What is a point of inflection? Is it a straight line on a graph?

Question

How do you sketch a graph with a inflection point?

Answer 1

A point of inflection is where the slope of curve becomes zero. Imagine a roller coaster (sine) the rail goes from bottom-up and then starts getting smoother and at some point it starts going down.

The change from going up to down happens in a single point (unless there is an interval that has a flat line).

VVVVVVVV

Answer 2

An inflection point, or point of inflection (or inflexion) can be defined in any of the following ways:

a point on a curve at which the tangent crosses the curve itself.
a point on a curve at which the curvature changes sign. The curve changes from being concave upwards (positive curvature) to concave downwards (negative curvature), or vice versa. If one imagines driving a vehicle along the curve, it is a point at which the steering-wheel is momentarily 'straight', being turned from left to right or vice versa.
a point on a curve at which the second derivative changes sign. This is very similar to the previous definition, since the sign of the curvature is always the same as the sign of the second derivative, but note that the curvature is not the same as the second derivative.
a point (x,y) on a function, f(x), at which the first derivative, f'(x), is at an extremum, i.e. a minimum or maximum. (This is not the same as saying that y is at an extremum, and in fact implies that y is not at an extremum).

Plot of y = x3, rotated, with tangent line at inflection point of (0,0).Note that since the first derivative is at an extremum, it follows that the second derivative, f''(x), is equal to zero, but the latter condition does not provide a sufficient definition of a point of inflection. One also needs the lowest-order non-zero derivative to be of odd order (third, fifth, etc.). If the lowest-order non-zero derivative is of even order, the point is not a point of inflection. (An example of such a function is y = x4 − x).

It follows from the definition that the sign of f'(x) on either side of the point (x,y) must be the same. If this is positive, the point is a rising point of inflection; if it is negative, the point is a falling point of inflection.

Points of inflection can also be categorised according to whether f'(x) is zero or not zero.

if f'(x) is zero, the point is a stationary point of inflection, also known as a saddle-point
if f'(x) is not zero, the point is a non-stationary point of inflection

Plot of y = x4 - x with tangent line at non-inflection point of (0,0).An example of a saddle point is the point (0,0) on the graph y=x³. The tangent is the x-axis, which cuts the graph at this point.

A non-stationary point of inflection can be visualised if the graph y=x³ is rotated slightly about the origin. The tangent at the origin still cuts the graph in two, but its gradient is non-zero.

Note that an inflection point is also called an ogee, although this term is sometimes applied to the entire curve which contains an inflection point.

Answer 3

It's where a curve goes from curving down to curving up. It may be vertical, horizontal, or even diagonal (y = sin x). Mathematically, it's where the second derivative of a function is equal to zero.

As an example, if you were graphing the distance between a car and the start line over time, and the car was accelerating, you would see the graph curve upward. If the car starts decelerating the curve would be curving in the other direction -- still increasing, but by less and less over time. The point at which the car goes from accelerating to decelerating is the inflection point.

Answer 4

Not a straight line exactly; it's when the curvature changes from positive to negative, or vice versa. (You may use the terms "upwards" and "downwards" curvature. Either way, it's where it changes from one to the other.)

Equivalently, it's where the first derivative has a maximum or minimum - that is, the slope stops increasing and starts decreasing, or vice versa.

It's where the graph seems to "straighten out" and starts to curve back in the other direction.

All of these statements are saying the same thing. ;-)

As far as sketching it, just plot the inflection point and make a note that you have to stop curving up and start curving down when you hit it. Or the other way around, of course.

Answer 5

Point of inflection (or inflexion) can be defined in any of the following ways:

1.a point on a curve at which the tangent crosses the curve itself.
2.a point on a curve at which the curvature changes sign. The curve changes from being concave upwards (positive curvature) to concave downwards (negative curvature), or vice versa. If one imagines driving a vehicle along the curve, it is a point at which the steering-wheel is momentarily 'straight', being turned from left to right or vice versa.
3.a point on a curve at which the second derivative changes sign. This is very similar to the previous definition, since the sign of the curvature is always the same as the sign of the second derivative, but note that the curvature is not the same as the second derivative.
4.a point (x,y) on a function, f(x), at which the first derivative, f'(x), is at an extremum, i.e. a minimum or maximum. (This is not the same as saying that y is at an extremum, and in fact implies that y is not at an extremum).

Answer 6

first of all, a point inflection when the concavity of the curve changes over it. So if the f'(x)=0 and f"=0, then (assuming x is a valid point) you should test the neighboorhood concavity. it's just a point

Answer 7

You plot the guy factors before you attempt to entice a line or curve between them. you will see that no matter if those factors fall on a promptly line or a curve once you draw the dots.

Answer 8

This is the point where the graphic of the function changes its way( from rising to decrease OR from decreasing to rise) ;)