Functions y k x and its graph. Linear function. Homework

Consider the function y=k/y. The graph of this function is a line, called a hyperbola in mathematics. General form hyperbola is shown in the figure below. (The graph shows a function y equals k divided by x, where k is equal to one.)

It can be seen that the graph consists of two parts. These parts are called branches of the hyperbola. It is also worth noting that each branch of the hyperbola comes closer and closer to the coordinate axes in one of the directions. The coordinate axes in this case are called asymptotes.

In general, any straight lines that the graph of a function infinitely approaches, but does not reach, are called asymptotes. A hyperbola, like a parabola, has axes of symmetry. For the hyperbola shown in the figure above, this is the straight line y=x.

Now let's deal with two general cases of hyperbolas. The graph of the function y = k/x, for k ≠ 0, will be a hyperbola, the branches of which are located either in the first and third coordinate angles, for k>0, or in the second and fourth coordinate angles, for k<0.

Main properties of the function y = k/x, for k>0

Graph of the function y = k/x, for k>0

5. y>0 for x>0; y6. The function decreases both on the interval (-∞;0) and on the interval (0;+∞).

10. The range of the function is two open intervals (-∞;0) and (0;+∞).

The main properties of the function y = k/x, for k<0

Graph of the function y = k/x, for k<0

1. The point (0;0) is the center of symmetry of the hyperbola.

2. Axes of coordinates - asymptotes of a hyperbola.

4. The scope of the function is all x, except x=0.

5. y>0 for x0.

6. The function increases both on the interval (-∞;0) and on the interval (0;+∞).

7. The function is not limited from below or from above.

8. The function has neither the largest nor the smallest values.

9. The function is continuous on the interval (-∞;0) and on the interval (0;+∞). Has a gap at the point x=0.

Linear function is called a function of the form y = kx + b, defined on the set of all real numbers. Here k– angular coefficient (real number), b – free member (real number), x is an independent variable.

In a particular case, if k = 0, we obtain a constant function y=b, whose graph is a straight line parallel to the Ox axis, passing through the point with coordinates (0;b).

If b = 0, then we get the function y=kx, which is in direct proportion.

b – segment length, which cuts off the line along the Oy axis, counting from the origin.

The geometric meaning of the coefficient k – tilt angle straight to the positive direction of the Ox axis is considered to be counterclockwise.

Linear function properties:

1) The domain of a linear function is the entire real axis;

2) If k ≠ 0, then the range of the linear function is the entire real axis. If k = 0, then the range of the linear function consists of the number b;

3) Evenness and oddness of a linear function depend on the values ​​of the coefficients k And b.

a) b ≠ 0, k = 0, hence, y = b is even;

b) b = 0, k ≠ 0, hence y = kx is odd;

c) b ≠ 0, k ≠ 0, hence y = kx + b is a general function;

d) b = 0, k = 0, hence y = 0 is both an even and an odd function.

4) A linear function does not have the property of periodicity;

5) Intersection points with coordinate axes:

Ox: y = kx + b = 0, x = -b/k, hence (-b/k; 0)- point of intersection with the abscissa axis.

Oy: y=0k+b=b, hence (0;b) is the point of intersection with the y-axis.

Note.If b = 0 And k = 0, then the function y=0 vanishes for any value of the variable X. If b ≠ 0 And k = 0, then the function y=b does not vanish for any value of the variable X.

6) The intervals of constancy of sign depend on the coefficient k.

a) k > 0; kx + b > 0, kx > -b, x > -b/k.

y = kx + b- positive at x from (-b/k; +∞),

y = kx + b- negative at x from (-∞; -b/k).

b) k< 0; kx + b < 0, kx < -b, x < -b/k.

y = kx + b- positive at x from (-∞; -b/k),

y = kx + b- negative at x from (-b/k; +∞).

c) k = 0, b > 0; y = kx + b positive throughout the domain of definition,

k = 0, b< 0; y = kx + b is negative throughout the domain of definition.

7) Intervals of monotonicity of a linear function depend on the coefficient k.

k > 0, hence y = kx + b increases over the entire domain of definition,

k< 0 , hence y = kx + b decreases over the entire domain of definition.

8) The graph of a linear function is a straight line. To draw a straight line, it is enough to know two points. The position of the straight line on the coordinate plane depends on the values ​​of the coefficients k And b. Below is a table that clearly illustrates this.

In this video tutorial, you will get acquainted with the function y = k / x, k is a coefficient that can take different meanings except 0. Let's consider the case where k = 1 => y = 1/x. To build a graph of this function, let's recall the material that was in the previous videos, namely: we select several arbitrary values ​​​​for x and substitute them into the formula y = k / x.

This will give us the opportunity to calculate the values ​​of the dependent variable y. We construct the selection of values ​​and calculations for y in two stages: first, we give the argument positive values, and then negative ones.

1. Using the formula y = k/x, we find the value of y. If x = 1 , then y = 1. Let's choose a few arguments ourselves.

In the case when x = 3, then y = 1/3; x = 5, then y = 1/5; x = 7, then y = 1/7.

And when x = 1/3, then y = 3; x = 1/5, then y = 5; x = 1/7, then y = 7.

Let's make a table:

1. In the case when x \u003d 1, then y \u003d -1, x \u003d -3, then y \u003d -1/3; x \u003d -5, then y \u003d -1/5; x = -7, then y = -1/7.

And when x = -1/3, then y = -3; x \u003d -1/5, then y \u003d 5; x = -1/7, then y = -7.

Let's make a table:

Let's build these points on the xOy coordinate plane and connect them.

You can see an example with other coordinates and the sequence of plotting in the video.

Also in the video tutorial you will get acquainted with the basic geometric properties of the hyperbola.

1. A hyperbola, like a parabola, has symmetry. If we draw a line through the origin 0, then it will intersect the hyperbola at two points that lie on the line on opposite sides of the point 0 and at equal distances from it. Thus, 0 will be the center of symmetry of the hyperbola, and it will be symmetrical with respect to the origin.
2. Symmetric, relative to the origin, parts of the hyperbola are called its branches.
3. One branch of the hyperbola is located near the abscissa axis, the other - near the ordinates. In such cases, the corresponding lines are usually called asymptotes. This means that the hyperbola has two asymptotes - the x-axis and the y-axis.
4. In addition to the center of symmetry, the hyperbola has axes of symmetry.

The graph of the function y = k/x, when k is not equal to 0, is a hyperbola, the branches of which are in the 1st and 3rd coordinate planes, in the case when k > 0, and in the 2nd and 4th k ​​> 0, and in the 2nd and 4th coordinate planes when k< 0. (0,0) - точка центра симметрии гиперболы, а осями координат являются её асимптоты. Функцию y = k/x называют обратно пропорциональной, в силу того, что её величины - x и у, являются обратно пропорциональными, а число k - это коэффициент обратной пропорциональности.

You can get examples and more detailed information on the topic by watching the video tutorial.