How to make a sign diagram

To make a sign diagram, follow these steps:

1. Identify the critical points: The critical points are the values of x where the function changes sign or is undefined. Find the values of x that make the function equal to zero or result in undefined values.

2. Place the critical points on a number line: Draw a horizontal number line and mark the critical points as vertical lines on the number line.

3. Test intervals: Select a test value from each interval created by the critical points. These test values should be easy to compute and belong to the interval but not on a critical point.

4. Evaluate the function at each test value: Plug in the test value into the function and determine the sign (+ or -) of the result.

5. Indicate the sign on the number line: Mark the sign for each interval on the number line using the result of the function evaluation from step 4. The sign can be indicated using a plus sign (+) for positive values and a minus sign (-) for negative values.

6. Determine the sign for the entire interval: Look at the signs for consecutive intervals. If the sign changes from positive to negative or negative to positive, write a "+" or "-" above the interval to indicate the sign of the function for that interval.

7. Finalize the sign diagram: Draw a final line underneath the number line and label each interval with the corresponding sign.

Remember to consider any limits or asymptotes as they may affect the sign diagram. The sign diagram helps in understanding where the function is positive or negative and aids in analyzing the behavior of the function.