Types of graphs of a linear function at k 0. Linear function, its properties and graph

A linear function is a function of the form

x-argument (independent variable),

y-function (dependent variable),

k and b are some constant numbers

The graph of a linear function is straight.

To create a graph it is enough two points, because through two points you can draw a straight line and, moreover, only one.

If k˃0, then the graph is located in the 1st and 3rd coordinate quarters. If k˂0, then the graph is located in the 2nd and 4th coordinate quarters.

The number k is called the slope of the straight graph of the function y(x)=kx+b. If k˃0, then the angle of inclination of the straight line y(x)= kx+b to the positive direction Ox is acute; if k˂0, then this angle is obtuse.

Coefficient b shows the point of intersection of the graph with the op-amp axis (0; b).

y(x)=k∙x-- a special case of a typical function is called direct proportionality. The graph is a straight line passing through the origin, so one point is enough to construct this graph.

Graph of a Linear Function

Where coefficient k = 3, therefore

The graph of the function will increase and have an acute angle with the Ox axis because coefficient k has a plus sign.

OOF linear function

OPF of a linear function

Except in the case where

Also a linear function of the form

Is a function of general form.

B) If k=0; b≠0,

In this case, the graph is a straight line parallel to the Ox axis and passing through the point (0; b).

B) If k≠0; b≠0, then the linear function has the form y(x)=k∙x+b.

Example 1 . Graph the function y(x)= -2x+5

Example 2 . Let's find the zeros of the function y=3x+1, y=0;

– zeros of the function.

Answer: or (;0)

Example 3 . Determine the value of the function y=-x+3 for x=1 and x=-1

y(-1)=-(-1)+3=1+3=4

Answer: y_1=2; y_2=4.

Example 4 . Determine the coordinates of their intersection point or prove that the graphs do not intersect. Let the functions y 1 =10∙x-8 and y 2 =-3∙x+5 be given.

If the graphs of functions intersect, then the values ​​of the functions at this point are equal

Substitute x=1, then y 1 (1)=10∙1-8=2.

Comment. You can also substitute the resulting value of the argument into the function y 2 =-3∙x+5, then we get the same answer y 2 (1)=-3∙1+5=2.

y=2- ordinate of the intersection point.

(1;2) - the point of intersection of the graphs of the functions y=10x-8 and y=-3x+5.

Answer: (1;2)

Example 5 .

Construct graphs of the functions y 1 (x)= x+3 and y 2 (x)= x-1.

You can see that the coefficient k=1 for both functions.

From the above it follows that if the coefficients of a linear function are equal, then their graphs in the coordinate system are located parallel.

Example 6 .

Let's build two graphs of the function.

The first graph has the formula

The second graph has the formula

In this case, we have a graph of two lines intersecting at the point (0;4). This means that the coefficient b, which is responsible for the height of the graph’s rise above the Ox axis, if x = 0. This means we can assume that the b coefficient of both graphs is equal to 4.

Editors: Ageeva Lyubov Aleksandrovna, Gavrilina Anna Viktorovna

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 $k > 0$.

1. $f"\left(x\right)=(\left(kx+b\right))"=k>0$. Consequently, this function increases over the entire domain of definition. There are no extreme points.
2. $(\mathop(lim)_(x\to -\infty ) kx\ )=-\infty $, $(\mathop(lim)_(x\to +\infty ) kx\ )=+\infty $
3. 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

1. The domain of definition is all numbers.
2. The range of values ​​is all numbers.
3. $f\left(-x\right)=-kx+b$. The function is neither even nor odd.
4. 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 $\left(0,\ b\right)$

1. $f"\left(x\right)=(\left(kx\right))"=k
2. $f^("")\left(x\right)=k"=0$. Therefore, the function has no inflection points.
3. $(\mathop(lim)_(x\to -\infty ) kx\ )=+\infty $, $(\mathop(lim)_(x\to +\infty ) kx\ )=-\infty $
4. 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).

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

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 of general form;

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 abscissa 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 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.

Function properties.

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.)<0.

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 look at two common cases of 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

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

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

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

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

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

2. Coordinate axes - asymptotes of the hyperbola.

4. The domain of definition of the function is all x except x=0.

5. y>0 at x0.

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

Learn to take derivatives of functions. The derivative characterizes the rate of change of a function at a certain point lying on the graph of this function. In this case, the graph can be either a straight or curved line. That is, the derivative characterizes the rate of change of a function at a specific point in time. Remember the general rules by which derivatives are taken, and only then proceed to the next step.

• Read the article.
• How to take the simplest derivatives, for example, the derivative of an exponential equation, is described. The calculations presented in the following steps will be based on the methods described therein.

Learn to distinguish problems in which the slope coefficient needs to be calculated through the derivative of a function. Problems do not always ask you to find the slope or derivative of a function. For example, you may be asked to find the rate of change of a function at point A(x,y). You may also be asked to find the slope of the tangent at point A(x,y). In both cases it is necessary to take the derivative of the function.