Curve sketching requires you to extract all the knowledge about the following points:
- Stationary points
- Points of inflexion
- x- and y- intercepts
- Domain and Range
- Asymptotes or Limits
- Symmetry, Odd or Even Functions
- Table of Values
These prerequisite knowledge are applied (not necessarily to all) in curve sketching. Sure, quadratics are not hard to sketch, but polynomials need more calculations to obtain sufficient information for sketches.
Let's start with one example: .
First, we could find the stationary points and points of inflexion.
First, find the stationary points:
So (3, -25) is a stationary point.
f(-1)=7 \\ So (-1, 7) is also a stationary point.
Then, the second derivative determine their nature of stationary points.
is a minimum turning point.
is a maximum turning point.
Are there any points of inflexion? Let's check it out with the second derivative = 0.
Check for change of concavity:
When x=0, f''(0)=-6
When x=1, f''(1)=0
When x=2, f''(2)=6
Since concavity changes, (1, -9) is a point of inflexion.
Next we check the intercept:
For y-intercept, x=0:
Therefore, the y-intercept is on (0, 2).
The x-intercepts are hard to solve, however we have enough information to sketch this curve.
We understand that the maximum and minimum points are not THE maximum or minimum point of the whole function, therefore we call the points relative maximum and relative minimum.