Sign Studies

Today you will conduct sign studies of ( ),

'( ), and ''( ) to organize characteristics of

the function when given the graph of the function.

f x

f x f x

Why do we use sign studies?

Each of our sign studies will tell us important

characteristics about the function of ( ). This

will be a concept that will help you organize the

first and second derivative tests.

f x

Example

1. Determine the equation of f(x) by identifying the x-intercepts (roots, solutions, zeros, factors) and using the end behavior to determine the leading coefficient.

2

3 2

3 2

( ) 2 1 3

2 3

2 5 6

2 5 6

f x x x x

x x x

x x x

x x x

Sign Study of ( )f x

f

2 1 3

Positive on , 2 1,3 Negative on 2,1 3,

0 0 0

What characteristics of ( ) does our sign

study of ( ) indicate?

f x

f x

Summarize Sign Study of ( )f x

Our sign study of ( ) indicates

the intervals on which the graph

of ( ) is above and below the

x-axis.

f x

f x

Sign Study of '( )f x

3 2Recall ( ) 2 5 6

5. Find '( )

6. Make a sign study of '( )

f x x x x

f x

f x

You will need to use your strategies for finding zeros (factoring, quadratic formula, synthetic division, etc.)

'f

2 19

3

2 19

3

0 0

Sign Study of '( )f x7. Determine the interval(s)

where ( ) is increasing by

examining the graph.

8. Determine the relative

maximum of ( ) by

examining the graph.

9. Determine the relative

minimum of ( ) by

examining th

f x

f x

f x

e graph.

10. Compare your answers

for # 7 - 9 with your sign

study for '( ). What do

you notice?

f x

What characteristics of ( ) does our sign

study of '( ) indicate?

f x

f x

Summarize Sign Study of '( )f x

Our sign study of '( ) indicates the intervals on

which the graph of ( ) is increasing or decreasing.

It also indicates where the relative maximums and

minimums (extremas) are located.

f x

f x

Where '( ) 0, the slope of the tangent to the graph of the function is horizontal,

so there exists a relative maximum or minimum.

f x

Sign Study of ''( )f x3 2

2

Recall ( ) 2 5 6

'( ) 3 4 5

11. Find ''( )

12. Make a sign study of ''( )

f x x x x

f x x x

f x

f x

Again, you will need to use your strategies for finding zeros (factoring, quadratic formula, synthetic division, etc.)

''f 0

23

Sign Study of ''( )f x

13. Determine the interval(s)

where ( ) changes concavity

by examining the graph.

14. Determine where ( ) is

concave up and concave down

by examining the graph.

15. Compare your answers

for # 13 - 14

f x

f x

with your sign

study for ''( ). What do

you notice?

f x

What characteristics of ( ) does our sign

study of ''( ) indicate?

f x

f x

Summarize Sign Study of ''( )f x

Our sign study of ''( ) indicates

the intervals on which the graph

of ( ) is concave up or concave

down. It also indicates where the

points of inflection (the points where

a graph changes concavity) ar

f x

f x

e located.

Summary

Sign StudyZeros

(Critical Pts) + sign study - sign study

x-intercepts of graph

lies above x-axis

lies below x-axis

Relative Extrema (if sign changes)(Max + to -)(Min - to +)

is increasing

is decreasing

Points of inflection (if sign changes)

is concave up

is concave down

( )f x

'( )f x

''( )f x

( )f x ( )f x

( )f x ( )f x

( )f x ( )f x

Let’s try another example:

Find the equation for a cubic polynomial which contains the following points:

(-1,0) (0,2) (1,0) (2,0)

Do a sign study of ( ).

-The places where ( ) 0 are "critical points"

-Without a graph you can "test" whether ( ) is

positive or negative by substituting values into

the equation.

f x

f x

f x

3 2( ) 2 2f x x x x

Do a sign study of '( ).

You will need to find the zeros using the quadratic

formula.

Without a graph you can "test" whether '( ) is

positive or negative in an interval by substituting

in an "easy" valu

f x

f x

e in each interval between the

zeros.

3 2( ) 2 2f x x x x Do a sign study of ''( ).

You will again need to find the zeros and test

points.

Name the point(s) of inflection.

Discuss the concavity of the graph.

f x

3 2( ) 2 2f x x x x

Using the characteristics, draw the graph of the function.

Assignment

HW I

Have a Good Weekend!