Table of power functions properties and graphs. Power function

Function where X – variable quantity, A– a given number is called Power function .

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

If then - quadratic function, its graph is a parabola (see paragraph 4.3, Fig. 4.8).

If then its graph is a cubic parabola (see paragraph 4.3, Fig. 4.9).

Power function

This inverse function For

1. Domain:

2. Multiple meanings:

3. Even and odd: function is odd.

4. Function frequency: non-periodic.

5. Function zeros: X= 0 – the only zero.

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

8. Graph of a function Symmetrical to the graph of a cubic parabola relative to a straight line Y=X and is shown in Fig. 5.1.

Power function

1. Domain:

2. Multiple meanings:

3. Even and odd: the function is even.

4. Function frequency: non-periodic.

5. Function zeros: 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. Increase and decrease intervals: the function is decreasing on the interval and increasing on the interval

8. Graph of a function(for each N Î N) is “similar” to the graph quadratic parabola(function graphs are shown in Fig. 5.2).

Power function

1. Domain:

2. Multiple meanings:

3. Even and odd: function is odd.

4. Function frequency: non-periodic.

5. Function zeros: X= 0 – the only zero.

6. Highest and lowest values:

7. Increase and decrease intervals: the function is increasing over the entire domain of definition.

8. Graph of a function(for each ) is “similar” to the graph of a cubic parabola (function graphs are shown in Fig. 5.3).

Power function

1. Domain:

2. Multiple meanings:

3. Even and odd: function is odd.

4. Function frequency: non-periodic.

5. Function zeros: 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. Increase and decrease intervals: the function is decreasing in its domain of definition.

8. Asymptotes:(axis OU) – vertical asymptote;

(axis Oh) – horizontal asymptote.

9. Graph of a function(for anyone N) is “similar” to the graph of a hyperbola (function graphs are shown in Fig. 5.4).

Power function

1. Domain:

2. Multiple meanings:

3. Even and odd: the function is even.

4. Function frequency: non-periodic.

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

6. Increase and decrease intervals: the function is increasing by and decreasing by

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

Y= 0 (axis Oh) – horizontal asymptote.

8. Function graphs They are quadratic hyperbolas (Fig. 5.5).

Power function

1. Domain:

2. Multiple meanings:

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

4. Function frequency: non-periodic.

5. Function zeros: X= 0 – the only zero.

6. The largest and smallest values of the function: the function takes the smallest value equal to 0 at the point X= 0; highest value does not have.

7. Increase and decrease intervals: the function is increasing over the entire domain of definition.

8. Each such function for a certain exponent is the inverse of the function provided

9. Graph of a function"resembles" the graph of a function for any N and is shown in Fig. 5.6.

Power function

1. Domain:

2. Multiple meanings:

3. Even and odd: function is odd.

4. Function frequency: non-periodic.

5. Function zeros: X= 0 – 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. Increase and decrease intervals: the function is increasing over the entire domain of definition.

8. Graph of a function Shown in Fig. 5.7.

Power function, its properties and graph Demo 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. Properties of functions. 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. Lesson summary. Homework assignment. Homework assignment.

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

Power function x y 1. The domain of definition and the range of values of power functions of the form, where n – natural number, are all real numbers. 2. These functions are odd. Their graph is symmetrical about the origin. Properties and graphs of power functions

Power functions with a rational positive exponent. The domain of definition is all positive numbers and the number 0. The range of values 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 power functions

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

Let us recall the properties and graphs of power functions with a negative integer exponent.

For even n, :

Example function:

All graphs of such functions pass through two fixed points: (1;1), (-1;1). The peculiarity of functions of this type is their parity; the graphs are symmetrical relative to the op-amp axis.

Rice. 1. Graph of a function

For odd n, :

Example function:

All graphs of such functions pass through two fixed points: (1;1), (-1;-1). The peculiarity of functions of this type is that they are odd; the graphs are symmetrical with respect to the origin.

Rice. 2. Graph of a function

Let us recall the basic definition.

The power of a non-negative number a with a rational positive exponent is called a number.

Degree positive number and with a rational negative exponent is called a number.

For the equality:

For example: ; - the expression does not exist by definition of a power with a negative rational indicator; exists because the exponent is integer,

Let's move on to considering power functions with a rational negative exponent.

For example:

To plot a graph of this function, you can create a table. We will do it differently: first we will build and study the graph of the denominator - it is known to us (Figure 3).

Rice. 3. Graph of a function

The graph of the denominator function passes through a fixed point (1;1). When plotting the original function given point remains, when the root also tends to zero, the function tends to infinity. And, conversely, as x tends to infinity, the function tends to zero (Figure 4).

Rice. 4. Function graph

Let's consider another function from the family of functions being studied.

It is important that by definition

Let's consider the graph of the function in the denominator: , the graph of this function is known to us, it increases in its domain of definition and passes through the point (1;1) (Figure 5).

Rice. 5. Graph of a function

When plotting the graph of the original function, the point (1;1) remains, while the root also tends to zero, the function tends to infinity. And, conversely, as x tends to infinity, the function tends to zero (Figure 6).

Rice. 6. Graph of a function

The considered examples help to understand how the graph flows and what are the properties of the function being studied - a function with a negative rational exponent.

The graphs of functions of this family pass through the point (1;1), the function decreases over the entire domain of definition.

Function scope:

The function is not limited from above, but is limited from below. The function has neither a greatest nor lowest value.

The function is continuous, accepts everything positive values from zero to plus infinity.

The function is convex downward (Figure 15.7)

Points A and B are taken on the curve, a segment is drawn through them, the entire curve is below the segment, this condition is satisfied for arbitrary two points on the curve, therefore the function is convex downward. Rice. 7.

Rice. 7. Convexity of function

It is important to understand that the functions of this family are bounded from below by zero, but do not have the smallest value.

Example 1 - find the maximum and minimum of a function on the interval \[(\mathop(lim)_(x\to +\infty ) x^(2n)\ )=+\infty \]

Graph (Fig. 2).

Figure 2. Graph of the function $f\left(x\right)=x^(2n)$

Properties of a power function with a natural odd exponent
The domain of definition is all real numbers.

$f\left(-x\right)=((-x))^(2n-1)=(-x)^(2n)=-f(x)$ -- the function is odd.

$f(x)$ is continuous over the entire domain of definition.

The range is all real numbers.

$f"\left(x\right)=\left(x^(2n-1)\right)"=(2n-1)\cdot x^(2(n-1))\ge 0$

The function increases over the entire domain of definition.

$f\left(x\right)0$, for $x\in (0,+\infty)$.

$f(""\left(x\right))=(\left(\left(2n-1\right)\cdot x^(2\left(n-1\right))\right))"=2 \left(2n-1\right)(n-1)\cdot x^(2n-3)$
\ \
The function is concave for $x\in (-\infty ,0)$ and convex for $x\in (0,+\infty)$.

Graph (Fig. 3).

Figure 3. Graph of the function $f\left(x\right)=x^(2n-1)$
Power function with integer exponent
First, let's introduce the concept of a degree with an integer exponent.
Definition 3

Degree real number$a$ with integer exponent $n$ is determined by the formula:

Figure 4.
Let us now consider a power function with an integer exponent, its properties and graph.
Definition 4

$f\left(x\right)=x^n$ ($n\in Z)$ is called a power function with an integer exponent.
If the degree Above zero, then we come to the case of a power function with natural indicator. We have already discussed it above. For $n=0$ we get linear function$y=1$. We will leave its consideration to the reader. It remains to consider the properties of a power function with a negative integer exponent
Properties of a power function with a negative integer exponent
The domain of definition is $\left(-\infty ,0\right)(0,+\infty)$.

If the exponent is even, then the function is even; if it is odd, then the function is odd.

$f(x)$ is continuous over the entire domain of definition.

Scope:

If the exponent is even, then $(0,+\infty)$; if it is odd, then $\left(-\infty ,0\right)(0,+\infty)$.

If not even indicator the function decreases as $x\in \left(-\infty ,0\right)(0,+\infty)$. If the exponent is even, the function decreases as $x\in (0,+\infty)$. and increases as $x\in \left(-\infty ,0\right)$.

$f(x)\ge 0$ over the entire domain of definition