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 a 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 a k ≠ 0, then the range of the linear function is the entire real axis. If a 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, Consequently, y = b is even;
b) b = 0, k ≠ 0, Consequently y = kx is odd;
c) b ≠ 0, k ≠ 0, Consequently y = kx + b is a general function;
d) b = 0, k = 0, Consequently 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, Consequently (-b/k; 0)- point of intersection with the abscissa axis.
Oy: y=0k+b=b, Consequently (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 a 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, Consequently y = kx + b increases over the entire domain of definition,
k< 0 , Consequently 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.
Linear function definition
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.
For $b=0$ the linear function is called the direct proportionality function $y=kx$.
Consider Figure 1.
Rice. 1. The geometric meaning of the slope of the straight line
Consider triangle ABC. We see that $BC=kx_0+b$. 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, the following conclusion can be drawn:
Conclusion
Geometric meaning of the coefficient $k$. The slope of the straight line $k$ is equal to the tangent of the slope of this straight line to the axis $Ox$.
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 $k > 0$.
- $f"\left(x\right)=(\left(kx+b\right))"=k>0$. Therefore, this function increases over the entire 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 scope is all numbers.
- The scope is all numbers.
- $f\left(-x\right)=-kx+b$. The function is neither even nor odd.
- For $x=0,f\left(0\right)=b$. For $y=0,0=kx+b,\ x=-\frac(b)(k)$.
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).
A linear function is a function of the form y=kx+b, where x is an 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 equation of the function, and calculate the corresponding y values from them.
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 the 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
The coefficient b shows the shift of the graph of the function 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 up along the OY axis
if b
The figure below shows the graphs of the functions y=2x+3; y= ½x+3; y=x+3
Note that in all these functions the coefficient k Above zero, and 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 the point (0;3)
Now consider the graphs of functions y=-2x+3; y=- ½ x+3; y=-x+3
This time, in all functions, the coefficient k less than zero and features decrease. The coefficient b=3, and the graphs, as in the previous case, cross the OY axis at the point (0;3)
Consider the graphs of functions y=2x+3; y=2x; y=2x-3
Now, in all equations of functions, 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) crosses the OY axis at the point (0;3)
The graph of the function y=2x (b=0) crosses the OY axis at the point (0;0) - the origin.
The graph of the function y=2x-3 (b=-3) crosses the OY axis at the 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 a k 0
If a k>0 and b>0, then the graph of the function y=kx+b looks like:
If a k>0 and b, then the graph of the function y=kx+b looks like:
If a k, then the graph of the function y=kx+b looks like:
If a k=0, then the function y=kx+b turns into a function y=b and its graph looks like:
The ordinates of all points of 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. Separately, we 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, since one value of the argument corresponds to different values of the function, which does not correspond to the definition of the 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. Intersection points 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 instead of x in the equation of the function. We get y=b. That is, the point of intersection with the OY axis has coordinates (0;b).
With the x-axis: The ordinate of any point belonging to the x-axis is zero. Therefore, to find the point of intersection with the OX axis, you need to substitute zero instead of y in the equation of the function. We get 0=kx+b. Hence x=-b/k. That is, the point of intersection with the OX axis has coordinates (-b / k; 0):
