Graphs

Shapes of graphs

Every exam for this course usually includes at least one question that requires you use your knowledge of the derivative to deduce some facts about a function. These facts might include:

• Where the function is increasing and decreasing
• Where the function is concave up and concave down
• Where any stationary points occur and their nature

Addressing each of these in order:

A function $f$ is increasing where its derivative $f'$ is positive and decreasing where $f'$ is negative.

A function $f$ is concave up where the slope of $f'$, the second derivative, is positive and concave down where the slope of $f'$ is negative.

Stationary points of $f$ occur where the derivative $f'$ is equal to 0. It will be a maximum where the derivative $f'$ is crossing from positive to negative and a minimum where $f'$ is crossing from negative to positive. If $f'$ is equal to 0 but does not cross the x-axis then that is a point of inflection.

Curve sketching

There are several factors to take into account when sketching a graph that can help you ensure it is accurate to even very complex functions.

1. Domain

2. Intercepts

3. Symmetry

4. Asymptotes

5. Intervals of increase or decrease

6. Local maximum and minimum values

7. Concavity and points of inflection

8. Sketch