Angol Matematika

Mathematics definitions and rules (to 11.a)

Absolute value function, quadratic function,
function transformations

Absolute value function

Absolute value can be described as a function in the form of f: R→R, f(x)=|x|
The graph of the function looks like a „V" with its vertex (0;0).
Its gradient is m=1 on the right side of the vertex, m=-1 on the left side of the vertex.
At ]-∞;0] the function grows or increases, at [0;∞[ the function goes down or decreases.
A function is called an even function if for all elements of the domain of the function f(-x)=f(x). With other words we can say that an even function's graph is symmetrical over y-axis.
The absolute value function is an even function.

Quadratic function

Function f: R→R, f(x)=ax2+bx+c is called a quadratic function.
If a=1, b=0 and c=0 then f(x)=x2.
The graph of the quadratic function is a parabola.
At ]-∞;0] the function grows or increases, at [0;∞[ the function goes down or decreases.
The quadratic function is even.

Function transformations

There are two types of transformations: variable and value transformations.
If y=f(x+a) then we move the graph to the left or to the right, in the opposite direction of the sign of a.
If y=f(a•x) then if a>1, we compress the graph in the x-direction to the reciprocal distance
If 0<a<1, we stretch it
If y=f(-x) then we reflect the graph over y-axis
If y=f(x)+a then we move the graph up or down according to the sign of a.
If y=a•f(x) then if a>1 then we stretch the graph in the direction of y-axis
If 0<a<1 then we compress it in the direction of y-axis
If y=-f(x) then we reflect the graph over x-axis
If y=|f(x)| then we reflect the negative part of the graph over x-axis


2012.03.09



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