Sketching Derivative & Second-Derivative Graphs
Learn sketching derivative and second-derivative graphs for NSW Year 12 Mathematics Advanced (NESA outcome MAV-12-06). Given only the graph of a function, you can work out the shape of its gradient graph, and then the graph of how that gradient changes.
You will learn to match turning points of the curve with zeros of the derivative, rising sections with a positive gradient graph, and inflections with turning points of the derivative graph, then sketch the second derivative — a strong test of real understanding.
Theory
From the graph of \(f\) you can sketch \(f'\) by reading gradients. This Year 12 Mathematics Advanced topic (MAV-12-06) builds derivative and second-derivative graphs.
A stationary point of \(f\) becomes an \(x\)-intercept of \(f'\); where \(f\) rises, \(f'\) is above the axis; where \(f\) falls, \(f'\) is below. A point of inflection of \(f\) is a turning point of \(f'\).
Method
- Mark the stationary points of \(f\) as zeros of \(f'\).
- Sign: \(f'\) is positive where \(f\) rises, negative where it falls.
- Repeat for \(f''\) from \(f'\).
| \(f'(x)\) | \(=\) | \(2x\) |
A line through the origin.
\(f'(3)=0\) — it crosses the axis.
Below the axis (\(f'<0\)).
| \(f'(x)\) | \(=\) | \(3x^{2}\) |
| \(f''(x)\) | \(=\) | \(6x\) |
A line through the origin.
Common pitfalls
Frequently asked questions
How do you sketch the derivative graph from f?
Mark zeros of f prime at the stationary points of f, put f prime above the axis where f increases and below where f decreases.
What happens to f prime at a maximum of f?
It is zero and crosses from positive to negative.
What does a point of inflection of f give on the f prime graph?
A turning point (maximum or minimum) of f prime.
How do you get the second-derivative graph?
Apply the same gradient reading to the graph of f prime.