5.05 Sign diagrams
Sign diagrams are useful when checking whether a function is above the $x$x-axis (positive) or below the $x$x-axis (negative).
- Draw a horizontal line (this represents the $x$x-axis).
- Mark the $x$x-intercepts or any other significant $x$x-values on the line.
- Draw dashed vertical lines to indicate any asymptotes.
- Put a positive or a negative sign on either side of these $x$x-values, depending on whether the $y$y-value is positive or negative.
Worked examples
example 1
Draw a sign diagram for the following function:
Think: The linear function crosses the $x$x-axis at $x=0$x=0 . So we need to mark $0$0 on the line in our sign diagram:
Do: Then since the function is below the $x$x-axis to the left of $x=0$x=0 we must put a negative sign to the left of $0$0 on the sign diagram. Since the function is above the $x$x-axis to the right of $x=0$x=0 we must put a positive sign to the right of $0$0 on the sign diagram:
This is the final sign diagram for the original linear function:
Example 2
Draw a sign diagram for the following function:
Think: This function crosses the $x$x-axis at $x=-4$x=−4 and $x=1$x=1, so we need to mark these values on our sign diagram:
Do: The function is above the $x$x-axis both to the left of $x=-4$x=−4 and to the right of $x=1$x=1. So we should put a positive sign to the left of $-4$−4 and to the right of $1$1 on our sign diagram:
And finally, since the function is below the $x$x-axis between $x=-4$x=−4 and $x=1$x=1, we should put a negative sign between $-4$−4 and $1$1 on our sign diagram:
Example 3
Draw a sign diagram for the following function:
Think: This function has no $x$x-intercepts, but it does have a vertical asymptote at $x=0$x=0. So, we should draw a vertical dashed line at $0$0 on our sign diagram:
Do: Since the function is above the $x$x-axis on both sides of the asymptote at $x=0$x=0, we should put a positive sign on either side of the dashed line on our sign diagram:
