Lesson power function its properties and graph. Summary of the lesson "Power function, its properties and graphs". Tasks for independent solution
Lesson topic: "Power functions, their properties and graphs"
Lesson Objectives:
Educational:
Create conditions for the formation of knowledge about the properties and features of graphs of power functions y = x r with different values r .
Developing:
Contribute to the development of information skills of students: the ability to work with the text of the slide, the ability to draw up a basic outline.
To promote the development of creative and mental activity of students.
Continue the formation of skills to clearly and clearly express their thoughts, analyze, draw conclusions.
Educational:
Continue the development of the culture of mathematical speech.
Contribute to the formation of communicative competence.
Lesson type: combined
Forms of organization learning activities: frontal, individual.
Methods: explanatory and illustrative, partially exploratory.
Means of education:
computer, media projector;
blackboard;
slide presentation (PowerPoint), (Appendix 1);
textbook "Algebra and the beginning of analysis" ed. A.G. Mordkovich;
workbook, drawing tools;
reference summary of the topic (Word document), (Appendix 3);
As a result of studying the topic, students should
Know: concept power function,
properties of a power function depending on the exponent.
Be able to: name the properties of a power function depending on the exponent,
build graphs (sketches of graphs) of power functions with rational
indicator,
perform simple graph transformations,
be able to write a summary
be able to clearly and clearly express their thoughts, analyze, draw conclusions.
During the classes: We continue to work on the formation of skills for plotting graphs of power functions. A number of such functions are familiar to us from the algebra course of grades 7-9, these are functions with a natural exponent, and power functions with a negative integer exponent. In the last lesson, we wrote down the theory on power functions with fractional exponents
y \u003d x p, where p is a given real number
The properties and graph of a power function depend on the properties of a power with a real exponent, and in particular on the values of x and p for which the exponent x p makes sense.
2.
Generalization of the properties of a power function. Work with reference summary.
1. Work on the board: construct graphs of functions. y=x 4, y=x 7, y=x -2, y=x -5, y=x 2/5, y=x 1.3, y=x -1/3
7 people work at the blackboard, remaining in the field, united in groups for further verification
We list the properties according to the plan.
Domain.
Range of values (set of values).
Even, odd functions.
Increasing, decreasing.
At the end of the work, check by the students who remained in their places (slides with graphs of functions are displayed on the screen).
2. "math lotto" Ready-made graphs of functions are displayed on the screen, sets of formulas are written on the board, it is necessary to establish a relationship.
Mutual check:
Correct answers: №1 578 643 192
3 oral work
1. Using the graphs of these functions, find the intervals at which the graph of the function y \u003d x π lies above (below) the graph of the function y \u003d x.
2. Using the graphs of these functions, find the intervals at which the graph of the function y \u003d x sin 45 lies above (below) the graph of the function y \u003d x.
3. Using the figure, find the intervals at which the graph of the function y \u003d x 1- π lies above (below) the graph of the function y \u003d x.
Graph Conversion
In many cases, function graphs can be constructed by some transformations of already known function graphs of a simpler form. Let's recall some of them.
Consider verbally transforming a graph of a power function, and then plot two graphs.
Independent work
Set your own power function, build its graph, describe the properties
In the last lesson, we repeated and summarized the knowledge on the topic "The concept of the exponent".
Recall that if - pe divided by ku is an ordinary fraction, and ku is not equal to one and a is greater than or equal to zero, then the expression a to the degree pe divided by ku is understood to be the root of the degree ku from a to the degree pe.
For example, the number one point three to the power of three sevenths can be written as the seventh root of one point three tenths cubed.
Functions of the form, where k is any real number, are usually called power functions.
Today we will consider the case if k is a rational (fractional) exponent.
In the course of algebra for grades 7-9, you studied the properties and graphs of power functions with a natural exponent. Function (k-any real number), power function.
For k=n (n∈N), -power function with natural exponent.
Recall the graphs of such functions.
The graph of the function or y \u003d x (y equals x to the first power or y equal to x) is a straight line.
The graph of the function (y equals x squared) is a parabola.
The graph of the function (y equals x cubed) is a cubic parabola.
The graph of a power function (y equals x to the power of ka) in the case of an even k is similar to a parabola. The figure shows a graph of a power function with k equal to six.
Graph of a power function (y equals x to the power of ka) in the case of an odd k is similar to a cubic parabola. The figure shows a graph of a power function with k equal to seven.
If the exponent of the exponential function has a negative integer, then we get a function of the form: y equal to x to the power minus en or y is equal to one divided by x to the nth power.
If n is an even number, then the graph has the form shown in the figure.
Where the function y \u003d x-2 is shown, or y \u003d
If n is an odd number, then the graph looks like this.
The drawing shows the function y=x-3, or y=
If the exponent of the exponential function zero, then the function will take the form: .The graph of such a function is a straight line passing through the ordinate one and parallel to the x-axis.
For k=-n (n∈Z), is a power function with a negative integer exponent.
Consider a power function (y equals x to the power of k), where k is a negative or positive fractional number.
As an example, let's plot a power function graph (y equals x to the power of two point three).
The area of its definition (that is, all the values taken by x) is a ray with the beginning at the zero point.
In this domain of definition, we will construct graphs of functions (y equal to x squared) - this is the branch of the parabola, highlighted lightly in green and (y equal to x cubed) - a branch of a cubic parabola, highlighted in dark green.
It is easy to verify that on the interval (0;1) the cubic parabola is located below the parabola, and on the open ray (1;+) it is above.
Note that the graphs of the functions (y equals x squared), (y equals x to the power of two point three) and (y equals x cubed) pass through the points (0;0) and (1;1).
For the remaining values of the x argument, the graph of the function (y equals x to the power of two point three) is between the graphs of the functions (y equals x squared) and (y equals x cubed).
The situation is similar with any power function, where is an improper fraction, that is, the numerator m is greater than the denominator n. The graph of this function is a curve similar to a parabola branch.
The greater the index of the function k, the "steeper" the branch is directed.
The figure shows the graph of the function y is equal to x to the power of seven second.
Thus, we can distinguish the following properties of the power function y is equal to x to the power of em divided by en, where the numerator m is greater than the denominator n.
1. The domain of definition is x values from zero to plus infinity.
4. From below it is limited by the abscissa axis, from above it is not limited.
5.Function accept smallest value zero; the greatest value does not have.
8. Convex down.
Let's build a graph of the function, where is a proper fraction (the numerator is less than the denominator) and 0< <1.
The properties and graph of the function considered earlier (y is equal to the root of the nth degree of x) or (y is equal to x to the power of one divided by en) are also applicable to the function, where is a proper fraction and 0< <1.
Let's remember these properties:
1. The domain of definition is all x values from zero to plus infinity.
2. The function is neither even nor odd.
3. The function increases over the entire domain of definition.
5. The function takes the smallest value zero; doesn't matter the most.
6. The function is continuous on the entire domain of definition.
7. The range of the function is the value of y from zero to plus infinity.
8. Convex up. function, where is a proper fraction (the numerator is less than the denominator) and 0<
2. Neither even nor odd.
3. Increasing by.
4. From below it is limited by the abscissa axis, from above it is not limited.
5. yname=0; doesn't matter the most.
6.Continuous.
8. Convex up.
Consider the following form of a power function - a function of the form: y is equal to x to the power minus em divided by en.
Earlier, we plotted a power function with a negative integer exponent, y equals x to the power minus k, where k is a natural number.
If x is greater than zero, the graph of this function is similar to a branch of a hyperbola.
Similarly, a graph of any power function with a negative rational (fractional) exponent is constructed.
It should be borne in mind that the graph of such a function has two asymptotes: horizontal - y is equal to zero and vertical asymptote - x is equal to zero.
So, the power function y is equal to x to the power minus em divided by en has the following properties (moreover, x is greater than zero, since in the case of a negative base with a negative exponent, the degree of expression does not make sense):
1) The domain of definition is an open beam from zero to infinity.
2) The function is neither even nor odd.
3) The function decreases over the entire domain of definition.
4) From below it is limited by the abscissa axis, from above it is not limited.
5) The function does not have a minimum or maximum value.
6) The function is continuous on the entire domain of definition.
7) The value area of the function is the value of y from zero to plus infinity.
8) Convex down.
Power function properties (x 0):
2). Neither even nor odd.
3). Descending.
4). From below it is limited by the abscissa axis, from above it is not limited.
5). It has no minimum or maximum value.
6). Continuous on
8). Curved down.
You already know that the derivative of a power function of the form y is x raised to the power of en, where n is a natural number, equals n times x raised to the power of n minus one.
Similarly, one can calculate the derivative of a power function with a rational exponent.
Thus, the following theorem is true:
If x is greater than zero and r is an arbitrary rational number, then the derivative of the power function y is equal to x to the power of r, and is calculated by the formula: the derivative of x to the power of r is equal to r times x to the power of r minus one.
For example, the derivative of a to a minus third power is equal to minus three and to the power of minus four.
The derivative of x to the power of minus two thirds is equal to minus two thirds of x to the power of minus five thirds.
Here minus one is represented as an improper fraction by three thirds, then the fractions minus two thirds and minus three thirds are added together.
Theorem: if x>0, r-rational number, then
It is not difficult to obtain the corresponding formula for integrating a power function when r is not equal to one. So, the indefinite integral of x to the power of r is x to the power of r plus one, divided by r plus one plus the constant ce.
It is not difficult to understand that the function is equal to x to the power of r plus one divided by r plus one is the antiderivative of the function y is equal to x to the power of r. The formula for integrating a power function is:
The function is antiderivative for a function.
Consider the application of the acquired knowledge in constructing a graph of a power function.
Construct a graph of the function y is equal to x plus two to the power of one second.
1. Let's build a graph of the x function to the power of one second. This is a function of the form where is a proper fraction (the numerator is less than the denominator) and 0< <1.График такой функции мы уже строили, на рисунке график выделен красным цветом.
2.Obviously, the graph of the function y is equal to x plus two to the power of one second is built using a parallel translation relative to the x-axis by two units to the left. In the figure, the graph is highlighted in green.
Plot a function
1. is a special case for a function of the form, where is a proper fraction (the numerator is less than the denominator) and 0< <1.
2. The graph was obtained by parallel translation along the X axis by 2 units to the left.
Abstract of a lesson in mathematics on the topic "Power function, its properties and graph"
Mathematics teacher GBPOU MO "STT"
Golubeva Natalya Borisovna
Lesson date
Formation of linguistic, communicative and informational general competencies.
OK 2. Organize your own activities, choose typical methods and ways to complete tasks.
OK 3. Make decisions in standard and non-standard situations and be responsible for them.
OK 4. Search and use the information necessary for the effective performance of tasks.
OK 5. Use information and communication technologies.
OK 6. Work in a team and team, communicate effectively with the teacher, comrades.
OK 7. Take responsibility for the work of team members (subordinates), the result of completing tasks.
OK 8. Independently determine the tasks of personal development, engage in self-education.
Lesson structure
1. Organizational stage (1 min)
Filling out the journal, marking those present at the lesson, checking the readiness of students for the lesson.
During the classes
2. Stage of preparation for active conscious assimilation of knowledge 5 min
Repetition of basic concepts.
1. Domain of definition.(the values that the x variable can take.) 2. Domain of values (set of values).(the set of values that the variable y can take.) 3. Even, odd functions. Graphic illustration of an even, odd function.(the graph of an even function is symmetrical about the op-y axis. the graph of an odd function is symmetrical about the origin, i.e. point O) Analytical record of the property of evenness, oddness.( - odd function - even function) 4. Intervals of increasing and decreasing functions.3. Stage of assimilation of new knowledge 14 min
Today in the lesson we will repeat and systematize our knowledge on the topic "Power function".Slide 3. Since the seventh grade, we have studied many functions, the graphs of which you see on the slide.
What unites all these functions?
All these functions are special cases of the power function.
We give the definition of a power function.
y = x p , where p is a given real number.
The properties and graph of a power function depend on the properties of a power with a real exponent, and in particular on the values of x and p for which the power of x makes sense. R .
Now each of you will draw up a basic abstract on the topic "Power Function". After completing this outline, it will be convenient for you to use it in preparing for the lesson. In the reference abstract, sketches of the graphs are already given. Your task: to formulate the properties of functions and make notes in the abstract.
Slides 5-17 . Frontal work with the class. Registration of entries in the "Reference Note" (Appendix 1). We list the properties of functions according to the following plan.
Domain.
Range of values (set of values).
Even, odd functions. Graphic illustration of an even, odd function. Analytical record of the property of evenness, oddness.
Write down the intervals of increase and decrease of the function.
During frontal work, I pay attention to possible options for recording answers in the form of gaps or inequalities. On slides 6, 8, 10, 12, 14, 15 I demonstrate how the appearance of the graph changes when the exponent p changes.
4. Stage of consolidation of new knowledge 10 min
Consolidation of the studied material. Solution of exercises from the textbook.
We remembered the functions that we are familiar with and saw new graphs. Let's check how you learned the new material. Let's do a compliance test.
Students have a set of cards with function formulas on their desks (Appendix 3).Thumbnails of graphs are reproduced using the presentation.7 students are invited to the board in turn, who must match the sketches of the graphs and cards with the formula, commenting on their choice. The student, using magnets, fixes the tablets with formulas next to the corresponding number of the graph.
A set of formulas for students.
y=x -0.4
Work with the textbook.№ 125 (1, 3)
Additional No. 126 (1). 5. Independent work - compliance test "Power function". 10 min
Independent work is carried out in the form of testing on laptops. The work is done in pairs. The students sit down at the laptop together, open the test, write down the names and take the test. The result of the test is recorded in a text document, after the lesson is copied and applied to the development of the lesson.
power function.mtf
Test "Power function"
Exercise 1
Question:
Final Part
6. Stage of debriefing the lesson 3 min
Summing up the lesson, grading
7. The stage of informing students about homework and instructing them to complete it 2 min
According to the reference abstract - learn the properties and graphs.№ 125(2,4), 128 .General lesson in algebra in grade 10
Lesson topic . Power function.
Lesson Objectives:
1) Generalize and systematize the knowledge, skills and abilities of students
on the topic "Power function".
2) To consolidate knowledge about the power function and its properties, the skills of applying the properties of the degree and root, the skills of solving irrational equations.
3) To develop students' thinking, attention, accuracy.
4) To instill in students a love for mathematics.
Lesson type: generalization of knowledge.
During the classes.
Message about the topic and purpose of the lesson. Write the date in a notebook.
Function and its properties.
Teacher's question: What is a power function?
(A function of the form , where p is a given real number, is called a power function.)
2) Sketches of functions are given. Which graph corresponds to the proposed formula. (Graphs are shown on the screen, formulas appear on the screen one by one)
Specify the scope and scope of the function.
Questions.
a) What is the redundant function?
b) Name an even function. Name the odd function. How do we determine?
Independent work of students.
Indicate to which formula the graph of the function corresponds: write the formula, and indicate the number of the function next to it.
IN 
option 1
1)
2)
3)
4)
5)
Option 2
1)
2)
3)
4)
5)
Students submit their work. Check answers on the screen.
3. Degree and its properties.
1) Repetition of the properties of the degree. (Properties appear one at a time on the screen, students formulate them).
Degree properties.
For any positive a and b and any rational m and n, the equalities are true:
2) Simplify expressions. The teacher dictates an example, the students write it down and decide with comments.
3) Checking solutions from the screen.
Check out the example solutions:
1.
2.
3.
4.
4) Task for students: find an error in the solution. (Tasks appear on the screen one at a time, students explain the errors in the solution. There is no error in the last example).
Find the error in the solution:
Define the arithmetic root of the nth degree.
What numbers are called non-negative?
Repetition of properties of roots. On the formula screen:
If a ≥0, b >0, m and n are natural numbers, and m ≥2, n ≥2, then
Questions.
What is the nth root of the product?
What is the nth root of a fraction?
Pay attention to formulas No. 6 and No. 7, they are used when solving irrational equations.
Task completion: simplify the expression (examples on the screen). Tasks 1 and 2 are solved by students on the board with an explanation, tasks 3 and 4 are explained orally and solved independently, followed by a screen check.
Simplify expressions:
.
5. Solving irrational equations.
1) What equation is called irrational? When solving equations, remember the key words: “equation - check!
6. Homework:
independent work on individual cards.
7. Summing up the lesson.
Lesson and presentation on the topic: "Power functions. Properties. Graphs"
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Power functions, domain of definition.
Guys, in the last lesson we learned how to work with numbers with a rational exponent. In this lesson, we will consider power functions and restrict ourselves to the case when the exponent is rational.We will consider functions of the form: $y=x^(\frac(m)(n))$.
Let us first consider functions whose exponent is $\frac(m)(n)>1$.
Let us be given a specific function $y=x^2*5$.
According to the definition we gave in the last lesson: if $x≥0$, then the domain of our function is the ray $(x)$. Let's schematically depict our function graph.
Properties of the function $y=x^(\frac(m)(n))$, $0 2. Is neither even nor odd.
3. Increases by $$,
b) $(2,10)$,
c) on the ray $$.
Solution.
Guys, do you remember how we found the largest and smallest value of a function on a segment in grade 10?
That's right, we used the derivative. Let's solve our example and repeat the algorithm for finding the smallest and largest value.
1. Find the derivative of the given function:
$y"=\frac(16)(5)*\frac(5)(2)x^(\frac(3)(2))-x^3=8x^(\frac(3)(2)) -x^3=8\sqrt(x^3)-x^3$.
2. The derivative exists on the entire domain of the original function, then there are no critical points. Let's find stationary points:
$y"=8\sqrt(x^3)-x^3=0$.
$8*\sqrt(x^3)=x^3$.
$64x^3=x^6$.
$x^6-64x^3=0$.
$x^3(x^3-64)=0$.
$x_1=0$ and $x_2=\sqrt(64)=4$.
Only one solution $x_2=4$ belongs to the given segment.
Let's build a table of values of our function at the ends of the segment and at the extremum point:
Answer: $y_(name)=-862.65$ with $x=9$; $y_(max)=38.4$ for $x=4$.
Example. Solve the equation: $x^(\frac(4)(3))=24-x$.
Solution. The graph of the function $y=x^(\frac(4)(3))$ is increasing, while the graph of the function $y=24-x$ is decreasing. Guys, you and I know: if one function increases and the other decreases, then they intersect at only one point, that is, we have only one solution.
Note:
$8^(\frac(4)(3))=\sqrt(8^4)=(\sqrt(8))^4=2^4=16$.
$24-8=16$.
That is, for $х=8$ we got the correct equality $16=16$, this is the solution of our equation.
Answer: $x=8$.
Example.
Plot the function: $y=(x-3)^\frac(3)(4)+2$.
Solution.
The graph of our function is obtained from the graph of the function $y=x^(\frac(3)(4))$, shifting it 3 units to the right and 2 units up. 
Example. Write the equation of the tangent to the line $y=x^(-\frac(4)(5))$ at the point $x=1$.
Solution. The tangent equation is determined by the formula known to us:
$y=f(a)+f"(a)(x-a)$.
In our case $a=1$.
$f(a)=f(1)=1^(-\frac(4)(5))=1$.
Let's find the derivative:
$y"=-\frac(4)(5)x^(-\frac(9)(5))$.
Let's calculate:
$f"(a)=-\frac(4)(5)*1^(-\frac(9)(5))=-\frac(4)(5)$.
Find the tangent equation:
$y=1-\frac(4)(5)(x-1)=-\frac(4)(5)x+1\frac(4)(5)$.
Answer: $y=-\frac(4)(5)x+1\frac(4)(5)$.
Tasks for independent solution
1. Find the largest and smallest value of the function: $y=x^\frac(4)(3)$ on the segment:a) $$.
b) $(4.50)$.
c) on the ray $$.
3. Solve the equation: $x^(\frac(1)(4))=18-x$.
4. Graph the function: $y=(x+1)^(\frac(3)(2))-1$.
5. Write the equation of the tangent to the line $y=x^(-\frac(3)(7))$ at the point $x=1$.
