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 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 exponent n = 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 the 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

A power function is called a function of the form y=x n (read as y equals x to the power of n), where n is some given number. Special cases of power functions are functions of the form y=x, y=x 2, y=x 3, y=1/x and many others. Let's tell you more about each of them.

Linear function y=x 1 (y=x)

The graph is a straight line passing through the point (0;0) at an angle of 45 degrees to the positive direction of the Ox axis.

The graph is presented below.

Basic properties of a linear function:

• The function is increasing and defined on the entire number line.
• It has no maximum or minimum values.

Quadratic function y=x 2

The graph of a quadratic function is a parabola.

Basic properties of a quadratic function:

• 1. At x =0, y=0, and y>0 at x0
• 2. The quadratic function reaches its minimum value at its vertex. Ymin at x=0; It should also be noted that the function does not have a maximum value.
• 3. The function decreases on the interval (-∞;0] and increases on the interval)