Properties of power functions and their graphs

Power function with an index 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 a constant equal to one:
y = x p = x 0 = 1, x ≠ 0.

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

Consider a power function y = x p = x n with a natural odd exponent n = 1, 3, 5, .... This exponent can also be written in the form: n = 2k + 1, where k = 0, 1, 2, 3, . .. – the whole is not negative. 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 < ∞

Extremes: no

Convex:

at –∞< x < 0 выпукла вверх

at 0< x < ∞ выпукла вниз

Inflection points: x = 0, y = 0

Private values:

at x = –1, y(–1) = (–1) n ≡ (–1) 2m+1 = –1

at x = 0, y(0) = 0 n = 0

for x = 1, y(1) = 1 n = 1

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

Consider a power function y = x p = x n with a natural even exponentn = 2, 4, 6, .... This exponent can also be written in the form: n = 2k, where k = 1, 2, 3, ... - natural . 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< ∞

Monotone:

at x< 0 монотонно убывает

for x > 0 monotonically increases

Extremes: minimum, x = 0, y = 0

Convex: convex down

Inflection points: no

Intersection points with coordinate axes: x = 0, y = 0
Private values:

at x = –1, y(–1) = (–1) n ≡ (–1) 2m = 1

at x = 0, y(0) = 0 n = 0

for x = 1, y(1) = 1 n = 1

Power function with negative integer 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 set 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, ....

Range of definition: x ≠ 0

Multiple values: y ≠ 0

Parity: odd, y(–x) = – y(x)

Extremes: no

Convex:

at x< 0: выпукла вверх

for x > 0: convex downward

Inflection points: no

Sign: at x< 0, y < 0

for x > 0, y > 0

Private values:

for x = 1, y(1) = 1 n = 1

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, ....

Range of definition: x ≠ 0

Multiple values: y > 0

Parity: even, y(–x) = y(x)

Monotone:

at x< 0: монотонно возрастает

for x > 0: monotonically decreases

Extremes: no

Convex: convex down

Inflection points: no

Intersection points with coordinate axes: no

Sign: y > 0

Private values:

at x = –1, y(–1) = (–1) n = 1

for x = 1, y(1) = 1 n = 1

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 values ​​of the argument. Let us consider the properties of such power functions when the exponent p is within certain limits.

The p-value is negative, p< 0

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

Graphs of power 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, ...

We present the properties of the 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 integer.

Range of definition: x ≠ 0

Multiple values: y ≠ 0

Parity: odd, y(–x) = – y(x)

Monotonicity: monotonically decreasing

Extremes: no

Convex:

at x< 0: выпукла вверх

for x > 0: convex downward

Inflection points: no

Intersection points with coordinate axes: no

at x< 0, y < 0

for x > 0, y > 0

Private values:

at x = –1, y(–1) = (–1) n = –1

for x = 1, y(1) = 1 n = 1

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

Properties of the 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 integer.

Range of definition: x ≠ 0

Multiple values: y > 0

Parity: even, y(–x) = y(x)

Monotone:

at x< 0: монотонно возрастает

for x > 0: monotonically decreases

Extremes: no

Convex: convex down

Inflection points: no

Intersection points with coordinate axes: no

Sign: y > 0

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

Power function graph with 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)

Monotonicity: monotonically increasing

Extremes: no

Convex:

at x< 0: выпукла вниз

for x > 0: convex upward

Inflection points: x = 0, y = 0

Intersection points with coordinate axes: x = 0, y = 0

at x< 0, y < 0

for x > 0, y > 0

Private values:

at x = –1, y(–1) = –1

at x = 0, y(0) = 0

for x = 1, y(1) = 1

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

The properties of the power function y = x p with a rational exponent 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: increases monotonically

Extremes: minimum at x = 0, y = 0

Convexity: convex upwards at x ≠ 0

Inflection points: no

Intersection points with coordinate axes: x = 0, y = 0

Sign: for x ≠ 0, y > 0

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 degree with a negative rational exponent; 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 the greatest nor the least value.

The function is continuous and takes all 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)