The graph of a function
      Stationary points and/or critical points
         Turning points (extremes, local or relative maximums or minimums)
         Inflection points and intervals of concavity
      Symmetry of a function, parity - odd and even functions
Stationary points and/or critical points
The gradient of a curve at a point on its graph, expressed as the slope of the tangent line at that point, represents the rate of change of the value of the function and is called derivative of the function at the point, written dy/dx or f '(x)
At points of the graph where function changes from increase to decrease, the slope of the tangent line changes from positive to negative values respectively, passing through zero value. 
The points of the graph of a function at which the tangent lines are parallel to the x-axis, and therefore the derivative at these points is zero, are called the stationary points
There are three different types of stationary points: maximum points, minimum points and horizontal points of inflection. On the above graph the stationary points are denoted as, PMax, I and Pmin.
The graph reaches a local (or relative) maximum when gradient changes from positive through zero to negative.
The graph reaches a local (or relative) minimum when gradient changes from negative through zero to positive.
A local maximum is a value of the function greater than any adjacent value, i.e., in its immediate area it is the highest point, but it may not be the greatest value of the function over its whole range.
The endpoints of intervals of monotonicity are places where function stops increasing and starts decreasing or vice versa.
A function  f (x) can change from increasing to decreasing (or vice versa) at values where f '(x) = 0 or f '(x) is undefined.
Note that these are only potential places where the graph can change from increasing to decreasing (or vice versa) since it is possible that the function may not change at those values, as for example at the point xI (where  f '(x) = 0), in the above figure or, as in case of the rational functions from the above two examples, at the vertical asymptotes (where  f '(x) is undefined).
If  f (x) is defined at x = c and either  f '(c) = 0 or  f '(c) is not defined, then x = c is called a critical value of the function  f (x), and its point (c, f (c)) is called a critical point.
Therefore, a critical point may be a local maximum, a local minimum, or neither.
The critical point is neither a maximum nor a minimum if the function does not change from increasing to decreasing (or vice versa) at the critical point, as at the point xI in the above figure.
For a given function and point (c, f (c)), the derivative of at x = c is the slope of the tangent line through the point (c, f (c)), i.e.,  f '(c) = tan at .
The function value f (c) in the right figure is defined and the derivative at x = c is undefined, therefore the point (c, f (c)) is a critical point.
As f (x) is increasing before x = c and decreasing after x = c, the point (c, f (c)) is a local maximum.
Turning points (extremes, local or relative maximums or minimums)
A stationary point at which the gradient (or the derivative) of a function changes sign, so that its graph does not cross a tangent line parallel to x-axis, is called the tuning point.
Thus, a turning point is a critical point where the function turns from being increasing to being decreasing (or vice versa), i.e., where its derivative changes sign. 
A local (or relative) maximum is a point where the function turns from being increasing to being decreasing, i.e., where its derivative changes sign from positive to negative.
Notice that, as we travel through the maximum turning point from left to right, the derivative (the slope of the tangent to the curve) is decreasing, i.e.,  f '(x) changes from positive through zero to negative as x increases.
Thus, if the derivative of a function is decreasing over an interval, the graph of the function is concave down.
A local (or relative) minimum is a point where the function turns from being decreasing to being increasing, i.e., where its derivative changes sign from negative to positive.
Thus, as we travel through the minimum turning point from left to right, the derivative is increasing, i.e.,  f '(x) changes from negative through zero to positive as x increases.
Therefore, if the derivative of a function is increasing over an interval, the graph of the function is concave up.
Inflection points and intervals of concavity
A point on a curve at which it crosses its tangent, and concavity changes from up to down or vice versa, is called the point of inflection, as shows the above figure.
The graph is concave up on an open interval where the slope increases and concave down on an open interval where the slope decreases.
Therefore, the points on a curve that join arcs of opposite concavity are points of inflection.
If the gradient of the function does not change sign at the stationary point, then it is a point of horizontal inflection. 
 
Symmetry of a function, parity - odd and even functions
A function ƒ that changes neither sign nor absolute value when the sign of the independent variable is changed is even, so that,  f (x)f (- x).
Therefore, the graph of such a function is symmetrical with respect to the y-axis, as is the graph shown in the left figure below.
The graph of an even function. The graph of an odd function.
     A function that changes sign but not absolute value when the sign of the independent variable is changed is odd, so that,  f (x) = - ƒ(- x). That is, for each x in the domain of ƒf (- x) = - f (x).  
Therefore, the graph of such a function is symmetrical with respect to the origin, as is the graph shown in the right figure above.
Intermediate algebra contents
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