# Elementary functions and their graphs

Proportional values. Linear function. Inverse proportionality.
Power function. Cubic parabola. Exponential function.
Logarithmic function. Trigonometric functions. Sinusoid.
Intervals of monotony. Inverse trigonometric functions.

 1 Proportional values. If variables y and x are direct proportional , then the functionaldependence between them is represented by the equation: y = kx , where k is a constant a factor of proportionality . A graph of a direct proportionality is a straight line, going through an origin of coordinatesand forming with an x -axis an angle , a tangent of which is equal to k :  tan = k ( Fig.8 ). Therefore, a factor of proportionality is called also a slope . There are shown three graphswith k = 1/3, k = 1 and k = – 3 on Fig.8. 2 Linear function. If variables y and x are tied by the 1-st degree equation: A x + B y = C , ( at least one of numbers A or B is non-zero ), then a graph of the functional dependence is a straight line. If C = 0, then it goes through an origin of coordinates, otherwise - not.Graphs of linear functions for different combinations of A, B, C are represented on Fig.9. 3 Inverse proportionality. If variables y and x are inverse proportional , then the functionaldependence between them is represented by the equation: y = k / x , where k is a constant. A graph of an inverse proportionality is a curve, having two branches ( Fig.10 ). This curveis called a hyperbola . These curves are received at crossing a circular cone by a plane(about conic sections see the paragraph "Cone" in the part "Stereometry (Solid geometry)" ).As shown on Fig.10, a product of coordinates of a hyperbola points is a constant value, equal in this case to 1. In general case this value is k , as it follows from a hyperbola equation: x y = k . The main characteristics and properties of hyperbola: -  the function domain: x 0,  and codomain: y 0 ; -  the function is monotone ( decreasing ) at x < 0 and at x > 0 , but it is not monotone on the whole, because of a point of discontinuity x = 0 ( think, please, why ? ); -  the function is unbounded, discontinuous at a point x = 0, odd, non-periodic; -  there are no zeros of the function. 4 Quadratic function. This is the function: y = ax 2 + bx + c ,  where a, b, c – constants, a 0. In the simplest case we have b = c = 0 and y = ax 2 . A graph of this function is a quadratic parabola - a curve, going through an origin of coordinates ( Fig.11 ). Every parabola has an axis of symmetry OY , which is called an axis of parabola. The point O of intersection of a parabola with its axis is a vertex of parabola. A graph of the function y = ax 2 + bx + c is also a quadratic parabola of the same shape, that y = ax 2 , but its vertex is not an origin of coordinates, this is a point with coordinates:

The form and location of a quadratic parabola in a coordinate system depends completely on two parameters: the coefficient a of x 2 and discriminant D = b 2 4 ac . These properties follow from analysis of the quadratic equation roots ( see the corresponding paragraph in the part "Algebra" ).
All possible different cases for a quadratic parabola are shown on Fig.12.

Show, please, a quadratic parabola for the case a > 0, D > 0 .

The main characteristics and properties of a quadratic parabola:
-  the function domain: < x < + ( i.e. x is any real number ) and codomain:
( answer, please, this question yourself  !) ;
-   the function is not monotone on the whole, but to the right or to the left of the vertex it
behaves as a monotone function;
-  the function is unbounded, continuous in everywhere, even at b = c = 0, and non-periodic;
-  the function has no zeros at D < 0. ( What about this at D 0 ? ) .

The graph of the function y = cos x is represented on  Fig.20 ; this is also a sinusoid,
received from the graph of y = sin x by its moving along an x -axis to the left for / 2.

From these graphs the following main characteristics and properties of the functions are obvious:
- the functions have as a domain: < x < + and a codomain:  – 1 y +1;
- these are periodic functions: their period is  2 ;
- the functions are bounded ( | y | 1 ), continuous in everywhere; they are not monotone
functions, but there are so called intervals of monotony , inside of which they behave as
monotone functions ( see graphs Fig.19 and Fig.20 );
- the functions have an innumerable set of zeros( see in details the section "Trigonometric equations" ).

Graphs of functions y = tan x and y = cot x are shown on Fig. 21 and Fig. 22 correspondingly.

The graphs show, that these functions are: periodic (their period is ), unbounded, not monotone on the whole, but they have the intervals of monotony (what intervals ?), discontinuous functions (what points of discontinuity these functions have ?). The domain and codomain of these functions: