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

2. Graphics of functions and equations
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Graphics of functions  and equations

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

4. 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.

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

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

7. Construct the graph of y = 2x+1.1
-1/2

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

9. 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.

10. 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 =>

11. 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

12. 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 )

13. 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

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

15. 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 .

16. 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.

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

18. 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.

19. 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.

20. 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 > f(m) < 0

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