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 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 general view;

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.

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Which is what its name is associated with. This concerns a real function of one real variable.

Encyclopedic YouTube

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    If all variables x 1 , x 2 , … , x n (\displaystyle x_(1),x_(2),\dots ,x_(n)) and odds a 0 , a 1 , a 2 , … , a n (\displaystyle a_(0),a_(1),a_(2),\dots ,a_(n)) are real numbers, then the graph of a linear function in (n + 1) (\displaystyle (n+1))-dimensional space of variables x 1 , x 2 , … , x n , y (\displaystyle x_(1),x_(2),\dots ,x_(n),y) is n (\displaystyle n)-dimensional hyperplane

    y = a 0 + a 1 x 1 + a 2 x 2 + ⋯ + a n x n (\displaystyle y=a_(0)+a_(1)x_(1)+a_(2)x_(2)+\dots +a_ (n)x_(n))

    in particular when n = 1 (\displaystyle n=1)- a straight line on a plane.

    Abstract algebra

    The term "linear function", or more precisely "linear homogeneous function", is often used to describe a linear representation of a vector space X (\displaystyle X) over some field k (\displaystyle k) into this field, that is, for such a display f: X → k (\displaystyle f:X\to k), which for any elements x , y ∈ X (\displaystyle x,y\in X) and any α , β ∈ k (\displaystyle \alpha ,\beta \in k) equality is true

    f (α x + β y) = α f (x) + β f (y) (\displaystyle f(\alpha x+\beta y)=\alpha f(x)+\beta f(y))

    Moreover, in this case, instead of the term “linear function”, the terms linear functional and linear form are also used - also meaning linear homogeneous function of a certain class.