Sign diagrams are useful when checking whether a function is above the x-axis (positive) or below the x-axis (negative).

To draw a sign diagram

Draw a horizontal line (this represents the x-axis).
Mark the x-intercepts or any other significant 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-values, depending on whether the 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-axis at x = 0 . So we need to mark 0 on the line in our sign diagram:

Do: Then since the function is below the x-axis to the left of x = 0 we must put a negative sign to the left of 0 on the sign diagram. Since the function is above the x-axis to the right of x = 0 we must put a positive sign to the right of 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-axis at x = - 4 and x = 1, so we need to mark these values on our sign diagram:

Do: The function is above the x-axis both to the left of x = - 4 and to the right of x = 1. So we should put a positive sign to the left of - 4 and to the right of 1 on our sign diagram:

And finally, since the function is below the x-axis between x = - 4 and x=1, we should put a negative sign between -4 and 1 on our sign diagram:

Example 3

Draw a sign diagram for the following function:

Think: This function has no x-intercepts, but it does have a vertical asymptote at x = 0. So, we should draw a vertical dashed line at 0 on our sign diagram:

Do: Since the function is above the x-axis on both sides of the asymptote at x = 0, we should put a positive sign on either side of the dashed line on our sign diagram:

5.05 Sign diagrams

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