How to find the coordinates of the points of intersection with the axes. Coordinates of the intersection point of function graphs. The case of two nonlinear functions

In practice and in textbooks, the most common methods listed below are for finding the intersection point of various function graphs.

The first and simplest is to take advantage of the fact that at this point the coordinates will be equal and equate the graphs, and from what you get you can find $x$. Then substitute the found $x$ into any of the two equations and find the coordinate of the game.

Example 1

Let's find the intersection point of two lines $y=5x + 3$ and $y=x-2$, equating the functions:

$x=-\frac(1)(2)$

Now let’s substitute the x we ​​received into any graph, for example, choose the one that is simpler - $y=x-2$:

$y=-\frac(1)(2) – 2 = - 2\frac12$.

The intersection point will be $(-\frac(1)(2);- 2\frac12)$.

The second method is that a system is compiled from existing equations, by means of transformations one of the coordinates is made explicit, that is, expressed through the other. After this expression in the given form is substituted into another.

Example 2

Find out at what points the graphs of the parabola $y=2x^2-2x-1$ and the straight line $y=x+1$ intersect.

Solution:

Let's create a system:

$\begin(cases) y=2x^2-2x-1 \\ y= x + 1 \\ \end(cases)$

The second equation is simpler than the first, so let’s substitute it for $y$:

$x+1 = 2x^2 – 2x-1$;

$2x^2 – 3x – 2 = 0$.

Let’s calculate what x is equal to, to do this we will find the roots that make the equality true, and write down the answers received:

$x_1=2; x_2 = -\frac(1)(2)$

Let’s substitute our results along the x-axis one by one into the second equation of the system:

$y_1= 2 + 1 = 3; y_2=1 - \frac(1)(2) = \frac(1)(2)$.

The intersection points will be $(2;3)$ and $(-\frac(1)(2); \frac(1)(2))$.

Let's move on to the third method - graphical, but keep in mind that the result it gives is not quite accurate.

To apply the method, both function graphs are plotted on the same scale on the same drawing, and then a visual search for the intersection point is performed.

This method is good only if an approximate result is sufficient, and also if there is no data on the patterns of the dependencies under consideration.

Consider two linear functions $ f(x) = k_1 x+m_1 $ and $ g(x) = k_2 x + m_2 $. These functions are called direct. It is quite easy to construct them; you need to take any two values ​​$ x_1 $ and $ x_2 $ and find $ f(x_1) $ and $ (x_2) $. Then repeat the same with the function $ g(x) $. Next, visually find the coordinate of the intersection point of the function graphs.

You should know that linear functions have only one intersection point and only when $ k_1 \neq k_2 $. Otherwise, in the case $ k_1=k_2 $ the functions are parallel to each other, since $ k $ is the slope coefficient. If $ k_1 \neq k_2 $, but $ m_1=m_2 $, then the intersection point will be $ M(0;m) $. It is advisable to remember this rule to quickly solve problems.

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Another New Year's Eve... frosty weather and snowflakes on the window glass... All this prompted me to write again about... fractals, and what Wolfram Alpha knows about it. There is an interesting article on this subject, which contains examples of two-dimensional fractal structures. Here we will look at more complex examples of three-dimensional fractals.

A fractal can be visually represented (described) as a geometric figure or body (meaning that both are a set, in this case, a set of points), the details of which have the same shape as the original figure itself. That is, this is a self-similar structure, examining the details of which when magnified, we will see the same shape as without magnification. Whereas in the case of an ordinary geometric figure (not a fractal), upon magnification we will see details that have a simpler shape than the original figure itself. For example, at a high enough magnification, part of an ellipse looks like a straight line segment. This does not happen with fractals: with any increase in them, we will again see the same complex shape, which will be repeated again and again with each increase.

Benoit Mandelbrot, the founder of the science of fractals, wrote in his article Fractals and Art in the Name of Science: “Fractals are geometric shapes that are as complex in their details as in their overall form. That is, if part of the fractal will be enlarged to the size of the whole, it will appear as a whole, either exactly, or perhaps with a slight deformation."