5. Curve Sketching
by M. Bourne
NOTES:
- There are now many tools for sketching functions (Mathcad, LiveMath, Scientific Notebook, graphics calculators, etc). It is important in this section to learn the basic shapes of each curve that you meet. An understanding of the nature of each function is important for your future learning. Most mathematical modelling starts with a sketch.
Need Graph Paper?
- You need to be able to sketch the curve, showing important features. Avoid drawing x-y boxes and just joining the dots.
- We will be using calculus to help find important points on the curve.
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: + → − |
| local minima | Use dy/dx = 0, sign: − → + |
| points of inflection | Use d2y/dx2 = 0, and sign of d2y/dx2 changes |
Finding Maxima and Minima
A local maximum occurs when y' = 0 and y' changes sign from positive to negative (as we go left to right).
A local minimum occurs when y' = 0 and y' changes sign from negative to positive.
The Second Derivative
The second derivative can tell us the shape of a curve at any point.
- If d2y/dx2 > 0, the curve will have a minimum-type shape (called concave up)
Example:
y = x2 + 3x - 2 has
and 
for
all values of x.
So it has a concave up shape for all x.
- If d2y/dx2 < 0, the curve will have a maximum-type shape (called concave down)
Example:
y = x3 − 2x + 5 has
and 
for
all values of x < 0.
So it has a concave down shape only for all 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 (d2y/dx2 < 0) to a minimum-type shape (d2y/dx2 > 0).
Clearly, the point of inflection will occur when
d2y/dx2 = 0 and when there is a change in sign
(from plus → minus or minus → plus) of d2y/dx2.
Example 1:
Sketch the following curve by finding intercepts, maxima and minima and points of inflection:
Let's get LiveMath to draw it for us. You can confirm all the information we obtained above by zooming in on different parts of the graph.
General Shapes
If we learn the general shapes of these curves, sketching becomes much easier.
| Quadratic | Cubic |
| Highest power of x: 2 | Highest power of x: 3 |
 |
 |
| 1 minimum, no maximum [if it has a positive x2 term] |
1 minimim, 1 maximum |
| no points of inflection | 1 point of inflection |
| Quartic | Pentic |
| Highest power of x: 4 | Highest power of x: 5 |
 |
 |
| 2 minimims, 1 maximim | 2 minimums, 2 maximums |
| 2 points of inflection | 3 points of inflection |
In LiveMath, we can play with these basic shapes (I've included LINEAR here).
Example 2:
Sketch the curve and show intercepts, maxima and minima and points of inflection:
Example 3:
Sketch the curve and show intercepts, maxima and minima and points of inflection:
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