Program of Lessons

# 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 x2 and discriminant D = b2 4ac. 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:
-   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 functions y = Arcsin x ( Fig. 23 ) and y = Arccos x ( Fig. 24 ) are multivalued, unbounded functions; their domain and codomain are correspondingly:  –1 x +1 and < y < + . Because of they are multi-valued functions, not considered in an elementary mathematics, their principal values y = arcsin x   and  y = arccos x are considered as inverse trigonometric functions; their graphs have been distinguished on  Fig. 23  and Fig. 24  as bold lines.

The functions y = arcsin x and y = arccos x have the following characteristics and properties:
- the both functions have the same domain: –1   x +1;  their codomains are:
– / 2   y / 2  for  y = arcsin x and  0   y  for  y = arccos x ;
- they are bounded, non-periodic, continuous and monotone functions ( y = arcsin x is an
increasing function;  y = arccos x – a decreasing function) ;
- each of the functions has one zero (x = 0 of  y = arcsin x; x = 1 of  y = arccosx).

The functions y = Arctan x ( Fig.25 ) and  y = Arccot x ( Fig.26 ) are multivalued, unbounded functions;  their domain is the same:   – x + . Their principal values y = arctan x and  y = arccot x are considered as inverse trigonometric functions; their graphs have been distinguished on Fig.25 and Fig.26 as bold branches.

The functions y = arctan x and y = arccot x have the following characteristics and properties:
- the both functions have the same domain: – x + ; their codomains are:
– / 2 < y < / 2  for  y = arctan x   and  0 < y <   for  y = arccos x ;
- they are bounded, non-periodic, continuous and monotone functions ( y = arctan is an increasing function,  y = arccot x   is a decreasing function) ;
- only  y = arctan x  has one zero ( x = 0 );   y = arccot x  has no zeros.

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