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Standard Level

5.05 Sign diagrams

Lesson

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

To draw a sign diagram
  1. Draw a horizontal line (this represents the $x$x-axis).
  2. Mark the $x$x-intercepts or any other significant $x$x-values on the line.
  3. Draw dashed vertical lines to indicate any asymptotes.
  4. 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:

 

 

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