#Graphs of $f$, $f'$ & $f''$

#Table of Contents

1. How to Sketch the Graph of a Function
2. Using Derivatives to Sketch Graphs
3. Worked Examples
4. Practice Questions
5. Glossary
6. Summary and Key Takeaways

#How to Sketch the Graph of a Function

You should be familiar with the general shapes of the graphs of common functions, including:

• Linear functions
• Quadratic functions
• Cubic and higher-order polynomial functions
• Trigonometric functions
• Exponential and logarithmic functions
• Reciprocals and reciprocal powers of $x$, e.g., $\frac{1}{x^2}$

When sketching a function, consider the following:

#Domain and Range

• Domain: Identify values of $x$ for which $f(x)$ is defined.
• Range: Sometimes useful to find using the candidates test for global extrema.

#Points of Undefined Values

• Identify where $f(x)$ is undefined, leading to vertical asymptotes.
  • Example: $x=0$ for $y=\frac{1}{x}$.

#Limits and Asymptotes

• Determine the limit of the function as $x \to \pm \infty$ to find horizontal asymptotes.
  • Example: $y=2$ for $y=\frac{1}{x}+2$.

#Symmetry

• Determine if the function is even ($f(-x)=f(x)$) or odd ($f(-x)=-f(x)$).

#Periodicity

• Check if the function is periodic: $f(x+a)=f(x)$ for some constant $a$.

#Using Derivatives to Sketch Graphs

Derivatives help identify key features and properties of the graph of a function. Reca...