plot and read the function graph:
$\begin (cases) -x, x 9. \end (cases)$
We need to build 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 $x=12$; $y>0$ for $хϵ(-∞;12)$; $y 3. The function is decreasing on the segments $(-∞;0)U(9;+∞)$. The function increases on the segment $(0;9)$.
4. The function is continuous on the entire domain of definition.
5. There is no maximum or minimum value.
6. Range of values: $(-∞;+∞)$.
Tasks for independent solution
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. Plot and read the graph of the function: $\begin (cases) 2-x, x 4. \end (cases)$
4. Build and read the graph of the function: $y=\sqrt(-x)$.
The square root as an elementary function.
Square root is an elementary function and a special case of a power function for . The arithmetic square root is smooth at , and at zero it is right continuous but not differentiable.
As a function, a complex variable root is a two-valued function whose sheets converge at zero.
Plotting the square root function.
- Fill in the data table:
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X |
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at |
2. Put the points that we got on the coordinate plane.
3. We connect these points and get a graph of the square root function:
Transformation of the graph of the square root function.
Let us determine what transformations of the function must be done in order to plot the graphs of the functions. Let us define the types of transformations.
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Type of transformation |
transformation |
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Move a function along an axis OY for 4 units up. |
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internal |
Move a function along an axis OX for 1 unit to the right. |
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internal |
The graph approaches the axis OY 3 times and shrinks along the axis OH. |
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The graph moves away from the axis OX OY. |
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internal |
The graph moves away from the axis OY 2 times and stretched along the axis OH. |
Often the transformations of functions are combined.
For example, you need to plot the function 
. This is a square root plot, to be moved one unit down the axis OY and one to the right along the axis OH and at the same time stretching it 3 times along the axis OY.
It happens that immediately before plotting a function graph, preliminary identical transformations or simplifications of functions are needed.