Derivatives & Graphs
How are derivatives and graphs connected?
- If the graph of a function is known, or can be sketched, then it is also possible to sketch the graphs of the derivatives and
- The key properties of a graph include
- the-axis intercept
- the-axis intercepts – the roots of the function; where
- stationary points; where
- turning points – (local) minimum and maximum points
- (horizontal) points of inflection
- (non-stationary,) 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
How do I sketch the graph of y = f'(x) from the graph of y = f(x)?
- The graph of will have its
- -axis intercepts at the-coordinates of the stationary points of
- turning points at the-coordinates of the points of inflection of
- For intervals where is concave up, will be increasing
- For intervals where is concave down , will be decreasing
- For intervals where is increasing, will be positive
- For intervals where is decreasing, will be negative
How do I sketch the graph of y = f''(x) from the graph of y = f(x)?
- First sketch the graph of from, as per the above process
- Then, using the same process, sketch the graph of from the graph of
- There are a couple of things you can deduce about the graph of directly from the graph of
- The graph of will have its -axis intercepts at the -coordinates of the points of inflection of
- For intervals where is concave up, will be positive
- For intervals where is concave down, will be negative
Is it possible to sketch the graph of y = f(x) from the graph of a derivative?
- It is possible to sketch a graph of by considering the reverse of the above
- For intervals where is positive, will be increasing but is not necessarily positive
- For intervals where is negative, will be decreasing but is not necessarily negative
- Roots of give the-coordinates of the stationary points of
- There are some properties of the graph of that cannot be determined from the graph of
- the-axis intercept
- the intervals for which is positive and negative
- the roots of
- 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,, it is easier to sketch the graph of first, then sketch
Worked Example
The graph of is shown in the diagram below.
On separate diagrams sketch the graphs of and, labelling any roots and turning points.