The function y=x^2 is called a quadratic function. The graph of a quadratic function is a parabola. The general view of the parabola is shown in the figure below.

Quadratic function

Fig 1. General view of the parabola

As can be seen from the graph, it is symmetrical about the Oy axis. The Oy axis is called the axis of symmetry of the parabola. This means that if you draw a straight line on the graph parallel to the Ox axis above this axis. Then it will intersect the parabola at two points. The distance from these points to the Oy axis will be the same.

The axis of symmetry divides the graph of a parabola into two parts. These parts are called branches of the parabola. And the point of a parabola that lies on the axis of symmetry is called the vertex of the parabola. That is, the axis of symmetry passes through the vertex of the parabola. The coordinates of this point are (0;0).

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 ;

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.

The properties and graphs of power functions are presented for different meanings exponent. 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, ...) 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 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.