Is the graph of the function y kx b. Linear function

Instructions

If the graph is a straight line passing through the origin of coordinates and forming an angle α with the OX axis (the angle of inclination of the straight line to the positive semi-axis OX). The function describing this line will have the form y = kx. The proportionality coefficient k is equal to tan α. If a straight line passes through the 2nd and 4th coordinate quarters, then k< 0, и является убывающей, если через 1-ю и 3-ю, то k >0 and the function is increasing. Let it be a straight line located in various ways relative to the coordinate axes. This is a linear function and has the form y = kx + b, where the variables x and y are to the first power, and k and b can be either positive or negative or equal to zero. The line is parallel to the line y = kx and cuts off at the axis |b| units. If the line is parallel to the abscissa axis, then k = 0, if the ordinate axis, then the equation has the form x = const.

A curve consisting of two branches located in different quarters and symmetrical relative to the origin of coordinates is a hyperbola. This chart inverse relationship variable y from x and is described by the equation y = k/x. Here k ≠ 0 is the proportionality coefficient. Moreover, if k > 0, the function decreases; if k< 0 - функция возрастает. Таким образом, областью определения функции является вся числовая прямая, кроме x = 0. Ветви приближаются к осям координат как к своим асимптотам. С уменьшением |k| ветки гиперболы все больше «вдавливаются» в coordinate angles.

The quadratic function has the form y = ax2 + bx + c, where a, b and c are constant quantities and a  0. If the condition b = c = 0 is met, the equation of the function looks like y = ax2 (the simplest case), and its the graph is a parabola passing through the origin. The graph of the function y = ax2 + bx + c has the same shape as the simplest case of the function, but its vertex (the point of intersection with the OY axis) does not lie at the origin.

A parabola is also the graph of a power function expressed by the equation y = xⁿ, if n is any even number. If n is any odd number, the graph of such a power function will look like a cubic parabola.
If n is any , the function equation takes the form. The graph of the function for odd n will be a hyperbola, and for even n their branches will be symmetrical with respect to the op axis.

Even in school years, functions are studied in detail and their graphs are constructed. But, unfortunately, they practically do not teach how to read the graph of a function and find its type from the presented drawing. It's actually quite simple if you remember the basic types of functions.

Instructions

If the presented graph is , which is through the origin of coordinates and with the OX axis the angle α (which is the angle of inclination of the straight line to the positive semi-axis), then the function describing such a straight line will be presented as y = kx. In this case, the proportionality coefficient k equal to tangent angle α.

If a given line passes through the second and fourth coordinate quarters, then k is equal to 0 and the function increases. Let the presented graph be a straight line located in any way relative to the coordinate axes. Then the function of such graphic arts will be linear, which is represented by the form y = kx + b, where the variables y and x are in the first, and b and k can take both negative and positive values ​​or.

If the line is parallel to the line with the graph y = kx and cuts off b units on the ordinate axis, then the equation has the form x = const, if the graph is parallel to the abscissa axis, then k = 0.

A curved line that consists of two branches, symmetrical about the origin and located in different quarters, is a hyperbola. Such a graph shows the inverse dependence of the variable y on the variable x and is described by an equation of the form y = k/x, where k should not be equal to zero, since it is a coefficient inverse proportionality. Moreover, if the value of k is greater than zero, the function decreases; if k is less than zero, it increases.

If the proposed graph is a parabola passing through the origin, its function, subject to the condition that b = c = 0, will have the form y = ax2. This is the simplest case quadratic function. The graph of a function of the form y = ax2 + bx + c will have the same form as the simplest case, however, the vertex (the point where the graph intersects the ordinate axis) will not be at the origin. In a quadratic function, represented by the form y = ax2 + bx + c, the values ​​of a, b and c are constant, while a is not equal to zero.

A parabola can also be the graph of a power function expressed by an equation of the form y = xⁿ only if n is any even number. If the value of n is an odd number, such a graph of a power function will be represented by a cubic parabola. In case the variable n is any negative number, the equation of the function takes the form .

Video on the topic

The coordinate of absolutely any point on the plane is determined by its two quantities: along the abscissa axis and the ordinate axis. The collection of many such points represents the graph of the function. From it you can see how the Y value changes depending on the change in the X value. You can also determine in which section (interval) the function increases and in which it decreases.

Instructions

What can you say about a function if its graph is a straight line? See if this line passes through the coordinate origin point (that is, the one where the X and Y values ​​​​are equal to 0). If it passes, then such a function is described by the equation y = kx. It is easy to understand that the larger the value of k, the closer to the ordinate axis this straight line will be located. And the Y axis itself actually corresponds infinitely of great importance k.

In this article we will look at linear function, graph of a linear function and its properties. And, as usual, we will solve several problems on this topic.

Linear function called a function of the form

In a function equation, the number we multiply by is called the slope coefficient.

For example, in the function equation ;

in the equation of the function;

in the equation of the function;

in the function equation.

The graph of a linear function is a straight line.

1 . To plot a function, 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 a function graph, it is convenient to take and , then the ordinates of these points will be equal to and .

We get points A(0;2) and B(3;3). Let's connect them and get a graph of the function:

2 . In a function equation, the coefficient is responsible for the slope of the function graph:

Title="k>0">!}

The coefficient is responsible for shifting the graph along the axis:

Title="b>0">!}

The figure below shows graphs of functions; ;

Note that in all these functions the coefficient Above zero right. Moreover, than more value, the steeper the straight line goes.

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

Now let's look at the graphs of functions; ;

This time in all functions the coefficient less than zero, and all function graphs are sloped left.

Note that the larger |k|, the steeper the straight line. The coefficient b is the same, 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 functions; ;

Now the coefficients in all function equations are equal. 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 (b=3) intersects the OY axis at point (0;3)

The graph of the function (b=0) intersects the OY axis at the point (0;0) - the origin.

The graph of the function (b=-2) intersects the OY axis at point (0;-2)

So, if we know the signs of the coefficients k and b, then we can immediately imagine what the graph of the function looks like.

If k<0 и b>0 , then the graph of the function looks like:

If k>0 and b>0 , then the graph of the function looks like:

If k>0 and b<0 , then the graph of the function looks like:

If k<0 и b<0 , then the graph of the function looks like:

If k=0 , then the function turns into a function and its graph looks like:

The ordinates of all points on the graph of the function are equal

If b=0, then the graph of the function passes through the origin:

This direct proportionality graph.

3. I would like to separately note the graph of the equation. The graph of this equation is a straight line parallel to the axis, all points of which have an abscissa.

For example, the graph of the equation looks like this:

Attention! The equation is not a function, since different values ​​of the argument correspond to the same value of the function, which does not correspond.

4 . Condition for parallelism of two lines:

Graph of a function parallel to the graph of the function, If

5. The condition for the perpendicularity of two straight lines:

Graph of a function perpendicular to the graph of the function, if or

6. Points of intersection of the graph of a function 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 equal to 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. From here. That is, the point of intersection with the OX axis has coordinates (;0):

Let's look at problem solving.

1 . Construct a graph of the function if it is known that it passes through the point A(-3;2) and is parallel to the straight line y=-4x.

The function equation has two unknown parameters: k and b. Therefore, the text of the problem must contain two conditions characterizing the graph of the function.

a) From the fact that the graph of the function is parallel to the straight line y=-4x, it follows that k=-4. That is, the function equation has the form

b) We just have to find b. It is known that the graph of the function passes through point A(-3;2). If a point belongs to the graph of a function, then when substituting its coordinates into the equation of the function, we obtain the correct equality:

hence b=-10

Thus, we need to plot the function

We know point A(-3;2), let’s take point B(0;-10)

Let's put these points in the coordinate plane and connect them with a straight line:

2. Write the equation of the line passing through the points A(1;1); B(2;4).

If a line passes through points with given coordinates, therefore, the coordinates of the points satisfy the equation of the line. That is, if we substitute the coordinates of the points into the equation of the straight line, we will get the correct equality.

Let's substitute the coordinates of each point into the equation and get the system linear equations.

Subtract the first from the second equation of the system and get . Let's substitute the value of k into the first equation of the system and get b=-2.

So, the equation of the line.

3. Graph the Equation

To find at what values ​​of the unknown the product of several factors equals zero, you need to equate each factor to zero and take into account each multiplier.

This equation has no restrictions on ODZ. Let's factorize the second bracket and set each factor equal to zero. We obtain a set of equations:

Let's construct graphs of all equations of the set in one coordinate plane. This is the graph of the equation :

4 . Construct a graph of the function if it is perpendicular to the line and passes through the point M(-1;2)

We will not build a graph, we will only find the equation of the line.

a) Since the graph of a function, if it is perpendicular to a line, therefore, hence. That is, the function equation has the form

b) We know that the graph of the function passes through the point M(-1;2). Let's substitute its coordinates into the equation of the function. We get:

From here.

Therefore, our function looks like: .

5 . Graph the Function

Let's simplify the expression on the right side of the function equation.

Important! Before simplifying the expression, let's find its ODZ.

The denominator of a fraction cannot be zero, so title="x1">, title="x-1">.!}

Then our function takes the form:

Title="delim(lbrace)(matrix(3)(1)((y=x+2) (x1) (x-1)))( )">!}

That is, we need to build a graph of the function and cut out two points on it: with abscissas x=1 and x=-1:

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There are a total of 25 presentations in the topic

Instructions

There are several ways to solve linear functions. Let's list the most of them. Most often used step by step method substitutions. In one of the equations it is necessary to express one variable in terms of another and substitute it into another equation. And so on until only one variable remains in one of the equations. To solve it, you need to leave a variable on one side of the equal sign (it can be with a coefficient), and on the other side of the equal sign all the numerical data, not forgetting to change the sign of the number to the opposite one when transferring. Having calculated one variable, substitute it into other expressions and continue calculations using the same algorithm.

For example, let's take a linear system functions, consisting of two equations:
2x+y-7=0;
x-y-2=0.
It is convenient to express x from the second equation:
x=y+2.
As you can see, when transferring from one part of the equality to another, the sign of y and variables changed, as was described above.
We substitute the resulting expression into the first equation, thus excluding the variable x from it:
2*(y+2)+y-7=0.
Expanding the brackets:
2y+4+y-7=0.
We put together variables and numbers and add them up:
3у-3=0.
Move to the right side of the equation and change the sign:
3y=3.
Divide by overall coefficient, we get:
y=1.
We substitute the resulting value into the first expression:
x=y+2.
We get x=3.

Another way to solve similar ones is to add two equations term by term to get a new one with one variable. The equation can be multiplied by a certain coefficient, the main thing is to multiply each member of the equation and not forget, and then add or subtract one equation from. This method is very economical when finding a linear functions.

Let’s take the already familiar system of equations with two variables:
2x+y-7=0;
x-y-2=0.
It is easy to notice that the coefficient of the variable y is identical in the first and second equations and differs only in sign. This means that when we add these two equations term by term, we get a new one, but with one variable.
2x+x+y-y-7-2=0;
3x-9=0.
We transfer the numerical data to the right side of the equation, changing the sign:
3x=9.
Finding the common factor equal to the coefficient, standing at x and divide both sides of the equation by it:
x=3.
The result can be substituted into any of the system equations to calculate y:
x-y-2=0;
3-у-2=0;
-y+1=0;
-y=-1;
y=1.

You can also calculate data by creating an accurate graph. To do this you need to find zeros functions. If one of the variables is equal to zero, then such a function is called homogeneous. Having solved such equations, you will get two points necessary and sufficient to construct a straight line - one of them will be located on the x-axis, the other on the y-axis.

We take any equation of the system and substitute the value x=0 there:
2*0+y-7=0;
We get y=7. Thus, the first point, let's call it A, will have coordinates A(0;7).
In order to calculate a point lying on the x-axis, it is convenient to substitute the value y=0 into the second equation of the system:
x-0-2=0;
x=2.
The second point (B) will have coordinates B (2;0).
We mark the obtained points on the coordinate grid and draw a straight line through them. If you plot it fairly accurately, other values ​​of x and y can be calculated directly from it.