Lesson and presentation on the topic: "Power functions. Cubic root. Properties of the cubic root"
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Definition of a power function - cube root
Guys, we continue to study power functions. Today we will talk about the "Cubic root of x" function.What is a cube root?
The number y is called a cube root of x (root of the third degree) if the equality $y^3=x$ holds.
Denoted as $\sqrt(x)$, where x is a radical number, 3 is an 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 $a=-b$. Using the notation for roots we obtain the desired identity.
Properties of cubic 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 $\sqrt(\frac(a)(b))$ cubed is equal to $\frac(a)(b)$ and then equals $\sqrt(\frac(a)(b))$, which and needed to be proven.
Guys, let's build a graph of our function.
1) Domain of definition is the set of real numbers.
2) The function is odd, since $\sqrt((-x))$=-$\sqrt(x)$. Next, consider our function for $x≥0$, then display the graph relative to the origin.
3) The function increases when $x≥0$. For our function, a larger value of the argument corresponds to a larger value of the function, which means increase.
4) The function is not limited from above. In fact, from an arbitrarily large number we can calculate the third root, and we can move upward indefinitely, 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 at x≥0.
Let's construct our graph of the function over 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 downward by (-∞;0), convex upward by (0;+∞).
Examples of solving power functions
Examples1. Solve the equation $\sqrt(x)=x$.
Solution. Let's construct 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. Construct 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 translation two units to the right and three units down. 
3. Graph the function and read it. $\begin(cases)y=\sqrt(x), x≥-1\\y=-x-2, x≤-1 \end(cases)$.
Solution. Let's construct two graphs of functions on the same coordinate plane, taking into account our conditions. For $x≥-1$ we build a graph of the cubic root, for $x≤-1$ we build 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 greatest value. The smallest value is minus one.
6) The function is continuous on the entire number line.
7) E(y)= (-1;+∞).
Problems to solve independently
1. Solve the equation $\sqrt(x)=2-x$.2. Construct a graph of the function $y=\sqrt((x+1))+1$.
3.Plot a graph of the function and read it. $\begin(cases)y=\sqrt(x), x≥1\\y=(x-1)^2+1, x≤1 \end(cases)$.
Municipal educational institution
secondary school No. 1
Art. Bryukhovetskaya
municipal formation Bryukhovetsky district
Mathematic teacher
Guchenko Angela Viktorovna
year 2014
Function y =
, its properties and graph
Lesson type: learning new material
Lesson objectives:
Problems solved in the lesson:
teach students to work independently;
make assumptions and guesses;
be able to generalize the factors being studied.
Equipment: board, chalk, multimedia projector, handouts
Timing of the lesson.
Determining the topic of the lesson together with students -1 min.
Determining the goals and objectives of the lesson together with students -1 min.
Updating knowledge (frontal survey) –3 min.
Oral work -3 min.
Explanation of new material based on creating problem situations -7min.
Fizminutka –2 minutes.
Plotting a graph together with the class, drawing up the construction in notebooks and determining the properties of a function, working with a textbook -10 min.
Consolidating acquired knowledge and practicing graph transformation skills –9min .
Summing up the lesson, providing feedback -3 min.
Homework -1 min.
Total 40 minutes.
During the classes.
Determining the topic of the lesson together with students (1 min).
The topic of the lesson is determined by students using guiding questions:
function- work performed by an organ, the organism as a whole.
function- possibility, option, skill of a program or device.
function- duty, range of activities.
function character in a literary work.
function- type of subroutine in computer science
function in mathematics - the law of dependence of one quantity on another.
Determining the goals and objectives of the lesson together with students (1 min).
The teacher, with the help of students, formulates and pronounces the goals and objectives of this lesson.
Updating knowledge (frontal survey – 3 min).
Oral work – 3 min.
Frontal work.
(A and B belong, C does not)
Explanation of new material (based on creating problem situations – 7 min).
Problem situation: describe the properties of an unknown function.
Divide the class into teams of 4-5 people, distribute forms for answering the questions asked.
Form No. 1
y=0, with x=?
The scope of the function.
Set of function values.
One of the team representatives answers each question, the rest of the teams vote “for” or “against” with signal cards and, if necessary, complement the answers of their classmates.
Together with the class, draw a conclusion about the domain of definition, the set of values, and the zeros of the function y=.
Problem situation : try to build a graph of an unknown function (there is a discussion in teams, searching for a solution).
The teacher recalls the algorithm for constructing function graphs. Students in teams try to depict the graph of the function y= on forms, then exchange forms with each other for self- and mutual testing.
Fizminutka (Clowning)
Constructing a graph together with the class with the design in notebooks – 10 min.
After a general discussion, the task of constructing a graph of the function y= is completed individually by each student in a notebook. At this time, the teacher provides differentiated assistance to students. After students complete the task, the graph of the function is shown on the board and students are asked to answer the following questions:
Conclusion: Together with the students, draw a conclusion about the properties of the function and read them from the textbook:
Consolidating acquired knowledge and practicing graph transformation skills – 9 min.
Students work on their card (according to the options), then change and check each other. Afterwards, graphs are shown on the board, and students evaluate their work by comparing it with the board.
Card No. 1
Card No. 2
Conclusion: about graph transformations
1) parallel transfer along the op-amp axis
2) shift along the OX axis.
9. Summing up the lesson, providing feedback – 3 min.
SLIDES – insert missing words
The domain of definition of this function, all numbers except ...(negative).
The graph of the function is located in... (I) quarters.
When the argument x = 0, the value... (functions) y = ... (0).
The greatest value of the function... (does not exist), smallest value - …(equals 0)
10. Homework (with comments – 1 min).
According to the textbook- §13
According to the problem book– No. 13.3, No. 74 (repetition of incomplete quadratic equations)
Consider the function y=√x. The graph of this function is shown in the figure below.
Graph of the function y=√x
As you can see, the graph resembles a rotated parabola, or rather one of its branches. We get a branch of the parabola x=y^2. It can be seen from the figure that the graph touches the Oy axis only once, at the point with coordinates (0;0).
Now it is worth noting the main properties of this function.
Properties of the function y=√x
1. The domain of definition of a function is a ray)
