We will present you one interesting presentation about a equations and functions….

Graphics of functions and equations Click me! Graphics of functions  and equations

Function (definition) Content Function (definition) Linear function Quadratic function Function y = k / x Graphical solution of equations

Function (definition) A function, in a mathematical sense, expresses the idea that one quantity (the argument of the function, also known as the input) completely determines another quantity (the value, or the output). A function assigns exactly one value to each input of a specified type. The argument and the value may be real numbers, but they can also be elements from any given sets: the domain and the codomain of the function.

Linear function Linear function is given by the formula y = kx + b, where k - is the angle tanges.

b - ordinate of the intersection of rights with vertical axis.

Construct the graph of y = 2x+1. 1 -1/2

Three special cases of a linear function of the type y = kx + b x =  - (b/k) 

In mathematics a function of the type Quadratic function In mathematics  a function of the type f(x) = ax2 + bx + c, where a ≠ 0,  b, c are random numbers.

The graphic of this function with real numbers is parabola which crosses the abscissa axis in the points with coordinates  A(x1,0) and B(x2,0), when the discriminant  D = b2 − 4ac of the quadratic equation  f(x) = 0 is favorable. The numbers x1 and x2 can be found with this formula =>

When we construct the graphical of the у = ах2 + bх + с We first find vertex then we find the intersections of the function as the axis OX find the roots of the equation ах2 + bх + с=0

D > 0 (with two different roots ) у < 0 when х є ( х1; х2) у > 0 when х є ( -∞; х1) υ ( х2; +∞) у < 0 when х є ( -∞; х1 ) υ ( х2; +∞ ) у > 0 when х є ( х1; х2 )

D = 0 у < 0 no roots у > 0 when х є ( ­∞; х0 ) υ ( х0; +∞) -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 1 2 3 4 5 6 7 8 9 10 -4 -3 -2 -1 у < 0 no roots у > 0 when х є ( ­∞; х0 ) υ ( х0; +∞) у < 0 when х є ( -∞; х0) υ ( х0; +∞ ) у > 0 no solution

D < 0 ( no real solutions ) у < 0 no solution у > 0 х є R у < 0 each х є R у > 0 no solution

k ≠ 0 is called the inverse. The graphics is hyperbola . Function у = k / х у = k/х, where k ≠ 0 is called the inverse. The graphics is hyperbola .

Graphical solution of equations The graphical method is useful in solving problems which  require to determine only the number of the root of the given equation.

Examples of problems with a theoretical character Method of solution Examples of problems with a theoretical character

у = f(х) = ах2 + bх + с, where а, b, с are numbers. The square trinomials are х1 and х2, and discriminate –D, D = b2 – 4ас и х1,2 = - b/2а ± D/ 2а. In this case а > 0. But if а < 0, we can make a correlation.

Lets to draw the graph of у = f(х). First it crosses the abscissa axis Under what conditions two roots of the quadratic equation ах2 + bх + с = 0 will be greater than some number m? Lets to draw the graph of у = f(х). First it crosses the abscissa axis or touches to it D ≥ 0 . Second, f(m) > 0, Third, we must show that the condition of the task is satisfied by the parabola. b/2а > m. two roots are greater than m in just in case: D ≥ 0, f(m) > 0, – b/2а > m.

у = ах2 + bх + с , when а > 0 : We get inequality Under what conditions two roots of the quadratic equation ах2 + bх + с = 0 are on different sides of the m? у = ах2 + bх + с , when а > 0 : We get inequality f(m) < 0, necessary condition is identical to the system : D > 0 f(m) < 0

Made by: Kalina Taneva SOU “Zheleznik” Stara Zagora Bulgaria