Construct a graph of the proportionality of the given function of 3x. Direct proportionality and its graph

How to build direct proportionality graphs?

Plot a direct proportionality graph given the formula y = 3x

Solution .

The function y = 3x is defined on the entire number line. Cm. .

We take any value of x, let it be 1, and find y by substituting x equal to 1 into the formula y = 3x

Y=3x=
3 * 1 = 3

that is, for x = 1 we get y = 3. The point with these coordinates belongs to the graph of the function y = 3x.

We know that the graph of direct proportionality is a straight line, and a straight line is defined by two points.

We just found one of them, and the second for direct proportionality is always the origin.

Now we are ready to graph the function y = 3x.

We mark a point on the coordinate plane with coordinates (1; 3).

Draw a straight line through this point and the origin

We have obtained a graph of direct proportionality given by the formula y = 3x.

Find from the graph the value of y corresponding to the value x = 2.

Find point 2 on the x-axis.

Draw a vertical line through it until it intersects with the graph.

We draw a horizontal line to the axis of the players. On the y-axis we go to point 6.

6 is the value of yk corresponding to the value x = 2.

Definition of direct proportionality

To begin with, recall the following definition:

Definition

Two quantities are called directly proportional if their ratio is equal to a specific non-zero number, that is:

\[\frac(y)(x)=k\]

From here we see that $y=kx$.

Definition

A function of the form $y=kx$ is called direct proportionality.

Direct proportionality is a special case of the linear function $y=kx+b$ for $b=0$. The number $k$ is called the proportionality coefficient.

An example of direct proportionality is Newton's second law: The acceleration of a body is directly proportional to the force applied to it:

Here mass is a coefficient of proportionality.

Study of the function of direct proportionality $f(x)=kx$ and its graph

First, consider the function $f\left(x\right)=kx$, where $k > 0$.

1. $f"\left(x\right)=(\left(kx\right))"=k>0$. 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. 1).

Rice. 1. Graph of the function $y=kx$, 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. $f\left(-x\right)=-kx=-f(x)$. The direct proportionality function is odd.
4. The function passes through the origin.
5. $f"\left(x\right)=(\left(kx\right))"=k
6. $f^("")\left(x\right)=k"=0$. Therefore, the function has no inflection points.
7. $(\mathop(lim)_(x\to -\infty ) kx\ )=+\infty $, $(\mathop(lim)_(x\to +\infty ) kx\ )=-\infty $
8. Graph (Fig. 2).

Rice. 2. Graph of the function $y=kx$, for $k

Important: to plot a graph of the function $y=kx$, it is enough to find one point $\left(x_0,\ y_0\right)$ different from the origin and draw a straight line through this point and the origin.

Let's build a graph of the function given by the formula y = 0.5x.

1. The domain of this function is the set of all numbers.

2. Let's find some corresponding values ​​of the variables X And at.

If x = -4, then y = -2.
If x = -3, then y = -1.5.
If x = -2, then y = -1.
If x = -1, then y = -0.5.
If x = 0, then y = 0.
If x = 1, then y = 0.5.
If x = 2, then y = 1.
If x = 3, then y = 1.5.
If x = 4, then y = 2.

3. Let us mark the points in the coordinate plane whose coordinates we determined in step 2. Note that the constructed points belong to a certain line.

4. Let's determine whether other points on the function graph belong to this line. To do this, we will find the coordinates of several more points on the graph.

If x = -3.5, then y = -1.75.
If x = -2.5, then y = -1.25.
If x = -1.5, then y = -0.75.
If x = -0.5, then y = -0.25.
If x = 0.5, then y = 0.25.
If x = 1.5, then y = 0.75.
If x = 2.5, then y = 1.25.
If x = 3.5, then y = 1.75.

Having constructed new points on the graph of the function, we notice that they belong to the same line.

If we reduce the step of our values ​​(take, for example, the values X through 0,1; through 0,01 etc.), we will receive other graph points belonging to the same line and located increasingly closer to each other from the drag. The set of all points on the graph of a given function is a straight line passing through the origin.

Thus, the graph of the function given by the formula y = khx, where k ≠ 0, is a straight line passing through the origin.

If the domain of definition of the function given by the formula y = khx, where k ≠ 0, does not consist of all numbers, then its graph is a subset of points on a line (for example, a ray, a segment, individual points).

To construct a straight line, it is enough to know the position of its two points. Therefore, a graph of direct proportionality defined on the set of all numbers can be constructed using any two of its points (it is convenient to take the origin of coordinates as one of them).

Let, for example, you want to plot a function given by the formula y = -1.5x. Let's choose some value X, not equal 0 , and calculate the corresponding value at.

If x = 2, then y = -3.

Let us mark a point on the coordinate plane with coordinates (2; -3) . Let's draw a straight line through this point and the origin. This straight line is the desired graph.

Based on this example, it can be proven that any straight line passing through the origin of coordinates and not coinciding with the axes is a graph of direct proportionality.

Proof.

Let a certain straight line be given, passing through the origin of coordinates and not coinciding with the axes. Let's take a point on it with abscissa 1. Let's denote the ordinate of this point by k. Obviously, k ≠ 0. Let us prove that this line is a graph of direct proportionality with coefficient k.

Indeed, from the formula y = kh it follows that if x = 0, then y = 0, if x = 1, then y = k, i.e. the graph of a function given by the formula y = kх, where k ≠ 0, is a straight line passing through the points (0; 0) and (1; k).

Because only one straight line can be drawn through two points, then this straight line coincides with the graph of the function given by the formula y = khx, where k ≠ 0, which was what needed to be proven.

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In grades 7 and 8, the graph of direct proportionality is studied.

How to construct a direct proportionality graph?

Let's look at examples of a direct proportionality graph.

Direct proportionality graph formula

A direct proportionality graph represents a function.

In general, direct proportionality has the formula

The inclination angle of the direct proportionality graph relative to the x-axis depends on the magnitude and sign of the coefficient of direct proportionality.

Direct proportionality graph goes through

A direct proportionality graph passes through the origin.

A direct proportionality graph is a straight line. A straight line is defined by two points.

Thus, when constructing a graph of direct proportionality, it is enough to determine the position of two points.

But we always know one of them - this is the origin of coordinates.

All that remains is to find the second one. Let's look at an example of constructing a graph of direct proportionality.

Graph direct proportionality y = 2x

Task .

Plot a graph of direct proportionality given by the formula

Solution .

All the numbers are there.

Take any number from the domain of direct proportionality, let it be 1.

Find the value of the function when x is equal to 1

Y=2x=
2 * 1 = 2

that is, for x = 1 we get y = 2. The point with these coordinates belongs to the graph of the function y = 2x.

We know that the graph of direct proportionality is a straight line, and a straight line is defined by two points.