In the previous chapter, it was shown that, by choosing a certain coordinate system on the plane, we can analytically express the geometric properties characterizing the points of the line under consideration by an equation between the current coordinates. Thus, we get the equation of the line. In this chapter, the equations of straight lines will be considered.

To formulate the equation of a straight line in Cartesian coordinates, you need to somehow set the conditions that determine its position relative to the coordinate axes.

First, we introduce the concept of the slope of a straight line, which is one of the quantities characterizing the position of a straight line on a plane.

Let's call the angle of inclination of the line to the Ox axis the angle by which the Ox axis must be rotated so that it coincides with the given line (or turns out to be parallel to it). As usual, we will consider the angle taking into account the sign (the sign is determined by the direction of rotation: counterclockwise or clockwise). Since an additional rotation of the Ox axis by an angle of 180 ° will again combine it with the straight line, the angle of inclination of the straight line to the axis can be chosen ambiguously (up to a multiple of ).

The tangent of this angle is uniquely determined (since changing the angle to does not change its tangent).

The tangent of the angle of inclination of a straight line to the x-axis is called the slope of the straight line.

The slope characterizes the direction of the straight line (here we do not distinguish between two mutually opposite directions of the straight line). If the slope of the line is zero, then the line is parallel to the x-axis. With a positive slope, the angle of inclination of the straight line to the Ox axis will be sharp (we are considering here the smallest positive value of the angle of inclination) (Fig. 39); in this case, the larger the slope, the greater the angle of its inclination to the Ox axis. If the slope is negative, then the angle of inclination of the straight line to the x-axis will be obtuse (Fig. 40). Note that a straight line perpendicular to the x-axis does not have a slope (the tangent of an angle does not exist).

Numerically equal to the tangent of the angle (constituting the smallest rotation from the Ox axis to the Oy axis) between the positive direction of the x-axis and the given straight line.

The tangent of an angle can be calculated as the ratio of the opposite leg to the adjacent one. k is always equal to , that is, the derivative of the straight line equation with respect to x.

With positive values ​​of the angular coefficient k and zero value of the shift coefficient b line will lie in the first and third quadrants (in which x And y both positive and negative). At the same time, large values ​​of the angular coefficient k a steeper straight line will correspond, and a smaller one - a flatter one.

Lines and are perpendicular if , and parallel when .

Notes

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See what the "Line Slope" is in other dictionaries:

slope (straight)- — Topics oil and gas industry EN slope … Technical Translator's Handbook

- (mathematical) number k in the equation of a straight line on the plane y \u003d kx + b (see Analytical geometry), characterizing the slope of the straight line relative to the abscissa axis. In a rectangular coordinate system U. to. k \u003d tg φ, where φ is the angle between ... ... Great Soviet Encyclopedia

A branch of geometry that studies the simplest geometric objects by means of elementary algebra based on the method of coordinates. The creation of analytical geometry is usually attributed to R. Descartes, who outlined its foundations in the last chapter of his ... ... Collier Encyclopedia

The measurement of reaction time (RT) is probably the most revered subject in empirical psychology. It originated in the field of astronomy, in 1823, with the measurement of individual differences in the speed at which a star was perceived to cross the telescope's line of sight. These … Psychological Encyclopedia

A branch of mathematics that gives methods for the quantitative study of various processes of change; deals with the study of the rate of change (differential calculus) and the determination of the lengths of curves, areas and volumes of figures bounded by curved contours and ... Collier Encyclopedia

This term has other meanings, see Direct (meanings). A straight line is one of the basic concepts of geometry, that is, it does not have an exact universal definition. In a systematic presentation of geometry, a straight line is usually taken as one ... ... Wikipedia

Representation of straight lines in a rectangular coordinate system A straight line is one of the basic concepts of geometry. In a systematic presentation of geometry, a straight line is usually taken as one of the initial concepts, which is only indirectly determined ... ... Wikipedia

Representation of straight lines in a rectangular coordinate system A straight line is one of the basic concepts of geometry. In a systematic presentation of geometry, a straight line is usually taken as one of the initial concepts, which is only indirectly determined ... ... Wikipedia

Not to be confused with the term "Ellipsis". Ellipse and its foci Ellipse (other Greek ἔλλειψις disadvantage, in the sense of lack of eccentricity up to 1) the locus of points M of the Euclidean plane, for which the sum of the distances from two given points F1 ... ... Wikipedia

Learn to take derivatives of functions. The derivative characterizes the rate of change of a function at a certain point lying on the graph of this function. In this case, the graph can be either a straight line or a curved line. That is, the derivative characterizes the rate of change of the function at a particular point in time. Remember the general rules by which derivatives are taken, and only then proceed to the next step.

• Read the article.
• How to take the simplest derivatives, for example, the derivative of an exponential equation, is described. The calculations presented in the following steps will be based on the methods described there.

Learn to distinguish between problems in which the slope needs to be calculated in terms of the derivative of a function. In tasks, it is not always suggested to find the slope or derivative of a function. For example, you may be asked to find the rate of change of a function at point A(x, y). You may also be asked to find the slope of the tangent at point A(x, y). In both cases, it is necessary to take the derivative of the function.

Take the derivative of the given function. You don't need to build a graph here - you only need the equation of the function. In our example, take the derivative of the function f (x) = 2 x 2 + 6 x (\displaystyle f(x)=2x^(2)+6x). Take the derivative according to the methods outlined in the article mentioned above:

Substitute the coordinates of the point given to you into the found derivative to calculate the slope. The derivative of the function is equal to the slope at a certain point. In other words, f "(x) is the slope of the function at any point (x, f (x)). In our example:

Let on a plane where there is a rectangular Cartesian coordinate system, a straight line l passes through the point M 0 parallel to the direction vector A (Fig. 96).

If straight l crosses the O axis X(at point N), then at an angle of a straight line l with O axis X we will understand the angle α by which it is necessary to rotate the axis O X around point N in the opposite direction of clockwise rotation so that the O axis X coincided with the line l. (This refers to an angle less than 180°.)

This corner is called tilt angle straight. If straight l parallel to the O axis X, then the angle of inclination is assumed to be zero (Fig. 97).

The tangent of the slope of a straight line is called slope of a straight line and is usually denoted by the letter k:

tgα = k. (1)

If α = 0, then k= 0; this means that the line is parallel to the o-axis X and its slope is zero.

If α = 90°, then k= tg α does not make sense: this means that the line perpendicular to the O axis X(i.e. parallel to the O axis at), has no slope.

The slope of a straight line can be calculated if the coordinates of any two points of this straight line are known. Let two points of a straight line be given: M 1 ( x 1 ; at 1) and M 2 ( x 2 ; at 2) and let, for example, 0< α < 90°, а x 2 > x 1 , at 2 > at 1 (Fig. 98).

Then from a right triangle M 1 PM 2 we find

$$ k=tga = \frac(|M_2 P|)(|M_1 P|) = \frac(y_2 - y_1)(x_2 - x_1) $$

$$ k=\frac(y_2 - y_1)(x_2 - x_1) \;\; (2)$$

Similarly, we prove that formula (2) is also true in the case of 90°< α < 180°.

Formula (2) loses its meaning if x 2 - x 1 = 0, i.e. if the line l parallel to the O axis at. For such lines, the slope does not exist.

Task 1. Determine the slope of the prima passing through the points

M 1 (3; -5) and M 2 (5; -7).

Substituting the coordinates of the points M 1 and M 2 into formula (2), we obtain

\(k=\frac(-7-(-5))(5-3) \) or k = -1

Task 2. Determine the slope of the straight line passing through the points M 1 (3; 5) and M 2 (3; -2).

Because x 2 - x 1 = 0, then equality (2) loses its meaning. For this direct slope does not exist. The straight line M 1 M 2 is parallel to the O axis at.

Task 3. Determine the slope of the straight line passing through the origin and point M 1 (3; -5)

In this case, the point M 2 coincides with the origin. Applying formula (2), we obtain

$$ k=\frac(y_2 - y_1)(x_2 - x_1)=\frac(0-(-5))(0-3)= -\frac(5)(3); \;\; k= -\frac(5)(3) $$

Compose the equation of a straight line with a slope k passing through the point

M 1 ( x 1 ; at 1). According to formula (2), the slope of a straight line is found from the coordinates of its two points. In our case, the point M 1 is given, and as the second point, you can take any point M( X; at) of the desired line.

If the point M lies on a straight line that passes through the point M 1 and has a slope k, then by formula (2) we have

$$ \frac(y-y_1)(x-x_1)=k \;\; (3) $$

If the point M does not lie on the line, then equality (3) does not hold. Therefore, equality (3) is the equation of a straight line passing through the point M 1 ( x 1 ; at 1) with slope k; this equation is usually written as

y- y 1 = k(x - x 1). (4)

If the line intersects the O axis at at some point (0; b), then equation (4) takes the form

at - b = k (X- 0),

y = kx + b. (5)

This equation is called equation of a straight line with slope k and initial ordinate b.

Task 4. Find the angle of inclination of a straight line √3 x + 3at - 7 = 0.

We bring this equation to the form

$$ y= =\frac(1)(\sqrt3)x + \frac(7)(3) $$

Hence, k= tg α = - 1 / √ 3 , whence α = 150°

Task 5. Compose the equation of a straight line passing through the point P (3; -4), with a slope k = 2 / 5

Substituting k = 2 / 5 , x 1 = 3, y 1 = - 4 in equation (4), we get

at - (- 4) = 2 / 5 (X- 3) or 2 X - 5at - 26 = 0.

Task 6. Compose the equation of a straight line passing through the point Q (-3; 4) and a component with a positive direction of the O axis X angle 30°.

If α = 30°, then k= tan 30° = √ 3 / 3 . Substituting into equation (4) the values x 1 , y 1 and k, we get

at -4 = √ 3 / 3 (x+ 3) or √3 x-3y + 12 + 3√3 = 0.

The topic "The angular coefficient of the tangent as the tangent of the angle of inclination" in the certification exam is given several tasks at once. Depending on their condition, the graduate may be required to provide both a full answer and a short one. When preparing for the exam in mathematics, the student should definitely repeat the tasks in which it is required to calculate the slope of the tangent.

The Shkolkovo educational portal will help you do this. Our experts have prepared and presented theoretical and practical material as accessible as possible. Having become acquainted with it, graduates with any level of training will be able to successfully solve problems related to derivatives, in which it is required to find the tangent of the slope of the tangent.

Basic moments

To find the correct and rational solution to such tasks in the USE, it is necessary to recall the basic definition: the derivative is the rate of change of the function; it is equal to the tangent of the slope of the tangent drawn to the graph of the function at a certain point. It is equally important to complete the drawing. It will allow you to find the correct solution to the USE problems on the derivative, in which it is required to calculate the tangent of the slope of the tangent. For clarity, it is best to plot a graph on the OXY plane.

If you have already familiarized yourself with the basic material on the topic of the derivative and are ready to start solving problems for calculating the tangent of the slope of the tangent, similar to the USE tasks, you can do this online. For each task, for example, tasks on the topic "Relationship of the derivative with the speed and acceleration of the body", we wrote down the correct answer and the solution algorithm. In this case, students can practice performing tasks of various levels of complexity. If necessary, the exercise can be saved in the "Favorites" section, so that later you can discuss the decision with the teacher.