5. Curve Sketching

by M. Bourne

NOTES:

Need Graph Paper?

rectangular grid
Download graph paper

The kinds of things we will be searching for in this section are:

x-intercepts Use `y = 0`
NOTE: In many cases, finding x-intercepts is not so easy. If so, delete this step.
y-intercepts Use `x = 0`
local maxima Use `(dy)/(dx)=0`, sign of first derivative changes `+ → −`
local minima Use `(dy)/(dx)=0`, sign of first derivative changes ` − → +`
points of inflection Use `(d^2 y)/(dx^2)=0`, and sign of `(d^2 y)/(dx^2)` changes

Finding Maxima and Minima

Local maximum

A local maximum occurs when `y^' = 0` and `y^'` changes sign from positive to negative (as we go left to right).

A local maximum curve changing from positive to negative slope

Local minimum

A local minimum occurs when `y^' = 0 and y^'` changes sign from negative to positive.

A local minimum curve changing from negative to positive slope

The Second Derivative

The second derivative can tell us the shape of a curve at any point.

Concave up

If `(d^2y)/(dx^2) > 0`, the curve will have a minimum-type shape (called concave up)

d2y/dx2 - second derivative positive curve

Example 1

The curve y = x2 + 3x − 2 has `(dy)/(dx)=2x+3`.

Now `(d^2y)/(dx^2)=2` and of course, this is `> 0` for all values of x.

So it has a concave up shape for all x.

Concave down

If `(d^2y)/(dx^2) < 0`, the curve will have a maximum-type shape (called concave down)

 

Second derivative negative curve (local maximum)


Example 2

The curve y = x3 − 2x + 5 has `(dy)/(dx)=3x^2-2`. The second derivative is `(d^2y)/(dx^2)=6x` and this is `< 0` for all values of `x < 0`.

So the curve has a concave down shape for all `x < 0` (and it is concave up if `x > 0`).

Finding Points of Inflection

A point of inflection is a point where the shape of the curve changes from a maximum-type shape `(d^2y)/(dx^2) < 0` to a minimum-type shape `(d^2y)/(dx^2) > 0`.

Clearly, the point of inflection will occur when

`(d^2y)/(dx^2) = 0` and when there is a change in sign

(from plus ` →` minus or minus ` →` plus) of `(d^2y)/(dx^2)`.

A curve changes from concave up to concave down across a point of inflection

Example 3

Sketch the following curve by finding intercepts, maxima and minima and points of inflection:

`y=x^3-9x`

General Shapes

If we learn the general shapes of these curves, sketching becomes much easier. Of course, the following are "ideal" shapes, and there are many other possibilities. But at least this helps get us started.

Quadratic Cubic
Highest power of x: 2 Highest power of x: 3
Typical quadratic shape (parabola) Typical cubic shape showing concave down to up
1 minimum, no maximum
[if it has a positive x2 term]
1 minimum, 1 maximum
no points of inflection 1 point of inflection
Quartic Pentic
Highest power of x: 4 Highest power of x: 5
Typical quartic shape showing local maxima and minima Typical quintic shape showing local maxima and minima
2 minimums, 1 maximum
[if it has a positive x4 term]
2 minimums, 2 maximums
2 points of inflection 3 points of inflection

Example 4

Sketch the curve and show intercepts, maxima and minima and points of inflection:

`y=x^4-6x^2`

Example 5

Sketch the curve and show intercepts, maxima and minima and points of inflection:

`y=x^5-5x^4`

Didn't find what you are looking for on this page? Try search:

Online Algebra Solver

This algebra solver can solve a wide range of math problems. (Please be patient while it loads.)

Ready for a break?

 

Play a math game.

(Well, not really a math game, but each game was made using math...)

The IntMath Newsletter

Sign up for the free IntMath Newsletter. Get math study tips, information, news and updates each fortnight. Join thousands of satisfied students, teachers and parents!

Given name: * required

Family name:

email: * required

See the Interactive Mathematics spam guarantee.

Share IntMath!

Calculus Lessons on DVD

 

Easy to understand calculus lessons on DVD. See samples before you commit.

More info: Calculus videos

Loading...
Loading...