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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
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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
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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
![Page 4: Sign Studies. Why do we use sign studies? Example 1. Determine the equation of f(x) by identifying the x-intercepts (roots, solutions, zeros, factors)](https://reader036.vdocuments.site/reader036/viewer/2022082821/5697bfd11a28abf838caae57/html5/thumbnails/4.jpg)
Sign Study of ( )f x
f
2 1 3
Positive on , 2 1,3 Negative on 2,1 3,
0 0 0
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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
![Page 6: Sign Studies. Why do we use sign studies? Example 1. Determine the equation of f(x) by identifying the x-intercepts (roots, solutions, zeros, factors)](https://reader036.vdocuments.site/reader036/viewer/2022082821/5697bfd11a28abf838caae57/html5/thumbnails/6.jpg)
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
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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
![Page 8: Sign Studies. Why do we use sign studies? Example 1. Determine the equation of f(x) by identifying the x-intercepts (roots, solutions, zeros, factors)](https://reader036.vdocuments.site/reader036/viewer/2022082821/5697bfd11a28abf838caae57/html5/thumbnails/8.jpg)
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
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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
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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
![Page 11: Sign Studies. Why do we use sign studies? Example 1. Determine the equation of f(x) by identifying the x-intercepts (roots, solutions, zeros, factors)](https://reader036.vdocuments.site/reader036/viewer/2022082821/5697bfd11a28abf838caae57/html5/thumbnails/11.jpg)
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.
![Page 12: Sign Studies. Why do we use sign studies? Example 1. Determine the equation of f(x) by identifying the x-intercepts (roots, solutions, zeros, factors)](https://reader036.vdocuments.site/reader036/viewer/2022082821/5697bfd11a28abf838caae57/html5/thumbnails/12.jpg)
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
![Page 13: Sign Studies. Why do we use sign studies? Example 1. Determine the equation of f(x) by identifying the x-intercepts (roots, solutions, zeros, factors)](https://reader036.vdocuments.site/reader036/viewer/2022082821/5697bfd11a28abf838caae57/html5/thumbnails/13.jpg)
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
![Page 14: Sign Studies. Why do we use sign studies? Example 1. Determine the equation of f(x) by identifying the x-intercepts (roots, solutions, zeros, factors)](https://reader036.vdocuments.site/reader036/viewer/2022082821/5697bfd11a28abf838caae57/html5/thumbnails/14.jpg)
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.
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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
![Page 16: Sign Studies. Why do we use sign studies? Example 1. Determine the equation of f(x) by identifying the x-intercepts (roots, solutions, zeros, factors)](https://reader036.vdocuments.site/reader036/viewer/2022082821/5697bfd11a28abf838caae57/html5/thumbnails/16.jpg)
3 2( ) 2 2f x x x x
Using the characteristics, draw the graph of the function.
![Page 17: Sign Studies. Why do we use sign studies? Example 1. Determine the equation of f(x) by identifying the x-intercepts (roots, solutions, zeros, factors)](https://reader036.vdocuments.site/reader036/viewer/2022082821/5697bfd11a28abf838caae57/html5/thumbnails/17.jpg)
Assignment
HW I
Have a Good Weekend!