Important!
A function of the form “y = kx + b” is called a linear function.
The letter factors "k" and "b" are called numerical coefficients.
Instead of “k” and “b” there can be any numbers (positive, negative or fractions).
In other words, we can say that “y = kx + b” is a family of all possible functions, where instead of “k” and “b” there are numbers.
Examples of functions like “y = kx + b”.
- y = 5x + 3
- y = −x + 1
- y = x − 2
k = 2 3 b = −2 y = 0.5x k = 0.5 b = 0 Pay special attention to the "y = 0.5x" function in the table. They often make the mistake of looking for the numerical coefficient “b”.
When considering the function “y = 0.5x”, it is incorrect to say that there is no numerical coefficient “b” in the function.
The numerical coefficient "b" is always present in a function like "y = kx + b" always. In the function “y = 0.5x” the numerical coefficient “b” is zero.
How to graph a linear function
"y = kx + b"Remember!
The graph of the linear function “y = kx + b” is a straight line.
Since the graph of the function “y = kx + b” is a straight line, the function is called linear function.
From geometry, let us recall the axiom (a statement that does not require proof) that through any two points you can draw a straight line and, moreover, only one.
Based on the axiom above, it follows that in order to plot a function of the form
“y = kx + b” it will be enough for us to find only two points.For example let's build a graph of the function"y = −2x + 1".
Let's find the value of the function "y" for two arbitrary values "x". Let us substitute, for example, instead of “x” the numbers “0” and “1”.
Important!
When choosing arbitrary numeric values instead of “x”, it is better to take the numbers “0” and “1”. It's easy to do calculations with these numbers.
The resulting values “x” and “y” are the coordinates of the points on the function graph.
Let's write the obtained coordinates of the points “y = −2x + 1” into the table.
Let us mark the obtained points on the coordinate system.
Now let's draw a straight line through the marked points. This straight line will be the graph of the function “y = −2x + 1”.
How to solve problems on
linear function “y = kx + b”Let's consider the problem.
Graph the function “y = 2x + 3”. Find by graph:
- the value “y” corresponding to the value “x” equal to −1; 2; 3; 5 ;
- the value of "x" if the value of "y" is 1; 4; 0; −1.
First, let's plot the function “y = 2x + 3”.
We use the rules by which we are superior. To graph the function “y = 2x + 3” it is enough to find only two points.
Let's choose two arbitrary numeric values for “x”. For convenience of calculations, we will choose the numbers “0” and “1”.
Let's carry out the calculations and write their results in the table.
Let us mark the obtained points on the rectangular coordinate system.
Let's connect the resulting points with a straight line. The drawn straight line will be a graph of the function “y = 2x + 3”.
Now we work with the constructed graph of the function “y = 2x + 3”.
You need to find the value “y” corresponding to the value “x”,
which is equal to −1; 2; 3; 5.- Ox" to zero (x = 0) ;
- substitute zero instead of “x” in the function formula and find the value “y”;
- Oy".
Instead of “x” in the formula of the function “y = −1.5x + 3”, let’s substitute the number zero.
Y(0) = −1.5 0 + 3 = 3
(0; 3) - coordinates of the point of intersection of the graph of the function “y = −1.5x + 3” with the axis “Oy”.Remember!
To find the coordinates of the intersection point of the graph of a function
with axis " Ox"(x axis) you need:- equate the coordinate of a point along the "" axis Oy" to zero (y = 0) ;
- substitute zero instead of “y” in the function formula and find the value “x”;
- write down the obtained coordinates of the point of intersection with the axis " Oy".
Instead of “y” in the formula of the function “y = −1.5x + 3”, let’s substitute the number zero.
0 = −1.5x + 3
1.5x = 3 | :(1.5)
x = 3: 1.5
x = 2
(2; 0) - coordinates of the point of intersection of the graph of the function “y = −1.5x + 3” with the “Ox” axis.To make it easier to remember which coordinate of a point should be equated to zero, remember the “rule of opposites.”
Important!
If you need to find the coordinates of the point of intersection of the graph with the axis " Ox", then we equate “y” to zero.
And vice versa. If you need to find the coordinates of the point of intersection of the graph with the "" axis Oy", then we equate “x” to zero.
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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 – inclination 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.
