Linearity of the function. Linear function and its graph

Let's consider the problem. A motorcyclist who left city A to present moment is located 20 km from it. At what distance s (km) from A will the motorcyclist be after t hours if he moves at a speed of 40 km/h?

Obviously, in t hours the motorcyclist will travel 50t km. Consequently, after t hours he will be at a distance of (20 + 50t) km from A, i.e. s = 50t + 20, where t ≥ 0.

Each value of t corresponds to a single value of s.

The formula s = 50t + 20, where t ≥ 0, defines the function.

Let's consider one more problem. For sending a telegram, a fee of 3 kopecks is charged for each word and an additional 10 kopecks. How many kopecks (u) should you pay for sending a telegram containing n words?

Since the sender must pay 3n kopecks for n words, the cost of sending a telegram of n words can be found using the formula u = 3n + 10, where n is any natural number.

In both considered problems, we encountered functions that are given by formulas of the form y = kx + l, where k and l are some numbers, and x and y are variables.

A function that can be specified by a formula of the form y = kx + l, where k and l are some numbers, is called linear.

Since the expression kx + l makes sense for any x, the domain of definition of a linear function can be the set of all numbers or any subset of it.

A special case of a linear function is the previously discussed direct proportionality. Recall that for l = 0 and k ≠ 0 the formula y = kx + l takes the form y = kx, and this formula, as is known, for k ≠ 0 specifies direct proportionality.

Let us need to plot a linear function f given by the formula
y = 0.5x + 2.

Let's get several corresponding values of the variable y for some values of x:

X	-6	-4	-2	0	2	4	6	8
y	-1	0	1	2	3	4	5	6

Let's mark the points with the coordinates we received: (-6; -1), (-4; 0); (-2; 1), (0; 2), (2; 3), (4; 4); (6; 5), (8; 6).

Obviously, the constructed points lie on a certain line. It does not follow from this that the graph of this function is a straight line.

To find out what form the graph of the function f in question looks like, let’s compare it with the familiar graph of direct proportionality x – y, where x = 0.5.

For any x, the value of the expression 0.5x + 2 is greater than the corresponding value of the expression 0.5x by 2 units. Therefore, the ordinate of each point on the graph of the function f is 2 units greater than the corresponding ordinate on the graph of direct proportionality.

Consequently, the graph of the function f in question can be obtained from the graph of direct proportionality by parallel translation by 2 units in the direction of the ordinate.

Since the graph of direct proportionality is a straight line, then the graph of the linear function f under consideration is also a straight line.

In general, the graph of a function given by a formula of the form y = kx + l is a straight line.

We know that to construct a straight line it is enough to determine the position of its two points.

Let, for example, you need to plot a function that is given by the formula
y = 1.5x – 3.

Let's take two arbitrary values of x, for example, x 1 = 0 and x 2 = 4. Calculate the corresponding values of the function y 1 = -3, y 2 = 3, construct points A (-3; 0) and B (4; 3) and draw a straight line through these points. This straight line is the desired graph.

If the domain of definition of a linear function is not fully represented numbers, then its graph will be a subset of points on a line (for example, a ray, a segment, a set of individual points).

The location of the graph of the function specified by the formula y = kx + l depends on the values of l and k. In particular, the angle of inclination of the graph of a linear function to the x-axis depends on the coefficient k. If k is a positive number, then this angle is acute; if k is a negative number, then the angle is obtuse. The number k is called the slope of the line.

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Consider the function y=k/y. The graph of this function is a line, called a hyperbola in mathematics. The general view of a hyperbola is shown in the figure below. (The graph shows the function y equals k divided by x, for which k equals 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 approaches in one of the directions closer and closer to the coordinate axes. The coordinate axes in this case are called asymptotes.

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

Now let's deal with two general cases hyperbole. 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.

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

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

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

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

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

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

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

2. Coordinate axes - asymptotes of the hyperbola.

4. Area function definitions all x except x=0.

5. y>0 at x0.

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

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

8. A function has neither a maximum nor a minimum value.

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

Definition of a Linear Function

Let us introduce the definition of a linear function

Definition

A function of the form $y=kx+b$, where $k$ is nonzero, is called a linear function.

The graph of a linear function is a straight line. The number $k$ is called the slope of the line.

When $b=0$ the linear function is called a function of direct proportionality $y=kx$.

Consider Figure 1.

Rice. 1. Geometric meaning of the slope of a line

Consider triangle ABC. We see that $ВС=kx_0+b$. Let's find the point of intersection of the line $y=kx+b$ with the axis $Ox$:

\ \

So $AC=x_0+\frac(b)(k)$. Let's find the ratio of these sides:

\[\frac(BC)(AC)=\frac(kx_0+b)(x_0+\frac(b)(k))=\frac(k(kx_0+b))((kx)_0+b)=k \]

On the other hand, $\frac(BC)(AC)=tg\angle A$.

Thus, we can draw the following conclusion:

Conclusion

Geometric meaning of the coefficient $k$. The angular coefficient of the straight line $k$ is equal to the tangent of the angle of inclination of this straight line to the $Ox$ axis.

Study of the linear function $f\left(x\right)=kx+b$ and its graph

First, consider the function $f\left(x\right)=kx+b$, where $k > 0$.

$f"\left(x\right)=(\left(kx+b\right))"=k>0$. Consequently, this function increases throughout domain of definition. There are no extreme points.
$(\mathop(lim)_(x\to -\infty ) kx\ )=-\infty $, $(\mathop(lim)_(x\to +\infty ) kx\ )=+\infty $
Graph (Fig. 2).

Rice. 2. Graphs of the function $y=kx+b$, for $k > 0$.

Now consider the function $f\left(x\right)=kx$, where $k

The domain of definition is all numbers.
The range of values is all numbers.
$f\left(-x\right)=-kx+b$. The function is neither even nor odd.
For $x=0,f\left(0\right)=b$. When $y=0.0=kx+b,\ x=-\frac(b)(k)$.

Intersection points with coordinate axes: $\left(-\frac(b)(k),0\right)$ and $\left(0,\ b\right)$

$f"\left(x\right)=(\left(kx\right))"=k
$f^("")\left(x\right)=k"=0$. Therefore, the function has no inflection points.
$(\mathop(lim)_(x\to -\infty ) kx\ )=+\infty $, $(\mathop(lim)_(x\to +\infty ) kx\ )=-\infty $
Graph (Fig. 3).

The concept of a numerical function. Methods for specifying a function. Properties of functions.

A numeric function is a function that acts from one numeric space (set) to another numeric space (set).

Three main ways to define a function: analytical, tabular and graphical.

1. Analytical.

The method of specifying a function using a formula is called analytical. This method is the main one in the mat. analysis, but in practice it is not convenient.

2. Tabular method of specifying a function.

A function can be specified using a table containing the argument values and their corresponding function values.

3. Graphical method of specifying a function.

A function y=f(x) is said to be given graphically if its graph is constructed. This method of specifying a function makes it possible to determine the function values only approximately, since constructing a graph and finding the function values on it is associated with errors.

Properties of a function that must be taken into account when constructing its graph:

1) The domain of definition of the function.

Function domain, that is, those values that the argument x of the function F =y (x) can take.

2) Intervals of increasing and decreasing functions.

The function is called increasing on the interval under consideration, if a larger value of the argument corresponds to a larger value of the function y(x). This means that if two arbitrary arguments x 1 and x 2 are taken from the interval under consideration, and x 1 > x 2, then y(x 1) > y(x 2).

The function is called decreasing on the interval under consideration, if a larger value of the argument corresponds to a smaller value of the function y(x). This means that if two arbitrary arguments x 1 and x 2 are taken from the interval under consideration, and x 1< х 2 , то у(х 1) < у(х 2).

3) Function zeros.

The points at which the function F = y (x) intersects the abscissa axis (they are obtained by solving the equation y(x) = 0) are called zeros of the function.

4) Even and odd functions.

The function is called even, if for all argument values from the scope

y(-x) = y(x).

Schedule even function symmetrical about the ordinate axis.

The function is called odd, if for all values of the argument from the domain of definition

y(-x) = -y(x).

The graph of an even function is symmetrical about the origin.

Many functions are neither even nor odd.

5) Periodicity of the function.

The function is called periodic, if there is a number P such that for all values of the argument from the domain of definition

y(x + P) = y(x).

Linear function, its properties and graph.

A linear function is a function of the form y = kx + b, defined on the set of all real numbers.

k – slope(real number)

b– dummy term (real number)

x– independent variable.

· In the special case, if k = 0, we obtain a constant function y = b, the graph of which 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 direct proportionality.

o The geometric meaning of the coefficient b is the length of the segment that the straight line cuts off along the Oy axis, counting from the origin.

o The geometric meaning of the coefficient k is the angle of inclination of the straight line to the positive direction of the Ox axis, calculated counterclockwise.

Properties of a linear function:

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

2) If k ≠ 0, then the range of values of the linear function is the entire real axis.

If k = 0, then the range of values 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, therefore, y = b – even;

b) b = 0, k ≠ 0, therefore y = kx – odd;

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

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

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

5) Points of intersection with coordinate axes:

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

Oy: y = 0k + b = b, therefore (0; b) is the point of intersection with the ordinate.

Comment. 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 constant 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 for 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 for x of (-b/k; +∞).

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

k = 0, b< 0; y = kx + b отрицательна на всей области определения.

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

k > 0, therefore y = kx + b increases throughout the entire domain of definition,

k< 0, следовательно y = kx + b убывает на всей области определения.

11. Function y = ax 2 + bx + c, its properties and graph.

The function y = ax 2 + bx + c (a, b, c are constants, a ≠ 0) is called quadratic In the simplest case, y = ax 2 (b = c = 0) the graph is a curved line passing through the origin. The curve serving as a graph of the function y = ax 2 is a parabola. Every parabola has an axis of symmetry called the axis of the parabola. The point O of the intersection of a parabola with its axis is called the vertex of the parabola.

The graph can be constructed according to the following scheme: 1) Find the coordinates of the vertex of the parabola x 0 = -b/2a; y 0 = y(x 0). 2) We construct several more points that belong to the parabola; when constructing, we can use the symmetries of the parabola relative to the straight line x = -b/2a. 3) Connect the indicated points with a smooth line. Example. Graph the function b = x 2 + 2x - 3. Solutions. The graph of the function is a parabola, the branches of which are directed upward. The abscissa of the vertex of the parabola x 0 = 2/(2 ∙1) = -1, its ordinates y(-1) = (1) 2 + 2(-1) - 3 = -4. So, the vertex of the parabola is point (-1; -4). Let's compile a table of values for several points that are located to the right of the axis of symmetry of the parabola - straight line x = -1.

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