You will need

- paper;
- - handle.

Instruction

1

From school course of mathematics the students become aware that the number of possible points

**of intersection of**two**graphs**depends on the function. For example, a linear function will have only one*point***of intersection**, linear and square – two, square – two or four, etc.2

Consider the General case of two linear functions (see Fig.1). Let y1=k1x+b1 and y2=k2x+b2. To find the

*point*of their**intersection**we need to solve the equation y1=y2 or k1x+b1=k2x+b2.Transforming the equation you will get: k1x-k2x=b2-b1.Express x as follows:x=(b2-b1)/(k1-k2).3

After finding the values of x – coordinates of the point

**of intersection of**two**graphs**on the x-axis (axis 0X), it remains to calculate the coordinate of the ordinate (axis 0У). It is necessary to substitute in any of the functions, the obtained value x. Thus, the point**of intersection of**U1 and U2 will have the following coordinates: ((b2-b1)/(k1-k2);k1(b2-b1)/(k1-k2)+b2).4

Analyze calculation example find the point

**of intersection of**two**graphs**(see Fig.2).You need to find*the point***of intersection****of the graphs**of the functions f1 (x)=0.5 x^2 and f2 (x)=0.6 x+1,2.Equating f1 (x) and f2 (x), we obtain the following equation:0.5 x y =0.6 x+1,2. Moving all terms to the left side, you will get a quadratic equation:0.5 x^2 x -0,6-1,2=0.The solution of this equation will be two values of x: x1≈2,26,x2≈-1,06.5

Substitute the values of x1 and x2 in any of the expression functions. For example, and f_2 (x1)=0,6•2,26+1,2=2,55, f_2 (x2)=0,6•(-1,06)+1,2=0,56.So, the required points are: t A (2,26;2,55) (-1,06;0,56).

# Advice 2: How to find the coordinates of the points of intersection of the graph of a function

The graph of the function y = f (x) is the set of all points in the plane, coordinates x, y which satisfy y = f(x). The graph of the function illustrates the behavior and properties of functions. To plot usually chosen several values of the argument x and for them calculate the corresponding values of the function y=f(x). For a more accurate and clear plotting, it is useful to find its point of intersection with the coordinate axes.

Instruction

1

To find the point of intersection of the function with the y-axis, you must calculate the value of the function at x=0, i.e. find f(0). For example, you will use the graph of the linear function depicted in Fig.1. Its value at x=0 (y=a*0+b) is equal to b therefore, the graph crosses the y-axis (Y-axis) at the point (0,b).

2

At the intersection of the abscissa (X-axis) the value of the function is 0, i.e., y=f(x)=0. To calculate x you need to solve the equation f(x)=0. In the case of linear functions we obtain the equation ax+b=0, whence we find x=-b/a.

Thus, the X-axis intersects at the point (-b/a,0).

Thus, the X-axis intersects at the point (-b/a,0).

3

In more complex cases, for example, in the case of quadratic dependence of y from x, the equation f(x)=0 has two roots, therefore, the x-axis is crossed twice. In the case of a periodic dependence y from x, for example y=sin(x), its graph has an infinite number of points of intersection with the axis X.

To check the correctness of the finding of the coordinates of the points of intersection of the function with the X-axis you need to substitute the values of x in the expression f(x). The value of the expression when any of the computed x should be 0.

To check the correctness of the finding of the coordinates of the points of intersection of the function with the X-axis you need to substitute the values of x in the expression f(x). The value of the expression when any of the computed x should be 0.