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Elementary functions and their graphs

Proportional values. Linear function. Inverse proportionality.
Hyperbola.  Quadratic function.  Quadratic parabola.
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: …
      ( 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 ? ) .

5. Power function. This is the function:  y = axn where a, n – constants. At  n = 1 we receive the function, called a direct proportionalityy = ax ; at  n = 2  - a quadratic parabola; at   n = – 1  -  an inverse proportionality or hyperbola. So, these  functions are particular casesof a power function. We know,  that a zero power of every non-zero number is 1, thus at   n = 0 the power function becomes a constant: y = a , i.e. its graph is a straight line, parallelto an  x-axis, except an origin of coordinates ( explain, please, why ? ).All these cases (at  a = 1 ) are shown on Fig.13 ( n 0 ) and  Fig.14 ( n < 0 ) .Negative values of   x are not considered here, because then some of functions:



If   n – integer, power functions have a meaning also at x < 0, but their graphs have different forms depending on that is  n  an even or an odd number. On Fig.15 two such power functions are shown: for  n = 2 and  n = 3. 

At  n = 2  the function is even and its graph is symmetric relatively an axis Y ; at   n = 3 the function is odd and its graph is symmetric relatively an origin of coordinates. The function y = x 3 is called a cubic parabola.

On Fig.16 the function is represented. This function is inverse to the quadratic parabola y = x 2, its graph is received by rotating the quadratic parabola graph around abisector of  the 1-st coordinate angle. (This is the way to receive a graph of every inverse function from its original function). We see by the graph, that this is the two-valued function(the sign ± before the square root symbol says about this). Such functions are not studied in an elementary mathematics, therefore we consider usually as a function one of its branches: either an upper or a lower branch.

6. Exponential function. The function y = ax, where a is a positive constant number, is called an exponential function.  The argument x adopts any real values; as the function values only positive numbers are considered, because otherwise we'll have a multi-valued function. So, the function y = 81 x has at x = 1/4 four different values: y = 3, y = – 3, y = 3i and y = – 3i ( check this, please ! ). But we consider as the function value only   y = 3. Graphs of an exponential function for  a = 2  and    a = 1 / 2 are shown on Fig.17. All they are goingthrough the point  ( 0, 1 ). At   a = 1 we have as a graph a straight line, parallel to x-axis,i.e. the function becomes a constant value, equal to 1. At  a > 1 an exponential function increases, and at  0 < a < 1 – decreases.


The main characteristics and properties of a exponential function:
  - the function domain: – < x < + ( i.e. x is any real number ) and its codomain: y > 0;
  - this is a monotone function: it increases at  a > 1 and decreases at 0 < a < 1;
  - the function is unbounded, continuous in everywhere, non-periodic;
  - the function has no zeros.


7. Logarithmic function. The function   y = log a x, where a is a positive constant number,not equal to 1, is called a   logarithmic function. This is an inverse function relatively to anexponential function; its graph  ( Fig.18 )  can be received by rotating a graph of an exponential function around of a bisector of the 1-st coordinate angle.


   The main characteristics and properties of a logarithmic function:
   -  the function domain: x > 0 and its codomain:  < y < + ( i.e. y is any real number );
   -  this is a monotone function: it increases at  a > 1 and decreases at 0 <a < 1;
   -  the function is unbounded, continuous in everywhere, non-periodic;
   -  the function has one zero:   x = 1.

8. Trigonometric functions. Building trigonometric functions we use a radian as a measure of angles. Then the function  y = sin x is represented by the graph ( Fig.19 ). This curve iscalled a sinusoid.

   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 areso called  intervals of monotony,inside of whichthey behave as
     monotone functions ( see graphs Fig.19 and Fig.20);
   - the functions havean 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:


9. Inverse trigonometric functions. Definitions of inverse trigonometric functions and their main properties have been written in the same named paragraph in the section "Trigonometry". So, we’ll give here only short comments concerning their graphs receivedby rotating the graphs of trigonometric functions around a bisector of the 1-st coordinate angle.

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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