It’s impossible to say that you know math if you don’t know how to draw graphs, depict inequalities on the coordinate line, and work with coordinate axes. The visual component in science is vital, because without illustrative examples in formulas and calculations one can sometimes get very confused. In this article, we will see how to work with coordinate axes, and learn how to build simple function graphs.
Application
The coordinate line is the basis of the simplest types of graphs that a student encounters on his educational path. It is used in almost every mathematical topic: when calculating speed and time, projecting the sizes of objects and calculating their area, in trigonometry when working with sines and cosines.
The main value of such a direct line is clarity. Since mathematics is a science that requires a high level of abstract thinking, graphics help in representing the object in the real world. How does he behave? At what point in space will it be in a few seconds, minutes, hours? What can be said about it in comparison with other objects? What speed does he have at a randomly selected point in time? How to characterize his movement?
And it’s not without reason about speed - this is what function graphics often display. And they can also display the change in temperature or pressure inside the object, its size, orientation relative to the horizon. Thus, the construction of a coordinate line is often required in physics.
One-dimensional graph
There is the concept of multidimensionality. In one-dimensional space , only one number is enough to determine the location of a point. This is precisely the case with the use of the coordinate line. If the space is two-dimensional, then two numbers are required. Charts of this type are used much more often, and a little further in the article we will definitely consider them.
What can be seen with the help of points on the axis, if it is only one? You can see the size of the object, its position in space relative to some "zero", that is, the point selected as the reference point.
Changing the parameters over time will not be possible to see, since all readings will be displayed for one specific moment. However, you have to start somewhere! So let's get started.
How to build a coordinate axis
First you need to draw a horizontal line - this will be our axis. On the right side, “sharpen” it so that it looks like an arrow. Thus, we denote the direction in which the numbers will increase. In the direction of decrease, the arrow is usually not set. Traditionally, the axis is directed to the right, so we just follow this rule.
Put a zero mark, which will display the origin. This is the very place from which the count is taken, be it size, weight, speed or anything else. In addition to zero, we must necessarily designate the so-called division price, that is, introduce the unit standard, in accordance with which we will postpone certain values ​​on the axis. This must be done in order to be able to find the length of the segment on the coordinate line.
Through an equal distance from each other, we put points or “notches” on the line, and below them we write 1,2,3 and so on. And now, everything is ready. But you still have to learn how to work with the resulting schedule.
Types of points on the coordinate line
At first glance, the drawings proposed in the textbooks become clear: the points on the axis can be filled or not filled. Do you think this is an accident? Not at all! A “solid” point is used for non-strict inequality - one that reads as “greater than or equal to”. If you need to strictly limit the interval (for example, “X” can take values ​​from zero to one, but does not include it), we will use the “hollow” point, that is, in fact, a small circle on the axis. It should be noted that students do not really like strict inequalities, because it is more difficult to work with them.
Depending on what points you use on the chart, the constructed intervals will also be called. If the inequality on both sides is not strict, then we get a segment. If on the one hand it turns out to be “open”, then it will be called a half-interval. Finally, if part of the line is bounded on two sides by hollow points, it will be called an interval.
Plane
When constructing two lines on the coordinate plane, we can already consider function graphs. Say the horizontal line will be the time axis, and the vertical line will be the distance. And now we are able to determine how far the object will travel in a minute or an hour. Thus, working with the plane makes it possible to monitor the change in the state of the object. This is much more interesting than investigating a static state.
The simplest graph on such a plane is a straight line, it reflects the function Y (X) = aX + b. Does the line bend? This means that the object changes its characteristics in the research process.
Imagine you are standing on the roof of a building and holding a stone in your outstretched hand. When you release it, it will fly down, starting its movement at zero speed. But in a second he will overcome 36 kilometers per hour. The stone will continue to accelerate further, and in order to draw its movement on the chart, you will need to measure its speed at several points in time by setting points on the axis in the appropriate places.
Marks on the horizontal coordinate line by default are called X1, X2, X3, and on the vertical - Y1, Y2, Y3, respectively. Projecting them onto a plane and finding intersections, we find fragments of the resulting drawing. Connecting them with one line, we get a graph of the function. In the case of a falling stone, the quadratic function will have the form: Y (X) = aX * X + bX + c.
Scale
Of course, it is not necessary to set integer values ​​next to divisions on a straight line. If you are considering the movement of a cochlea that crawls at a speed of 0.03 meters per minute, set the fraction on the coordinate line as values. In this case, set the division price to 0.01 meters.
It is especially convenient to carry out such drawings in a notebook in a cage - here you can immediately see whether there is enough space on the sheet for your schedule, whether you go beyond the fields. It’s easy to calculate your strength, because the width of the cell in such a notebook is 0.5 centimeters. It took - reduced the picture. From a change in the scale of the graph, it will not lose or change its properties.
Point and line coordinates
When a math problem is given in a lesson, it can contain parameters of various geometric shapes, both in the form of side lengths, perimeter, area, and in the form of coordinates. In this case, you may need to both build a figure and get some data related to it. The question arises: how to find the required information on the coordinate line? And how to build a figure?
For example, we are talking about a point. Then the capital letter will appear in the condition of the problem, and several numbers will appear in brackets, most often two (this means we will count in two-dimensional space). If there are three numbers in brackets, written through a semicolon or through a comma, then this is three-dimensional space. Each of the values ​​is a coordinate on the corresponding axis: first along the horizontal (X), then along the vertical (Y).
Remember how to build a line? You went through it on geometry. If there are two points, then a straight line can be drawn between them. Their coordinates are indicated in brackets if a segment appears in the problem. For example: A (15, 13) - B (1, 4). To build such a line, you need to find and mark the points on the coordinate plane, and then connect them. That's all!
And any polygons, as you know, can be drawn using segments. The problem is solved.
Calculations
Suppose there is some object whose position along the X axis is characterized by two numbers: it starts at a point with coordinate (-3) and ends at (+2). If we want to know the length of this item, we must subtract the smaller from the larger number. Please note that a negative number absorbs the subtraction sign, because “minus minus gives plus”. So, we add (2 + 3) and get 5. This is the desired result.
Another example: we are given the end point and the length of the object, but the initial is not given (and we need to find it). Let the position of the known point be (6), and the size of the studied subject - (4). Subtracting the length from the final coordinate, we get the answer. Total: (6 - 4) = 2.
Negative numbers
It is often required in practice to work with negative values. In this case, we will go to the left along the coordinate axis. For example, an object 3 centimeters high floats in water. A third of it is immersed in liquid, two-thirds is in the air. Then, choosing the surface of the water as the axis, we use the simplest arithmetic calculations to get two numbers: the top point of the object has the coordinate (+2), and the bottom - (-1) centimeter.
It is easy to see that in the case of a plane, we form four quarters of the coordinate line. Each of them has its own number. In the first (upper right) part, there will be points having two positive coordinates, in the second - from the top left - the values ​​along the "x" axis will be negative, and along the "igra" - positive. The third and fourth count down further counterclockwise.
Important property
You know that a straight line can be represented as an infinite number of points. We can look at any number of values ​​on each side of the axis arbitrarily carefully, but we will not meet duplicate ones. This seems naive and understandable, but this statement stems from an important fact: each number corresponds to one and only one point on the coordinate line.
Conclusion
Remember that any axis, shapes and, if possible, graphics must be built on a ruler. Units of measurements were not invented by a person by chance - if you make a mistake in drawing, you run the risk of seeing not the same image that should have turned out.
Be careful and accurate when plotting and calculating. Like any science studied at school, mathematics loves accuracy. Put a little effort, and good grades will not take long.