Derivatives & Graphs

If the graph of a function $y = f (x)$ is known, or can be sketched, then it is also possible to sketch the graphs of the derivatives $y = f' (x)$ and $y = f'' (x)$
The key properties of a graph include
- the $y$ -axis intercept
- the $x$ -axis intercepts – the roots of the function; where $f (x) = 0$
- stationary points; where $f' (x) = 0$
  - turning points – (local) minimum and maximum points
  - (horizontal) points of inflection
- (non-stationary, $f' (x) \neq 0$ ) points of inflection
- asymptotes – vertical and horizontal
- intervals where the graph is increasing and decreasing
- intervals where the graph is concave down and concave up
Not all graphs have all of these properties and not all can be determined without knowing the expression of the function
However questions will provide enough information to sketch
- the shape of the graph
- some of the key properties such as roots or turning points

The graph of $y = f' (x)$ will have its
- $x$ -axis intercepts at the $x$ -coordinates of the stationary points of $y = f (x)$
- turning points at the $x$ -coordinates of the points of inflection of $y = f (x)$
For intervals where $y = f (x)$ is concave up, $y = f' (x)$ will be increasing
For intervals where $y = f (x)$ is concave down , $y = f' (x)$ will be decreasing
For intervals where $y = f (x)$ is increasing, $y = f' (x)$ will be positive
For intervals where $y = f (x)$ is decreasing, $y = f' (x)$ will be negative

First sketch the graph of $y = f' (x)$ from $y = f (x)$ , as per the above process
Then, using the same process, sketch the graph of $y = f'' (x)$ from the graph of $y = f' (x)$
There are a couple of things you can deduce about the graph of $y = f'' (x)$ directly from the graph of $y = f (x)$
- The graph of $y = f'' (x)$ will have its $x$ -axis intercepts at the $x$ -coordinates of the points of inflection of $y = f (x)$
- For intervals where $y = f (x)$ is concave up, $y = f'' (x)$ will be positive
- For intervals where $y = f (x)$ is concave down, $y = f'' (x)$ will be negative

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It is possible to sketch a graph of $y = f (x)$ by considering the reverse of the above
- For intervals where $y = f' (x)$ is positive, $y = f (x)$ will be increasing but is not necessarily positive
- For intervals where $y = f' (x)$ is negative, $y = f (x)$ will be decreasing but is not necessarily negative
- Roots of $y = f' (x)$ give the $x$ -coordinates of the stationary points of $y = f (x)$
There are some properties of the graph of $y = f (x)$ that cannot be determined from the graph of $y = f' (x)$
- the $y$ -axis intercept
- the intervals for which $y = f (x)$ is positive and negative
- the roots of $y = f (x)$
Unless a specific point the curve passes through is known, the constant of integration cannot be determined
- the exact location of the curve will remain unknown
- but it will still be possible to sketch its shape
If starting from the graph of the second derivative, $y = f'' (x)$ , it is easier to sketch the graph of $y = f' (x)$ first, then sketch $y = f (x)$

The graph of $y = f (x)$ is shown in the diagram below.

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On separate diagrams sketch the graphs of $y = f' (x)$ and $y = f'' (x)$ , labelling any roots and turning points.

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Derivatives & Graphs (DP IB Maths: AA HL)