Power function graphics of all different powers. Power function, its properties and graph Demonstration material Lesson-lecture Function concept
Are you familiar with the features y=x, y=x 2 , y=x 3 , y=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 x p. Let us proceed to a similar consideration of various cases, depending on
exponent p.
- Index p=2n is an even natural number.
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.
2. Indicator p=2n-1- odd natural number
In this case, the power function y=x 2n-1, where is a natural number, has the following properties:
- domain of definition - set R;
- set of values - set R;
- function y=x 2n-1 odd because (- x) 2n-1=x 2n-1 ;
- the function is increasing on the entire real axis.
3.Indicator p=-2n, where n- natural number.
In this case, the power function y=x -2n=1/x2n has the following properties:
- domain of definition - set R, except for x=0;
- 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 x<0 и убывающей на промежутке x>0.
Lesson and presentation on the topic: "Power functions. Properties. Graphs"
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Power functions, domain of definition.
Guys, in the last lesson we learned how to work with numbers with a rational exponent. In this lesson, we will consider power functions and restrict ourselves to the case when the exponent is rational.We will consider functions of the form: $y=x^(\frac(m)(n))$.
Let us first consider functions whose exponent is $\frac(m)(n)>1$.
Let us be given a specific function $y=x^2*5$.
According to the definition we gave in the last lesson: if $x≥0$, then the domain of our function is the ray $(x)$. Let's schematically depict our function graph.
Properties of the function $y=x^(\frac(m)(n))$, $0 2. Is neither even nor odd.
3. Increases by $$,
b) $(2,10)$,
c) on the ray $$.
Solution.
Guys, do you remember how we found the largest and smallest value of a function on a segment in grade 10?
That's right, we used the derivative. Let's solve our example and repeat the algorithm for finding the smallest and largest value.
1. Find the derivative of the given function:
$y"=\frac(16)(5)*\frac(5)(2)x^(\frac(3)(2))-x^3=8x^(\frac(3)(2)) -x^3=8\sqrt(x^3)-x^3$.
2. The derivative exists on the entire domain of the original function, then there are no critical points. Let's find stationary points:
$y"=8\sqrt(x^3)-x^3=0$.
$8*\sqrt(x^3)=x^3$.
$64x^3=x^6$.
$x^6-64x^3=0$.
$x^3(x^3-64)=0$.
$x_1=0$ and $x_2=\sqrt(64)=4$.
Only one solution $x_2=4$ belongs to the given segment.
Let's build a table of values of our function at the ends of the segment and at the extremum point:
Answer: $y_(name)=-862.65$ with $x=9$; $y_(max)=38.4$ for $x=4$.
Example. Solve the equation: $x^(\frac(4)(3))=24-x$.
Solution. The graph of the function $y=x^(\frac(4)(3))$ is increasing, while the graph of the function $y=24-x$ is decreasing. Guys, you and I know: if one function increases and the other decreases, then they intersect at only one point, that is, we have only one solution.
Note:
$8^(\frac(4)(3))=\sqrt(8^4)=(\sqrt(8))^4=2^4=16$.
$24-8=16$.
That is, for $х=8$ we got the correct equality $16=16$, this is the solution of our equation.
Answer: $x=8$.
Example.
Plot the function: $y=(x-3)^\frac(3)(4)+2$.
Solution.
The graph of our function is obtained from the graph of the function $y=x^(\frac(3)(4))$, shifting it 3 units to the right and 2 units up. 
Example. Write the equation of the tangent to the line $y=x^(-\frac(4)(5))$ at the point $x=1$.
Solution. The tangent equation is determined by the formula known to us:
$y=f(a)+f"(a)(x-a)$.
In our case $a=1$.
$f(a)=f(1)=1^(-\frac(4)(5))=1$.
Let's find the derivative:
$y"=-\frac(4)(5)x^(-\frac(9)(5))$.
Let's calculate:
$f"(a)=-\frac(4)(5)*1^(-\frac(9)(5))=-\frac(4)(5)$.
Find the tangent equation:
$y=1-\frac(4)(5)(x-1)=-\frac(4)(5)x+1\frac(4)(5)$.
Answer: $y=-\frac(4)(5)x+1\frac(4)(5)$.
Tasks for independent solution
1. Find the largest and smallest value of the function: $y=x^\frac(4)(3)$ on the segment:a) $$.
b) $(4.50)$.
c) on the ray $$.
3. Solve the equation: $x^(\frac(1)(4))=18-x$.
4. Graph the function: $y=(x+1)^(\frac(3)(2))-1$.
5. Write the equation of the tangent to the line $y=x^(-\frac(3)(7))$ at the point $x=1$.
Lecture: Power function with a natural exponent, its graph
We are constantly dealing with functions in which the argument has some power:
y \u003d x 1, y \u003d x 2, y \u003d x 3, y \u003d x -1, etc.
Graphs of Power Functions
So, now we will consider several possible cases of a power function.
1) y = x 2 n .
This means that now we will consider functions in which the exponent is an even number.
Feature Feature:
1. All real numbers are accepted as the range.
2. The function can take all positive values and the number zero.
3. The function is even because it does not depend on the sign of the argument, but only on its modulus.
4. For a positive argument, the function is increasing, and for a negative one, it is decreasing.
The graphs of these functions resemble a parabola. For example, below is a graph of the function y \u003d x 4. 
2) The function has an odd exponent: y \u003d x 2 n +1.
1. The domain of the function is the entire set of real numbers.
2. Function range - can take the form of any real number.
3. This function is odd.
4. Monotonically increases over the entire interval of considering the function.
5.
The graph of all power functions with an odd exponent is identical to the function y \u003d x 3. 
3) The function has an even negative natural exponent: y \u003d x -2 n.
We all know that a negative exponent allows you to drop the exponent into the denominator and change the sign of the exponent, that is, you get the form y \u003d 1 / x 2 n.
1. The argument of this function can take any value except zero, since the variable is in the denominator.
2. Since the exponent is an even number, the function cannot take negative values. And since the argument cannot be equal to zero, then the value of the function equal to zero should also be excluded. This means that the function can only take positive values.
3. This function is even.
4. If the argument is negative, the function is monotonically increasing, and if it is positive, it is decreasing.
View of the graph of the function y \u003d x -2: 
4) Function with negative odd exponent y \u003d x - (2 n + 1) .
1. This function exists for all values of the argument, except for the number zero.
2. The function accepts all real values, except for the number zero.
3. This function is odd.
4. Decreases on the two considered intervals.
Consider an example of a graph of a function with a negative odd exponent using the example y \u003d x -3. 
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 a natural odd exponent n = 1, 3, 5, .... Such an exponent 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 the power function y = x n with a natural odd exponent at different values exponent n = 1, 3, 5, ....
Definition area: –∞< x < ∞
Set of 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
for 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, .... Such an exponent can also be written as: n = 2k, where k = 1, 2, 3, ... is a 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, ....
Definition area: –∞< x < ∞
Set of values: 0 ≤ y< ∞
Monotone:
at x< 0 монотонно убывает
for x > 0 monotonically increases
Extremes: minimum, x = 0, y = 0
Convexity: convex down
Knee points: no
Intersection points with coordinate axes: x = 0, y = 0
Private values:
at x = –1, y(–1) = (–1) n ≡ (–1) 2m = 1
for x = 0, y(0) = 0 n = 0
for x = 1, y(1) = 1 n = 1
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 of definition: x ≠ 0
Set of values: y ≠ 0
Parity: odd, y(–x) = – y(x)
Extremes: no
Convex:
at x< 0: выпукла вверх
for x > 0: convex down
Knee 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, ....
Domain of definition: x ≠ 0
Set of values: y > 0
Parity: even, y(–x) = y(x)
Monotone:
at x< 0: монотонно возрастает
for x > 0: monotonically decreasing
Extremes: no
Convexity: convex down
Knee 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.
p 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 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 of definition: x ≠ 0
Set of values: y ≠ 0
Parity: odd, y(–x) = – y(x)
Monotonicity: monotonically decreasing
Extremes: no
Convex:
at x< 0: выпукла вверх
for x > 0: convex down
Knee 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, ...
Power function properties y = x p with rational negative exponent , where n = -2, -4, -6, ... is an even negative integer, m = 3, 5, 7 ... is an odd natural number.
Domain of definition: x ≠ 0
Set of values: y > 0
Parity: even, y(–x) = y(x)
Monotone:
at x< 0: монотонно возрастает
for x > 0: monotonically decreasing
Extremes: no
Convexity: convex down
Knee 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 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.
Definition area: –∞< x < +∞
Set of values: –∞< y < +∞
Parity: odd, y(–x) = – y(x)
Monotonicity: monotonically increasing
Extremes: no
Convex:
at x< 0: выпукла вниз
for x > 0: convex up
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
for 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 , being within 0 are presented.< p < 1, где n = 2, 4, 6, ... - четное натуральное, m = 3, 5, 7 ... - нечетное натуральное.
Definition area: –∞< x < +∞
Set of 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 upward at x ≠ 0
Knee points: no
Intersection points with coordinate axes: x = 0, y = 0
Sign: for x ≠ 0, y > 0
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 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 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.