Sign Diagrams

Often we feel it is too tedious to draw a time-consuming graph of a function but wish to know when the function is positive, negative, zero or undefined. A sign diagram allows us to do this and is relatively easy to construct.

For the function $f(x)$, the sign diagram consists of the following.
  • a horizontal line which is really the $x$-axis
  • positive ($+$) and negative ($-$) signs indicating that the graph is above and below the $x$-axis respectively
  • the zeros of the function, which are the $x$-intercepts of the graph of $y=f(x)$, and the roots of the equation $f(x)=0$
  • value of $x$ where the graph is undefined, represented as a dotted line

Consider the following three functions. Sign Diagrams A sign change occurs about a zero of the function $y=(x+1)(x-2)$ for linear factors $(x+1)$ and $(x-2)$. This indicates curring of the $x$-axis.

Sign Diagrams No sign change occurs about a zero of the function $y=(x+1)^2$ for squared linear factor $(x+1)^2$. This indicates touching of the $x$-axis.

Sign Diagrams The dotted line indicates that a function is undefined at $x=0$.

Example 1

Draw a sign diagram for the following graph. Sign Diagrams

Example 2

Draw a sign diagram for the following graph. Sign Diagrams

Example 3

Draw a sign diagram for $(x+3)(x-1)$.

Example 4

Draw a sign diagram for $\dfrac{x+2}{x-1}$.

Example 5

Draw a sign diagram for $\dfrac{x(x+1)}{x-2}$.