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Linear function and its graph. Linear function Linear function y x 3

A linear function is a function of the form y=kx+b, where x is the independent variable, k and b are any numbers.
The graph of a linear function is a straight line.

1. To plot a function graph, we need the coordinates of two points belonging to the graph of the function. To find them, you need to take two x values, substitute them into the function equation, and use them to calculate the corresponding y values.

For example, to plot the function y= x+2, it is convenient to take x=0 and x=3, then the ordinates of these points will be equal to y=2 and y=3. We get points A(0;2) and B(3;3). Let's connect them and get a graph of the function y= x+2:

2. In the formula y=kx+b, the number k is called the proportionality coefficient:
if k>0, then the function y=kx+b increases
if k
Coefficient b shows the displacement of the function graph along the OY axis:
if b>0, then the graph of the function y=kx+b is obtained from the graph of the function y=kx by shifting b units upward along the OY axis
if b
The figure below shows the graphs of the functions y=2x+3; y= ½ x+3; y=x+3

Note that in all these functions the coefficient k Above zero, and the functions are increasing. Moreover, the greater the value of k, the greater the angle of inclination of the straight line to the positive direction of the OX axis.

In all functions b=3 - and we see that all graphs intersect the OY axis at point (0;3)

Now consider the graphs of the functions y=-2x+3; y=- ½ x+3; y=-x+3

This time in all functions the coefficient k less than zero and functions are decreasing. Coefficient b=3, and the graphs, as in the previous case, intersect the OY axis at point (0;3)

Let's look at the graphs of the functions y=2x+3; y=2x; y=2x-3

Now in all function equations the coefficients k are equal to 2. And we got three parallel lines.

But the coefficients b are different, and these graphs intersect the OY axis at different points:
The graph of the function y=2x+3 (b=3) intersects the OY axis at point (0;3)
The graph of the function y=2x (b=0) intersects the OY axis at the point (0;0) - the origin.
The graph of the function y=2x-3 (b=-3) intersects the OY axis at point (0;-3)

So, if we know the signs of the coefficients k and b, then we can immediately imagine what the graph of the function y=kx+b looks like.
If k 0

If k>0 and b>0, then the graph of the function y=kx+b looks like:

If k>0 and b, then the graph of the function y=kx+b looks like:

If k, then the graph of the function y=kx+b looks like:

If k=0, then the function y=kx+b turns into the function y=b and its graph looks like:

The ordinates of all points on the graph of the function y=b are equal to b If b=0, then the graph of the function y=kx (direct proportionality) passes through the origin:

3. Let us separately note the graph of the equation x=a. The graph of this equation is a straight line parallel to the OY axis, all points of which have an abscissa x=a.

For example, the graph of the equation x=3 looks like this:
Attention! The equation x=a is not a function, so one value of the argument corresponds to different values ​​of the function, which does not correspond to the definition of a function.


4. Condition for parallelism of two lines:

The graph of the function y=k 1 x+b 1 is parallel to the graph of the function y=k 2 x+b 2 if k 1 =k 2

5. The condition for two straight lines to be perpendicular:

The graph of the function y=k 1 x+b 1 is perpendicular to the graph of the function y=k 2 x+b 2 if k 1 *k 2 =-1 or k 1 =-1/k 2

6. Points of intersection of the graph of the function y=kx+b with the coordinate axes.

With OY axis. The abscissa of any point belonging to the OY axis is equal to zero. Therefore, to find the point of intersection with the OY axis, you need to substitute zero in the equation of the function instead of x. We get y=b. That is, the point of intersection with the OY axis has coordinates (0; b).

With OX axis: The ordinate of any point belonging to the OX axis is zero. Therefore, to find the point of intersection with the OX axis, you need to substitute zero in the equation of the function instead of y. We get 0=kx+b. Hence x=-b/k. That is, the point of intersection with the OX axis has coordinates (-b/k;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 $k > 0$.

  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).

Linear function called a function of the form y = kx + b, defined on the set of all real numbers. Here k– slope (real number), b – free 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.

b – segment length, which is cut off by a straight line along the Oy axis, counting from the origin.

Geometric meaning of the coefficient k – tilt angle straight to the positive direction of the Ox axis, considered 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, hence, y = b – even;

b) b = 0, k ≠ 0, hence y = kx – odd;

c) b ≠ 0, k ≠ 0, hence y = kx + b – function of general form;

d) b = 0, k = 0, hence y = 0 – both even and odd functions.

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)– point of intersection with the ordinate axis.

Note: If b = 0 And k = 0, then the function y = 0 goes to zero 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 when x from (-b/k; +∞),

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

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

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

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

c) k = 0, b > 0; y = kx + b positive over the entire definition range,

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

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

k > 0, hence y = kx + b increases throughout 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 construct 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.