Power function graphics of all different powers. Functions and Graphs

Graph (Fig. 2).

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

Properties of a power function with natural odd exponent

The domain of definition is all real numbers.

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

$f(x)$ is continuous on 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

To begin with, we introduce the concept of a degree with an integer exponent.

Definition 3

The degree of a real number $a$ with an integer exponent $n$ is determined by the formula:

Figure 4

Consider now 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 integer exponent.

If the degree is greater than zero, then we come to the case of a power function with a natural exponent. We have already discussed it above. For $n=0$ we get a linear function $y=1$. We 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 scope 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 on the entire domain of definition.

Range of value:

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

If the exponent is odd, the function decreases as $x\in \left(-\infty ,0\right)(0,+\infty)$. For an even exponent, 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

The properties and graphs of power functions are presented for various values ​​of the exponent. Basic formulas, domains and sets of values, parity, monotonicity, increase and decrease, extrema, convexity, inflections, points of intersection with coordinate axes, limits, particular values.

Power Function Formulas

On the domain of the power function y = x p, the following formulas hold:
; ;
;
; ;
; ;
; .

Properties of power functions and their graphs

Power function with exponent equal to zero, p = 0

If the exponent of the power function y = x p is equal to zero, p = 0 , then the power function is defined for all x ≠ 0 and is constant, equal to one:
y \u003d x p \u003d x 0 \u003d 1, x ≠ 0.

Power function with natural odd exponent, p = n = 1, 3, 5, ...

Consider a power function y = x p = x n with natural odd exponent n = 1, 3, 5, ... . Such an indicator can also be written as: n = 2k + 1, where k = 0, 1, 2, 3, ... is a non-negative integer. Below are the properties and graphs of such functions.

Graph of a power function y = x n with a natural odd exponent for various values ​​of the exponent n = 1, 3, 5, ... .

Domain: -∞ < x < ∞
Multiple values: -∞ < y < ∞
Parity: odd, y(-x) = - y(x)
Monotone: increases monotonically
Extremes: No
Convex:
at -∞< x < 0 выпукла вверх
at 0< x < ∞ выпукла вниз
Breakpoints: x=0, y=0
x=0, y=0
Limits:
;
Private values:
at x = -1,
y(-1) = (-1) n ≡ (-1) 2k+1 = -1
for x = 0, y(0) = 0 n = 0
for x = 1, y(1) = 1 n = 1
Reverse function:
for n = 1 , the function is inverse to itself: x = y
for n ≠ 1, the inverse function is a root of degree n:

Power function with natural even exponent, p = n = 2, 4, 6, ...

Consider a power function y = x p = x n with natural even exponent n = 2, 4, 6, ... . Such an indicator can also be written as: n = 2k, where k = 1, 2, 3, ... is a natural number. The properties and graphs of such functions are given below.

Graph of a power function y = x n with a natural even exponent for various values ​​of the exponent n = 2, 4, 6, ... .

Domain: -∞ < x < ∞
Multiple values: 0 ≤ y< ∞
Parity: even, y(-x) = y(x)
Monotone:
for x ≤ 0 monotonically decreases
for x ≥ 0 monotonically increases
Extremes: minimum, x=0, y=0
Convex: convex down
Breakpoints: No
Intersection points with coordinate axes: x=0, y=0
Limits:
;
Private values:
for x = -1, y(-1) = (-1) n ≡ (-1) 2k = 1
for x = 0, y(0) = 0 n = 0
for x = 1, y(1) = 1 n = 1
Reverse function:
for n = 2, square root:
for n ≠ 2, root of degree n:

Power function with integer negative exponent, p = n = -1, -2, -3, ...

Consider a power function y = x p = x n with a negative integer exponent n = -1, -2, -3, ... . If we put n = -k, where k = 1, 2, 3, ... is a natural number, then it can be represented as:

Graph of a power function y = x n with a negative integer exponent for various values ​​of the exponent n = -1, -2, -3, ... .

Odd exponent, n = -1, -3, -5, ...

Below are the properties of the function y = x n with an odd negative exponent n = -1, -3, -5, ... .

Domain: x ≠ 0
Multiple values: y ≠ 0
Parity: odd, y(-x) = - y(x)
Monotone: decreases monotonically
Extremes: No
Convex:
at x< 0 : выпукла вверх
for x > 0 : convex down
Breakpoints: No
Intersection points with coordinate axes: No
Sign:
at x< 0, y < 0
for x > 0, y > 0
Limits:
; ; ;
Private values:
for x = 1, y(1) = 1 n = 1
Reverse function:
for n = -1,
for n< -2 ,

Even exponent, n = -2, -4, -6, ...

Below are the properties of the function y = x n with an even negative exponent n = -2, -4, -6, ... .

Domain: x ≠ 0
Multiple values: y > 0
Parity: even, y(-x) = y(x)
Monotone:
at x< 0 : монотонно возрастает
for x > 0 : monotonically decreasing
Extremes: No
Convex: convex down
Breakpoints: No
Intersection points with coordinate axes: No
Sign: y > 0
Limits:
; ; ;
Private values:
for x = 1, y(1) = 1 n = 1
Reverse function:
for n = -2,
for n< -2 ,

Power function with rational (fractional) exponent

Consider a power function y = x p with a rational (fractional) exponent , where n is an integer, m > 1 is a natural number. Moreover, n, m do not have common divisors.

The denominator of the fractional indicator is odd

Let the denominator of the fractional exponent be odd: m = 3, 5, 7, ... . In this case, the power function x p is defined for both positive and negative x values. Consider the properties of such power functions when the exponent p is within certain limits.

p is negative, p< 0

Let the rational exponent (with odd denominator m = 3, 5, 7, ... ) be less than zero: .

Graphs of exponential functions with a rational negative exponent for various values ​​of the exponent , where m = 3, 5, 7, ... is odd.

Odd numerator, n = -1, -3, -5, ...

Here are the properties of a power function y = x p with a rational negative exponent , where n = -1, -3, -5, ... is an odd negative integer, m = 3, 5, 7 ... is an odd natural number.

Domain: x ≠ 0
Multiple values: y ≠ 0
Parity: odd, y(-x) = - y(x)
Monotone: decreases monotonically
Extremes: No
Convex:
at x< 0 : выпукла вверх
for x > 0 : convex down
Breakpoints: No
Intersection points with coordinate axes: No
Sign:
at x< 0, y < 0
for x > 0, y > 0
Limits:
; ; ;
Private values:
for x = -1, y(-1) = (-1) n = -1
for x = 1, y(1) = 1 n = 1
Reverse function:

Even numerator, n = -2, -4, -6, ...

Properties of a power function y = x p with a rational negative exponent , where n = -2, -4, -6, ... is an even negative integer, m = 3, 5, 7 ... is an odd natural number.

Domain: x ≠ 0
Multiple values: y > 0
Parity: even, y(-x) = y(x)
Monotone:
at x< 0 : монотонно возрастает
for x > 0 : monotonically decreasing
Extremes: No
Convex: convex down
Breakpoints: No
Intersection points with coordinate axes: No
Sign: y > 0
Limits:
; ; ;
Private values:
for x = -1, y(-1) = (-1) n = 1
for x = 1, y(1) = 1 n = 1
Reverse function:

The p-value is positive, less than one, 0< p < 1

Graph of a power function with a rational exponent (0< p < 1 ) при различных значениях показателя степени , где m = 3, 5, 7, ... - нечетное.

Odd numerator, n = 1, 3, 5, ...

< p < 1 , где n = 1, 3, 5, ... - нечетное натуральное, m = 3, 5, 7 ... - нечетное натуральное.

Domain: -∞ < x < +∞
Multiple values: -∞ < y < +∞
Parity: odd, y(-x) = - y(x)
Monotone: increases monotonically
Extremes: No
Convex:
at x< 0 : выпукла вниз
for x > 0 : convex up
Breakpoints: x=0, y=0
Intersection points with coordinate axes: x=0, y=0
Sign:
at x< 0, y < 0
for x > 0, y > 0
Limits:
;
Private values:
for x = -1, y(-1) = -1
for x = 0, y(0) = 0
for x = 1, y(1) = 1
Reverse function:

Even numerator, n = 2, 4, 6, ...

The properties of the power function y = x p with a rational exponent , being within 0 are presented.< p < 1 , где n = 2, 4, 6, ... - четное натуральное, m = 3, 5, 7 ... - нечетное натуральное.

Domain: -∞ < x < +∞
Multiple values: 0 ≤ y< +∞
Parity: even, y(-x) = y(x)
Monotone:
at x< 0 : монотонно убывает
for x > 0 : monotonically increasing
Extremes: minimum at x = 0, y = 0
Convex: convex upward at x ≠ 0
Breakpoints: No
Intersection points with coordinate axes: x=0, y=0
Sign: for x ≠ 0, y > 0
Limits:
;
Private values:
for x = -1, y(-1) = 1
for x = 0, y(0) = 0
for x = 1, y(1) = 1
Reverse function:

The exponent p is greater than one, p > 1

Graph of a power function with a rational exponent (p > 1 ) for various values ​​of the exponent , where m = 3, 5, 7, ... is odd.

Odd numerator, n = 5, 7, 9, ...

Properties of a power function y = x p with a rational exponent greater than one: . Where n = 5, 7, 9, ... is an odd natural number, m = 3, 5, 7 ... is an odd natural number.

Domain: -∞ < x < ∞
Multiple values: -∞ < y < ∞
Parity: odd, y(-x) = - y(x)
Monotone: increases monotonically
Extremes: No
Convex:
at -∞< x < 0 выпукла вверх
at 0< x < ∞ выпукла вниз
Breakpoints: x=0, y=0
Intersection points with coordinate axes: x=0, y=0
Limits:
;
Private values:
for x = -1, y(-1) = -1
for x = 0, y(0) = 0
for x = 1, y(1) = 1
Reverse function:

Even numerator, n = 4, 6, 8, ...

Properties of a power function y = x p with a rational exponent greater than one: . Where n = 4, 6, 8, ... is an even natural number, m = 3, 5, 7 ... is an odd natural number.

Domain: -∞ < x < ∞
Multiple values: 0 ≤ y< ∞
Parity: even, y(-x) = y(x)
Monotone:
at x< 0 монотонно убывает
for x > 0 monotonically increases
Extremes: minimum at x = 0, y = 0
Convex: convex down
Breakpoints: No
Intersection points with coordinate axes: x=0, y=0
Limits:
;
Private values:
for x = -1, y(-1) = 1
for x = 0, y(0) = 0
for x = 1, y(1) = 1
Reverse function:

The denominator of the fractional indicator is even

Let the denominator of the fractional exponent be even: m = 2, 4, 6, ... . In this case, the power function x p is not defined for negative values ​​of the argument. Its properties coincide with those of a power function with an irrational exponent (see the next section).

Power function with irrational exponent

Consider a power function y = x p with an irrational exponent p . The properties of such functions differ from those considered above in that they are not defined for negative values ​​of the x argument. For positive values ​​of the argument, the properties depend only on the value of the exponent p and do not depend on whether p is integer, rational, or irrational.

y = x p for different values ​​of the exponent p .

Power function with negative p< 0

Domain: x > 0
Multiple values: y > 0
Monotone: decreases monotonically
Convex: convex down
Breakpoints: No
Intersection points with coordinate axes: No
Limits: ;
private value: For x = 1, y(1) = 1 p = 1

Power function with positive exponent p > 0

The indicator is less than one 0< p < 1

Domain: x ≥ 0
Multiple values: y ≥ 0
Monotone: increases monotonically
Convex: convex up
Breakpoints: No
Intersection points with coordinate axes: x=0, y=0
Limits:
Private values: For x = 0, y(0) = 0 p = 0 .
For x = 1, y(1) = 1 p = 1

The indicator is greater than one p > 1

Domain: x ≥ 0
Multiple values: y ≥ 0
Monotone: increases monotonically
Convex: convex down
Breakpoints: No
Intersection points with coordinate axes: x=0, y=0
Limits:
Private values: For x = 0, y(0) = 0 p = 0 .
For x = 1, y(1) = 1 p = 1

References:
I.N. Bronstein, K.A. Semendyaev, Handbook of Mathematics for Engineers and Students of Higher Educational Institutions, Lan, 2009.

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.

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1. Power function, its properties and graph;

2. Transformations:

Parallel transfer;

Symmetry about the coordinate axes;

Symmetry about the origin;

Symmetry about the line y = x;

Stretching and shrinking along the coordinate axes.

3. An exponential function, its properties and graph, similar transformations;

4. Logarithmic function, its properties and graph;

5. Trigonometric function, its properties and graph, similar transformations (y = sin x; y = cos x; y = tg x);

Function: y = x\n - its properties and graph.

Power function, its properties and graph

y \u003d x, y \u003d x 2, y \u003d x 3, y \u003d 1 / x etc. All these functions are special cases of the power function, i.e., the function y = xp, where p is a given real number.
The properties and graph of a power function essentially depend on the properties of a power with a real exponent, and in particular on the values ​​for which x and p makes sense xp. Let us proceed to a similar consideration of various cases, depending on
exponent p.

1. Index p = 2n is an even natural number.

y=x2n, where n is a natural number and has the following properties:

• the domain of definition is all real numbers, i.e., the set R;
• set of values ​​- non-negative numbers, i.e. y is greater than or equal to 0;
• function y=x2n even, because x 2n = (-x) 2n
• the function is decreasing on the interval x< 0 and increasing on the interval x > 0.

Function Graph y=x2n has the same form as, for example, the graph of a function y=x4.

2. Indicator p = 2n - 1- odd natural number

In this case, the power function y=x2n-1, where is a natural number, has the following properties:

• domain of definition - set R;
• set of values ​​- set R;
• function y=x2n-1 odd because (- x) 2n-1= x 2n-1 ;
• the function is increasing on the entire real axis.

Function Graph y=x2n-1 y=x3.

3. Indicator p=-2n, where n- natural number.

In this case, the power function y=x-2n=1/x2n has the following properties:

• set of values ​​- positive numbers y>0;
• function y = 1/x2n even, because 1/(-x) 2n= 1/x2n;
• the function is increasing on the interval x0.

Graph of the function y = 1/x2n has the same form as, for example, the graph of the function y = 1/x2.

4. Indicator p = -(2n-1), where n- natural number.
In this case, the power function y=x-(2n-1) has the following properties:

• the domain of definition is the set R, except for x = 0;
• set of values ​​- set R, except for y = 0;
• function y=x-(2n-1) odd because (- x)-(2n-1) = -x-(2n-1);
• the function is decreasing on the intervals x< 0 and x > 0.

Function Graph y=x-(2n-1) has the same form as, for example, the graph of the function y = 1/x3.