Power function, its properties and graph Demonstration material Lesson-lecture Concept of function. Function properties. Power function, its properties and graph. Grade 10 All rights reserved. Copyright with Copyright with

Lesson progress: Repetition. Function. Function properties. Learning new material. 1. Definition of a power function. Definition of a power function. 2. Properties and graphs of power functions. Properties and graphs of power functions. Consolidation of the studied material. Verbal counting. Verbal counting. Summary of the lesson. Homework. Homework.

Domain and range of the function All values ​​of the independent variable form the domain of the function x y=f(x) f Domain of the function Domain of the function All values ​​that the dependent variable takes form the domain of the function Function. Function Properties

Graph of a function Let a function be given where xY y x.75 3 0.6 4 0.5 The graph of a function is the set of all points of the coordinate plane, the abscissas of which are equal to the values ​​of the argument, and the ordinates are equal to the corresponding values ​​of the function. Function. Function Properties

Y x Domain of definition and range of the function 4 y=f(x) Domain of the function: Domain of the function: Function. Function Properties

Even function y x y=f(x) Graph of an even function is symmetrical with respect to the y-axis The function y=f(x) is called even if f(-x) = f(x) for any x from the domain of the function Function. Function Properties

Odd function y x y \u003d f (x) The graph of the odd function is symmetrical about the origin O (0; 0) The function y \u003d f (x) is called odd if f (-x) \u003d -f (x) for any x from the region function definitions Function. Function Properties

Definition of a power function A function, where p is a given real number, is called a power function. p y \u003d x p P \u003d x y 0 Lesson progress

Power function x y 1. The domain of definition and the domain of values ​​of power functions of the form, where n is a natural number, are all real numbers. 2. These functions are odd. Their graph is symmetrical with respect to the origin. Power Function Properties and Plots

Power functions with a rational positive exponent The domain of definition is all positive numbers and the number 0. The range of functions with such an exponent is also all positive numbers and the number 0. These functions are neither even nor odd. y x Properties and Graphs of the Power Function

Power function with rational negative exponent. The domain of definition and range of such functions are all positive numbers. The functions are neither even nor odd. Such functions decrease over their entire domain of definition. y x Properties and graphs of the power function Lesson progress

The functions y \u003d ax, y \u003d ax 2, y \u003d a / x - are special types of a power function for n = 1, n = 2, n = -1 .

If n fractional number p/ q with an even denominator q and odd numerator R, then the value can have two signs, and the graph has one more part at the bottom of the x-axis X, and it is symmetrical to the upper part.

We see a graph of a two-valued function y \u003d ± 2x 1/2, i.e. represented by a parabola with a horizontal axis.

Function Graphs y = xn at n = -0,1; -1/3; -1/2; -1; -2; -3; -10 . These graphs pass through the point (1; 1).

When n = -1 we get hyperbole. At n < - 1 the graph of the power function is first located above the hyperbola, i.e. between x = 0 and x = 1, and then below (at x > 1). If a n> -1 the graph runs in reverse. Negative values X and fractional values n similar for positive n.

All graphs approach indefinitely as to the x-axis X, as well as to the y-axis at without coming into contact with them. Because of their resemblance to a hyperbola, these graphs are called hyperbolas. n th order.

Function where X- variable, A- a given number is called power function .

If then is a linear function, its graph is a straight line (see Section 4.3, Figure 4.7).

If then is a quadratic function, its graph is a parabola (see Section 4.3, Figure 4.8).

If then its graph is a cubic parabola (see Section 4.3, Figure 4.9).

Power function

This is the inverse function for

1. Domain:

2. Multiple values:

3. Even and Odd: odd function.

4. Function periodicity: non-periodic.

5. Function nulls: X= 0 is the only zero.

6. The function does not have a maximum or minimum value.

7.

8. Function Graph Symmetric to the graph of a cubic parabola with respect to a straight line Y=X and shown in Fig. 5.1.

Power function

1. Domain:

2. Multiple values:

3. Even and Odd: the function is even.

4. Function periodicity: non-periodic.

5. Function nulls: single zero X = 0.

6. The largest and smallest values ​​of the function: takes the smallest value for X= 0, it is equal to 0.

7. Ascending and descending intervals: the function is decreasing on the interval and increasing on the interval

8. Function Graph(for everybody N Î N) "looks" like a graph of a quadratic parabola (the graphs of the functions are shown in Fig. 5.2).

Power function

1. Domain:

2. Multiple values:

3. Even and Odd: odd function.

4. Function periodicity: non-periodic.

5. Function nulls: X= 0 is the only zero.

6. Maximum and minimum values:

7. Ascending and descending intervals: the function is increasing over the entire domain of definition.

8. Function Graph(for each ) "looks" like a graph of a cubic parabola (the function graphs are shown in Fig. 5.3).

Power function

1. Domain:

2. Multiple values:

3. Even and Odd: odd function.

4. Function periodicity: non-periodic.

5. Function nulls: has no zeros.

6. The largest and smallest values ​​of the function: the function does not have the largest and smallest values ​​for any

7. Ascending and descending intervals: the function is decreasing in the domain of definition.

8. Asymptotes:(axis OU) is the vertical asymptote;

(axis Oh) is the horizontal asymptote.

9. Function Graph(for anyone N) "looks" like a graph of a hyperbola (the graphs of the functions are shown in Fig. 5.4).

Power function

1. Domain:

2. Multiple values:

3. Even and Odd: the function is even.

4. Function periodicity: non-periodic.

5. The largest and smallest values ​​of the function: the function does not have the largest and smallest values ​​for any

6. Ascending and descending intervals: the function is increasing on and decreasing on

7. Asymptotes: X= 0 (axis OU) is the vertical asymptote;

Y= 0 (axis Oh) is the horizontal asymptote.

8. Function graphs Are quadratic hyperbolas (Fig. 5.5).

Power function

1. Domain:

2. Multiple values:

3. Even and Odd: the function does not have the property of even and odd.

4. Function periodicity: non-periodic.

5. Function nulls: X= 0 is the only zero.

6. The largest and smallest values ​​of the function: the smallest value equal to 0, the function takes at the point X= 0; doesn't matter the most.

7. Ascending and descending intervals: the function is increasing over the entire domain of definition.

8. Each such function with a certain indicator is inverse for the function, provided

9. Function Graph"looks" like a graph of a function for any N and shown in Fig. 5.6.

Power function

1. Domain:

2. Multiple values:

3. Even and Odd: odd function.

4. Function periodicity: non-periodic.

5. Function nulls: X= 0 is the only zero.

6. The largest and smallest values ​​of the function: the function does not have the largest and smallest values ​​for any

7. Ascending and descending intervals: the function is increasing over the entire domain of definition.

8. Function Graph Shown in fig. 5.7.