Tip 1: How to calculate the equation of a straight line
Tip 1: How to calculate the equation of a straight line
The equation straight can uniquely determine its position in thespace. A straight line can be given by two points, like the line of intersection of two planes, a point and a collinear vector. Depending on this, find the equation straight There are several ways.
Instructions
1
If a straight line is given by two points, find itequation by the formula (x-x1) / (x2-x1) = (y-y1) / (y2-y1) = (z-z1) / (z2-z1). Substitute the coordinates of the first point (x1, y1, z1) and the second point (x2, y2, z2) in the equation and simplify the expression.
2
Perhaps the points are given to you only by two coordinates, for example, (x1, y1) and (x2, y2), in which case the equation straight find by the simplified formula (x-x1) / (x2-x1) = (y-y1) / (y2-y1). To make it more visual and convenient, express y in terms of x - bring the equation to the form y = kx + b.
3
In order to find the equation straight, which is the line of intersection of two planes,make the equations of these planes in the system and solve it. As a rule, the plane is given by an expression of the form Ax + Boo + Cz + D = 0. Thus, solving the system A1x + B1y + C1z + D1 = 0 and A2x + B2y + C2z + D2 = 0 with respect to the unknown x and y (that is, you take z as a parameter or a number), you get two reduced equations: x = mz + a and y = nz + b.
4
If necessary, from the above equations get the canonical equation straight. To do this, express z from each equation and equate the resulting expressions: (x-a) / m = (y-b) / n = z / 1. A vector with coordinates (m, n, 1) will be the directing vector of this straight.
5
A line can also be given by a point u(co-directional) vector, in this case, to find the equation, use the formula (x-x1) / m = (y-y1) / n = (z-z1) / p, where (x1, y1, z1) are the coordinates of the point , and (m, n, p) is a collinear vector.
6
In order to determine the equation straight, given graphically on the plane, find the pointIts intersection with the axes of coordinates and substitute in the equation. If the angle of its inclination to the axis is known, it will be sufficient for you to find the tangent of this angle (this is the coefficient before x in the equation) and the point of intersection with the axis oy (this is the free term of the equation).
Tip 2: How to calculate the formula for a function
One of the most common ways of studying functions is to construct their graphs. However, knowing the basic properties of the graphical display of functions, you can calculate the formula from the graph.
Instructions
1
The easiest way is to calculate the formula of a straight line, in generalit corresponds to the equation y = kx + b. Find the coordinates of any two points belonging to a straight line, and substitute them in the equation (abscissa instead of x, ordinate instead of y). You will get a system of two equations, deciding which, you will find the coefficients k and b. Substituting values in the general form of the equation, you will see a formula corresponding to your schedule.
2
Look at how the charts look like standardquadratic functions, and compare them with your drawing. If the graph is symmetrical with respect to some line and the shape resembles a parabola or hyperbola, you will need three points to determine the coefficients of the equation. For example, the parabola equation in general form looks like y = ax ^ 2 + bx + c. Substituting the values of three points and obtaining a system of three equations, you can find the coefficients a, b, c.
3
If the graph is similar to a sinusoid or a cosine wave,try to find the equation in the following way. Determine how different the schedule is from the standard. If it is compressed along the ordinate by n times, then in the equation before the sign of sin or cos there is a factor less than one (if stretched along the axis oy, then the factor is greater than one).
4
If the graph is stretched or compressed along the x axis, conclude that there is a number in front of the variable inside the trigonometric function (if the number is greater than 1, the graph is compressed, if less than 1, it is stretched).
5
When constructing a trigonometric function inthe degree of its graph becomes either more flat (with a degree less than 1), or steeper (with a degree greater than 1). In addition, when the diagram is raised to an even power, the axis ox will be symmetrically displayed upward.
6
The graph can simply be moved up or downfor some distance. In this case, add this number to the value of the function, for example, y = tgx + 2. If the graph is moved to the left or right, add a number to the value of the argument, for example, y = tg (x + p).
