What is a point of inflection?

Question

Can anyone help?

Answer 1

A point of inflection is where the concavity of a graph changes.

Answer 2

Point of Inflection.

Let's start with "slope" first. You must understand slope to understand "point of inflection" -- which is the point where the slope that has been increasing starts decreasing, or vise versa.

Assume we are talking about some single-valued function of x, call it f(x), and let's plot the function on an x-y graph showing y=f(x). OK? [[[[ If not, let me know, and I'll start at the beginning. ]]]]

Now, for a straight line, the "slope" is the same along the line, and it represents the steepness of the line: a steep uphill climb (as we go along increasing x) has a large, positive slope; fractional positive slopes are wimpy uphill lines; negative slopes are downhill.
OK???? [[[[ If not, I'll say this more simply. ]]]]

Straight lines do not change slope, and therefore have no point of inflection.

If the graph of f(x) is not a line, but curves somehow, then we can still try to look at the slope (or "tangent" to the point) at any single point on the curve. (If you like calculus, you may say "derivative".)

Now, if we move the point -- or the x for f(x)) along a rising curve (with slope greater than zero), the slope (rise-over-run) will change and get larger if the curve goes more uphill. Or else, the slope gets smaller if the curve gets less steep. (There is a third case, where slope does not change at all, for a while -- but that section of the curve, with constant slope, must be a straight line!)

As we move along the curve (increasing x), sometimes the slope gets larger (i.e. steeper), and sometimes it gets smaller (less steep). If the slope (or first derivative) has been getting larger for a while, and then starts getting smaller, the point where this change occurs is the "point of inflection".

In terms of "derivatives", the point where the uphill/downhill slope or derivative goes for positive (uphill) to negative (downhill), or vise versa, is called an "extrema" (or maximum or minimum). It is the point where the slope of a tangent line is zero -- neither uphill nor downhill!

The point of inflection is where the increasing slope starts decreasing, or vise versa. Since the slope is the first derivative, the change in slope is the second derivative, and when it goes from positive to negative (or vise versa), from increasing slope to decreasing slope (or vise versa), there is some change point where it is neither increasing nor decreasing -- at that point, the second derivative is zero, and we call that a "point of inflection".

Answer 3

This is the point on a curve where the curve changes direction. You will find more about this at this web site:

www.answers.com/topic/inflection-point

Answer 4

the point of inflection is the point in a graph where the curvature changes direction from upward to downward or vice versa which is not the maximum or minimum point.