Graphical Methods (Roots)

Make a plot of function and observe where it crosses the $x$-axis.

For example, to find the roots of:

$$f(x)=3x^2+4x+1$$

We can first find its minimum at $f^{\prime }(x)=0$ and finding that $x=-2/3$ for this minimum point. We also observe from the graph that it crosses the horizontal axis at $x=-1,-1/3$.

This basic graphical method is useful for predicting the number of roots and their approximate location, which can serve as a useful starting point for analysis.

Rules of Thumb

Here are some rules of thumb for root prediction based on graphical methods:

• If $f(x_i)$ and $f(x_{i+1})$ have the same sign, interval has no roots (a) or an even number of roots (c)
• If $f(x_i)$ and $f(x_{i+1})$ have the different signs, there is at least one root (b) or an odd number of roots (d)

• Exceptions:
  • $f(x)$ is tangent to the $x$-axis as minimum or maximum
  • Function of discontinuous
  • Even number of roots in both cases