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

Formulas with power functions

On the domain of definition 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 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 indicator can also be written in the form: 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 meanings: -∞ < y < ∞
Parity: odd, y(-x) = - y(x)
Monotone: monotonically increases
Extremes: No
Convex:
at -∞< x < 0 выпукла вверх
at 0< x < ∞ выпукла вниз
Inflection points: x = 0, y = 0
x = 0, y = 0
Limits:
;
Private values:
at x = -1,
y(-1) = (-1) n ≡ (-1) 2k+1 = -1
at x = 0, y(0) = 0 n = 0
for x = 1, y(1) = 1 n = 1
Reverse function:
for n = 1, the function is its inverse: x = y
for n ≠ 1, the inverse function is the 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 a natural even exponent n = 2, 4, 6, ... . This indicator 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 meanings: 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
Inflection points: No
Intersection points with coordinate axes: x = 0, y = 0
Limits:
;
Private values:
at x = -1, y(-1) = (-1) n ≡ (-1) 2k = 1
at 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 negative integer exponent, p = n = -1, -2, -3, ...

Consider a power function y = x p = x n with an integer negative 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 meanings: y ≠ 0
Parity: odd, y(-x) = - y(x)
Monotone: monotonically decreases
Extremes: No
Convex:
at x< 0 : выпукла вверх
for x > 0: convex downward
Inflection points: 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:
when n = -1,
at 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 meanings: 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
Limits:
; ; ;
Private values:
for x = 1, y(1) = 1 n = 1
Reverse function:
at n = -2,
at 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 values ​​of the argument x. 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, ... - 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.

Domain: x ≠ 0
Multiple meanings: y ≠ 0
Parity: odd, y(-x) = - y(x)
Monotone: monotonically decreases
Extremes: No
Convex:
at x< 0 : выпукла вверх
for x > 0: convex downward
Inflection points: No
Intersection points with coordinate axes: No
Sign:
at x< 0, y < 0
for x > 0, y > 0
Limits:
; ; ;
Private values:
at 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 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.

Domain: x ≠ 0
Multiple meanings: 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
Limits:
; ; ;
Private values:
at 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 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 meanings: -∞ < y < +∞
Parity: odd, y(-x) = - y(x)
Monotone: monotonically increases
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
Sign:
at x< 0, y < 0
for x > 0, y > 0
Limits:
;
Private values:
at x = -1, y(-1) = -1
at 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 within 0 are presented< p < 1 , где n = 2, 4, 6, ... - четное натуральное, m = 3, 5, 7 ... - нечетное натуральное.

Domain: -∞ < x < +∞
Multiple meanings: 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
Convex: convex upward for x ≠ 0
Inflection points: No
Intersection points with coordinate axes: x = 0, y = 0
Sign: for x ≠ 0, y > 0
Limits:
;
Private values:
at x = -1, y(-1) = 1
at x = 0, y(0) = 0
for x = 1, y(1) = 1
Reverse function:

The p index 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, ... - odd.

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

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

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

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

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

Domain: -∞ < x < ∞
Multiple meanings: 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
Inflection points: No
Intersection points with coordinate axes: x = 0, y = 0
Limits:
;
Private values:
at x = -1, y(-1) = 1
at 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 the properties 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 discussed above in that they are not defined for negative values ​​of the argument x. 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 exponent p< 0

Domain: x > 0
Multiple meanings: y > 0
Monotone: monotonically decreases
Convex: convex down
Inflection points: No
Intersection points with coordinate axes: No
Limits: ;
Private meaning: For x = 1, y(1) = 1 p = 1

Power function with positive exponent p > 0

Indicator less than one 0< p < 1

Domain: x ≥ 0
Multiple meanings: y ≥ 0
Monotone: monotonically increases
Convex: convex upward
Inflection points: 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 meanings: y ≥ 0
Monotone: monotonically increases
Convex: convex down
Inflection points: 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 college students, “Lan”, 2009.

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The section contains reference material on the main elementary functions and their properties. A classification of elementary functions is given. Below are links to subsections that discuss the properties of specific functions - graphs, formulas, derivatives, antiderivatives (integrals), series expansions, expressions through complex variables.

Reference pages for basic functions

Classification of elementary functions

Algebraic function is a function that satisfies the equation:
,
where is a polynomial in the dependent variable y and the independent variable x. It can be written as:
,
where are polynomials.

Algebraic functions are divided into polynomials (entire rational functions), rational functions and irrational functions.

Entire rational function, which is also called polynomial or polynomial, is obtained from the variable x and a finite number of numbers using the arithmetic operations of addition (subtraction) and multiplication. After opening the brackets, the polynomial is reduced to canonical form:
.

Fractional rational function, or simply rational function, is obtained from the variable x and a finite number of numbers using the arithmetic operations of addition (subtraction), multiplication and division. The rational function can be reduced to the form
,
where and are polynomials.

Irrational function is an algebraic function that is not rational. As a rule, an irrational function is understood as roots and their compositions with rational functions. A root of degree n is defined as the solution to the equation
.
It is designated as follows:
.

Transcendental functions are called non-algebraic functions. These are exponential, trigonometric, hyperbolic and their inverse functions.

Overview of basic elementary functions

All elementary functions can be represented as a finite number of addition, subtraction, multiplication and division operations performed on an expression of the form:
z t .
Inverse functions can also be expressed in terms of logarithms. The basic elementary functions are listed below.

Power function :
y(x) = x p ,
where p is the exponent. It depends on the base of the degree x.
The inverse of the power function is also the power function:
.
For an integer non-negative value of the exponent p, it is a polynomial. For an integer value p - a rational function. With a rational meaning - an irrational function.

Transcendental functions

Exponential function :
y(x) = a x ,
where a is the base of the degree. It depends on the exponent x.
The inverse function is the logarithm to base a:
x = log a y.

Exponent, e to the x power:
y(x) = e x ,
This is an exponential function whose derivative is equal to the function itself:
.
The base of the exponent is the number e:
≈ 2,718281828459045... .
The inverse function is the natural logarithm - the logarithm to the base of the number e:
x = ln y ≡ log e y.

Trigonometric functions:
Sine: ;
Cosine: ;
Tangent: ;
Cotangent: ;
Here i is the imaginary unit, i 2 = -1.

Inverse trigonometric functions:
Arcsine: x = arcsin y, ;
Arc cosine: x = arccos y, ;
Arctangent: x = arctan y, ;
Arc tangent: x = arcctg y, .

To understand this topic, let's consider a function depicted on a graph // Let's show how a graph of a function allows you to determine its properties.

Let's look at the properties of a function using an example

The domain of definition of the function is span [ 3.5; 5.5].

The range of values ​​of the function is span [ 1; 3].

1. At x = -3, x = - 1, x = 1.5, x = 4.5, the value of the function is zero.

The argument value at which the function value is zero is called function zero.

//those. for this function the numbers are -3;-1;1.5; 4.5 are zeros.

2. At intervals [ 4.5; 3) and (1; 1.5) and (4.5; 5.5] the graph of the function f is located above the abscissa axis, and in the intervals (-3; -1) and (1.5; 4.5) below the axis abscissa, this is explained as follows: on the intervals [ 4.5; 3) and (1; 1.5) and (4.5; 5.5] the function takes positive values, and on the intervals (-3; -1) and ( 1.5; 4.5) negative.

Each of the indicated intervals (where the function takes values ​​of the same sign) is called the interval of constant sign of the function f.//i.e. for example, if we take the interval (0; 3), then it is not an interval of constant sign of this function.

In mathematics, when searching for intervals of constant sign of a function, it is customary to indicate intervals of maximum length. //Those. the interval (2; 3) is interval of constancy of sign function f, but the answer should include the interval [ 4.5; 3) containing the interval (2; 3).

3. If you move along the x-axis from 4.5 to 2, you will notice that the function graph goes down, that is, the function values ​​decrease. //In mathematics it is customary to say that on the interval [ 4.5; 2] the function decreases.

As x increases from 2 to 0, the graph of the function goes up, i.e. the function values ​​increase. //In mathematics it is customary to say that on the interval [ 2; 0] the function increases.

A function f is called if for any two values ​​of the argument x1 and x2 from this interval such that x2 > x1, the inequality f (x2) > f (x1) holds. // or the function is called increasing over some interval, if for any values ​​of the argument from this interval, a larger value of the argument corresponds to a larger value of the function.//i.e. the more x, the more y.

The function f is called decreasing over some interval, if for any two values ​​of the argument x1 and x2 from this interval such that x2 > x1, the inequality f(x2) is decreasing on some interval, if for any values ​​of the argument from this interval the larger value of the argument corresponds to the smaller value of the function. //those. the more x, the less y.

If a function increases over the entire domain of definition, then it is called increasing.

If a function decreases over the entire domain of definition, then it is called decreasing.

Example 1. graph of increasing and decreasing functions respectively.

Example 2.

Define the phenomenon. Is the linear function f(x) = 3x + 5 increasing or decreasing?

Proof. Let's use the definitions. Let x1 and x2 be arbitrary values ​​of the argument, and x1< x2., например х1=1, х2=7

Functions and their properties

Function is one of the most important mathematical concepts.Function They call such a dependence of the variable y on the variable x in which each value of the variable x corresponds to a single value of the variable y.

Variable X called independent variable or argument. Variable at called dependent variable. They also say thatthe variable y is a function of the variable x. The values ​​of the dependent variable are calledfunction values.

If the dependence of the variableat from variableX is a function, then it can be written briefly as follows:y= f( x ). (Read:at equalsf fromX .) Symbolf( x) denote the value of the function corresponding to the value of the argument equal toX .

All values ​​of the independent variable formdomain of a function . All values ​​that the dependent variable takes formfunction range .

If a function is specified by a formula and its domain of definition is not specified, then the domain of definition of the function is considered to consist of all values ​​of the argument for which the formula makes sense.

Methods for specifying a function:

1.analytical method (the function is specified using a mathematical formula;

2.tabular method (the function is specified using a table)

3.descriptive method (the function is specified by verbal description)

4. graphical method (the function is specified using a graph).

Function graph name the set of all points of the coordinate plane, the abscissas of which are equal to the values ​​of the argument, and the ordinates - corresponding function values.

BASIC PROPERTIES OF FUNCTIONS

1. Function zeros

Zero of a function is the value of the argument at which the value of the function is equal to zero.

2. Intervals of constant sign of a function

Intervals of constant sign of a function are sets of argument values ​​on which the function values ​​are only positive or only negative.

3. Increasing (decreasing) function.

Increasing in a certain interval, a function is a function for which a larger value of the argument from this interval corresponds to a larger value of the function.

Function y = f ( x ) called increasing on the interval (A; b ), if for any x 1 And x 2 from this interval such thatx 1 < x 2 , inequality is truef ( x 1 )< f ( x 2 ).

Descending in a certain interval, a function is a function for which a larger value of the argument from this interval corresponds to a smaller value of the function.

Function at = f ( x ) called decreasing on the interval (A; b ) , if for any x 1 And x 2 from this interval such that x 1 < x 2 , inequality is truef ( x 1 )> f ( x 2 ).

4. Even (odd) function

Even function - a function whose domain of definition is symmetrical with respect to the origin and for anyX from the domain of definition the equalityf (- x ) = f ( x ) . The graph of an even function is symmetrical about the ordinate.

For example, y = x 2 - even function.

Odd function- a function whose domain of definition is symmetrical with respect to the origin and for any X from the domain of definition the equality is true f (- x ) = - f (x ). The graph of an odd function is symmetrical about the origin.

For example: y = x 3 - odd function .

A function of general form is not even or odd (y = x 2 +x ).

Properties of some functions and their graphics

1. Linear function called a function of the form , Where k And b – numbers.

The domain of definition of a linear function is a setR real numbers.

Graph of a linear functionat = kx + b ( k ≠ 0) is a straight line passing through the point (0;b ) and parallel to the lineat = kx .

Straight, not parallel to the axisOU, is the graph of a linear function.

Properties of a linear function.

1. When k > 0 function at = kx + b

2. When k < 0 function y = kx + b decreasing in the domain of definition.

y = kx + b ( k ≠ 0 ) is the entire number line, i.e. a bunch ofR real numbers.

At k = 0 set of function valuesy = kx + b consists of one numberb .

3. When b = 0 and k = 0 the function is neither even nor odd.

At k = 0 linear function has the formy = b and at b ≠ 0 it is even.

At k = 0 and b = 0 linear function has the formy = 0 and is both even and odd.

Graph of a linear functiony = b is a straight line passing through the point (0; b ) and parallel to the axisOh. Note that when b = 0 function graphy = b coincide with the axis Oh .

5. When k > 0 we have that at> 0, if and at< 0 if . At k < 0 we have that y > 0 if and at< 0, если .

2. Function y = x 2

Rreal numbers.

Giving a variableX several values ​​from the function's domain and calculating the corresponding valuesat according to the formula y = x 2 , we depict the graph of the function.

Graph of a function y = x 2 called parabola.

Properties of the function y = x 2 .

1. If X= 0, then y = 0, i.e. The parabola has a common point with the coordinate axes (0; 0) - the origin of coordinates.

2. If x ≠ 0 , That at > 0, i.e. all points of the parabola, except the origin, lie above the x-axis.

3. Set of function valuesat = X 2 is the span functionat = X 2 decreases.

X

3.Function

The domain of this function is the span functiony = | x | decreases.

7. The function takes its smallest value at the pointX, it equals 0. There is no greatest value.

6. Function

Function scope: .

Function range: .

The graph is a hyperbole.

1. Function zeros.

y ≠ 0, no zeros.

2. Intervals of constancy of signs,

If k > 0, then at> 0 at X > 0; at < 0 при X < О.

If k < 0, то at < 0 при X > 0; at> 0 at X < 0.

3. Intervals of increasing and decreasing.

If k > 0, then the function decreases as .

If k < 0, то функция возрастает при .

4. Even (odd) function.

The function is odd.

Square trinomial

Equation of the form ax 2 + bx + c = 0, where a , b And With - some numbers, anda≠ 0, called square.

In a quadratic equationax 2 + bx + c = 0 coefficient A called the first coefficient b - second coefficients, with - free member.

The formula for the roots of a quadratic equation is:

.

The expression is called discriminant quadratic equation and is denoted byD .

If D = 0, then there is only one number that satisfies the equation ax 2 + bx + c = 0. However, we agreed to say that in this case the quadratic equation has two equal real roots, and the number itself called double root.

If D < 0, то квадратное уравнение не имеет действительных корней.

If D > 0, then the quadratic equation has two different real roots.

Let a quadratic equation be givenax 2 + bx + c = 0. Since a≠ 0, then dividing both sides of this equation byA, we get the equation . Believing And , we arrive at the equation , in which the first coefficient is equal to 1. Such an equation is calledgiven.

The formula for the roots of the above quadratic equation is:

.

Equations of the form

A x 2 + bx = 0, ax 2 + s = 0, A x 2 = 0

are called incomplete quadratic equations. Incomplete quadratic equations are solved by factoring the left side of the equation.

Vieta's theorem .

The sum of the roots of a quadratic equation is equal to the ratio of the second coefficient to the first, taken with the opposite sign, and the product of the roots is the ratio of the free term to the first coefficient, i.e.

Converse theorem.

If the sum of any two numbersX 1 And X 2 equal to , and their product is equal, then these numbers are the roots of the quadratic equationOh 2 + b x + c = 0.

Function of the form Oh 2 + b x + c called square trinomial. The roots of this function are the roots of the corresponding quadratic equationOh 2 + b x + c = 0.

If the discriminant of a quadratic trinomial is greater than zero, then this trinomial can be represented as:

Oh 2 + b x + c = a(x-x 1 )(x-x 2 )

Where X 1 And X 2 - roots of the trinomial

If the discriminant of a quadratic trinomial is zero, then this trinomial can be represented as:

Oh 2 + b x + c = a(x-x 1 ) 2

Where X 1 - the root of the trinomial.

For example, 3x 2 - 12x + 12 = 3(x - 2) 2 .

Equation of the form Oh 4 + b X 2 + s= 0 is called biquadratic. Using variable replacement using the formulaX 2 = y it reduces to a quadratic equationA y 2 + by + c = 0.

Quadratic function

Quadratic function is a function that can be written by a formula of the formy = ax 2 + bx + c , Where x – independent variable,a , b And c – some numbers, anda ≠ 0.

The properties of the function and the type of its graph are determined mainly by the values ​​of the coefficienta and discriminant.

Properties of a quadratic function

Domain:R;

Range of values:

at A > 0 [- D/(4 a); ∞)

at A < 0 (-∞; - D/(4 a)];

Even, odd:

at b = 0 even function

at b ≠ 0 function is neither even nor odd

at D> 0 two zeros: ,

at D= 0 one zero:

at D < 0 нулей нет

Sign constancy intervals:

if a > 0, D> 0, then

if a > 0, D= 0, then

e if a > 0, D < 0, то

if a< 0, D> 0, then

if a< 0, D= 0, then

if a< 0, D < 0, то

- Intervals of monotony

for a > 0

at a< 0

The graph of a quadratic function isparabola – a curve symmetrical about a straight line , passing through the vertex of the parabola (the vertex of the parabola is the point of intersection of the parabola with the axis of symmetry).

To graph a quadratic function, you need:

1) find the coordinates of the vertex of the parabola and mark it in the coordinate plane;

2) construct several more points belonging to the parabola;

3) connect the marked points with a smooth line.

The coordinates of the vertex of the parabola are determined by the formulas:

; .

Converting function graphs

1. Stretching graphic artsy = x 2 along the axisat V|a| times (at|a| < 1 is a compression of 1/|a| once).

If, and< 0, произвести, кроме того, зеркальное отражение графика отно­сительно оси X (the branches of the parabola will be directed downwards).

Result: graph of a functiony = ah 2 .

2. Parallel transfer function graphicsy = ah 2 along the axisX on| m | (to the right when

m > 0 and to the left whenT< 0).

Result: function graphy = a(x - t) 2 .

3. Parallel transfer function graphics along the axisat on| n | (up atp> 0 and down atP< 0).

Result: function graphy = a(x - t) 2 + p.

Quadratic inequalities

Inequalities of the formOh 2 + b x + c > 0 andOh 2 + bx + c< 0, whereX - variable,a , b AndWith - some numbers, anda≠ 0 are called inequalities of the second degree with one variable.

Solving a second degree inequality in one variable can be thought of as finding the intervals in which the corresponding quadratic function takes positive or negative values.

To solve inequalities of the formOh 2 + bx + c > 0 andOh 2 + bx + c< 0 proceed as follows:

1) find the discriminant of the quadratic trinomial and find out whether the trinomial has roots;

2) if the trinomial has roots, then mark them on the axisX and through the marked points a parabola is drawn schematically, the branches of which are directed upward atA > 0 or down whenA< 0; if the trinomial has no roots, then schematically depict a parabola located in the upper half-plane atA > 0 or lower atA < 0;

3) found on the axisX intervals for which the points of the parabola are located above the axisX (if inequality is solvedOh 2 + bx + c > 0) or below the axisX (if inequality is solvedOh 2 + bx + c < 0).

Example:

Let's solve the inequality .

Consider the function

Its graph is a parabola, the branches of which are directed downward (since ).

Let's find out how the graph is located relative to the axisX. Let's solve the equation for this . We get thatx = 4. The equation has a single root. This means that the parabola touches the axisX.

By schematically depicting a parabola, we find that the function takes negative values ​​for anyX, except 4.

The answer can be written like this:X - any number not equal to 4.

Solving inequalities using the interval method

solution diagram

1. Find zeros function on the left side of the inequality.

2. Mark the position of the zeros on the number axis and determine their multiplicity (Ifk i is even, then zero is of even multiplicity ifk i odd is odd).

3. Find the signs of the function in the intervals between its zeros, starting from the rightmost interval: in this interval the function on the left side of the inequality is always positive for the given form of inequalities. When moving from right to left through the zero of a function from one interval to an adjacent one, one should take into account:

if zero is odd multiplicity, the sign of the function changes,

if zero is even multiplicity, the sign of the function is preserved.

4. Write down the answer.

Example:

(x + 6) (x + 1) (X - 4) < 0.

Function zeros found. They are equal:X 1 = -6; X 2 = -1; X 3 = 4.

Let us mark the zeros of the function on the coordinate linef ( x ) = (x + 6) (x + 1) (X - 4).

Let's find the signs of this function in each of the intervals (-∞; -6), (-6; -1), (-1; 4) and

It is clear from the figure that the set of solutions to the inequality is the union of the intervals (-∞; -6) and (-1; 4).

Answer: (-∞ ; -6) and (-1; 4).

The considered method for solving inequalities is calledinterval method.