function characteristics – end behavior
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Function Characteristics – End Behavior. - PowerPoint PPT PresentationTRANSCRIPT
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Function Characteristics – End Behavior
AII.7 - The student will investigate and analyze functions algebraically and graphically. Key concepts include
a) domain and range, including limited and discontinuous domains and ranges; b) zeros; c) x- and y-intercepts;
d) intervals in which a function is increasing or decreasing; e) asymptotes; f) end behavior; g) inverse of a function;
and h) composition of multiple functions. Graphing calculators will be used as a tool to assist in investigation
of functions.
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The arrows at the end of a graph tell us the image goes on forever. In what direction would you say these graphs continue indefinitely?
End Behavior
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The end behavior tells us about the far ends of the graph, when the x or y values are infinitely large or small.
Most graphs have two ends so we talk about the left-hand end behavior and the right-hand end behavior.
There are typically two dimensions to the end behavior: left/right and up/down. Most graphs do not have strictly horizontal or vertical end behavior.
End Behavior
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Terminology – infinite end behavior◦ If a graph continues to the LEFT indefinitely:
“x approaches -∞" or symbolically ◦ If a graph continues to the RIGHT indefinitely:
“x approaches ∞" or symbolically
◦ If a graph continues down indefinitely: “y approaches -∞" or symbolically
◦ If a graph continues up indefinitely: “y approaches ∞" or symbolically
End Behavior
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Direction of Infinite Continuance We say Symbolic
Notation
Left x approaches negative infinity
Right x approaches infinity
Up y approaches infinity
Down y approaches negative infinity
End Behavior Terminology
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End Behavior – visual approach
Let’s see what happens to the graph if we ‘zoom’ out a bit. (note the change in the scales on the graph)
Notice that the ends continue to extend in the same directions as we zoom out. What directions do they go?
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End Behavior – visual approach
Right End Behavior◦ The right-hand side of this
graph goes up indefinitely.◦ Our two directions are right ()
and up ().◦ So the right-hand end
behavior is “as approaches , approaches ”
Left End Behavior◦ The left-hand side of this
graph goes down indefinitely.◦ Our two directions are left ()
and down ().◦ So the left-hand end behavior
is “as approaches , approaches ”
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End Behavior – numerical approach
We said the right end behavior of this graph was “as approaches , approaches ”.
Let’s examine this numerically by checking out some large values of x and seeing the y value that go with them.
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End Behavior – numerical approach
x y10
100
1,000
10,000
100,000
1,000,000
Right End Behavior: Notice what happens the y values as x gets exponentially larger.
x y10 19
100
1,000
10,000
100,000
1,000,000
x y10 19
100 199
1,000
10,000
100,000
1,000,000
x y10 19
100 199
1,000 1,999
10,000
100,000
1,000,000
x y10 19
100 199
1,000 1,999
10,000 19,999
100,000
1,000,000
x y10 19
100 199
1,000 1,999
10,000 19,999
100,000 199,999
1,000,000
x y10 19
100 199
1,000 1,999
10,000 19,999
100,000 199,999
1,000,000 1,999,999
As x gets exponentially larger, y also continues to get exponentially larger, thus confirming the right end behavior “as approaches , approaches ”.
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End Behavior – numerical approach
x y-10
-100
-1,000
-10,000
-100,000
-1,000,000
Left End Behavior: Notice what happens the y values as x gets exponentially smaller (approaches -∞).
As x gets exponentially smaller, y also continues to get exponentially smaller, thus confirming the left end behavior “as approaches , approaches ”.
x y-10 -21
-100
-1,000
-10,000
-100,000
-1,000,000
x y-10 -21
-100 -201
-1,000
-10,000
-100,000
-1,000,000
x y-10 -21
-100 -201
-1,000 -2,001
-10,000
-100,000
-1,000,000
x y-10 -21
-100 -201
-1,000 -2,001
-10,000 -20,001
-100,000
-1,000,000
x y-10 -21
-100 -201
-1,000 -2,001
-10,000 -20,001
-100,000 -200,001
-1,000,000
x y-10 -21
-100 -201
-1,000 -2,001
-10,000 -20,001
-100,000 -200,001
-1,000,000 -2,000,001
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End Behavior – visual approach
What would you predict is the end behavior for the quartic graph we looked at earlier?
Left-hand end behavior:◦ As x approaches negative
infinity (goes to the left), y approaches infinity (goes up)
Right-hand end behavior:◦ As x approaches infinity (goes
to the right), y approaches infinity (goes up)
End behavior:◦ As ◦ Since both ends continued
up, we combined the end behaviors into one statement.
Goes up to the left
Goes up to the right
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End Behavior – visual approach
End behavior: As Let’s visually confirm this by ‘zooming out’ on the
graph.
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x y-10 476.2-100-1,000-10,000-100,000-1,000,000
x y-10 476.2-100 5,088,383.2-1,000-10,000-100,000-1,000,000
End Behavior – numerical approachLeft End Behavior
As x gets exponentially larger or smaller, y continues to rise exponentially confirming
the end behavior: as
x y101001,00010,000100,0001,000,000
x y10 300.2100 4,888,623.21,00010,000100,0001,000,000
x y10 300.2100 4,888,623.21,00010,000100,0001,000,000
x y10 300.2100 4,888,623.21,00010,000100,0001,000,000
x y10 300.2100 4,888,623.21,00010,000100,0001,000,000
x y10 300.2100 4,888,623.21,00010,000100,0001,000,000
x y101001,00010,000100,0001,000,000
Right End Behavior
x y-10 476.2-100 5,088,383.2-1,000-10,000-100,000-1,000,000
x y-10-100-1,000-10,000-100,000-1,000,000
x y-10 476.2-100 5,088,383.2-1,000-10,000-100,000-1,000,000
x y-10 476.2-100 5,088,383.2-1,000-10,000-100,000-1,000,000
x y-10 476.2-100 5,088,383.2-1,000-10,000-100,000-1,000,000
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Predict the end behavior of the following functions. What difference do you notice about their shape compared to the functions we have been exploring?
End Behavior – non-infinite
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These functions level off as they go to the left and/or right. The y-values do not necessarily approach ±∞.
End Behavior – non-infinite
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End Behavior – non-infinite
x y
-10
-30
-100
-1,000
Fill in the following tables. Use the data you find to determine the end behavior of this exponential function.
x y
-10 -.99987
-30 -.99999999988
-100 -1*
-1,000 -1*
𝑓 (𝑥 )=2𝑥− 3−1
* These values are rounded because the decimal exceeds the capabilities of the calculator.
Left End Behavior
Left End Behavior: As x approaches −∞, y approaches -1
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End Behavior – non-infinite
x y
10
30
100
1,000
Fill in the following tables. Use the data you find to determine the end behavior of this exponential function.
x y
10 127
30 134,217,727
100
1,000 Error*
𝑓 (𝑥 )=2𝑥− 3−1
* This value was so large that it exceeded the capabilities of my calculator.
Right End Behavior
Right End Behavior: As x approaches ∞, y approaches ∞
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End Behavior – non-infinite
𝑓 (𝑥 )=2𝑥− 3−1 Recap:
Left End Behavior: As x approaches −∞, y
approaches -1
Right End Behavior: As x approaches ∞, y
approaches ∞OR
As ,and as
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End Behavior – non-infinite
x y
-10
-100
-1,000
-10,000
Fill in the following tables. Use the data you find to determine the end behavior of this rational function.
x y
-10 1.875
-100 1.9897
-1,000 1.9989
-10,000 1.9998
𝑔 (𝑥 )= 1𝑥+2
+2Left End Behavior
Left End Behavior: As x approaches −∞, y approaches 2
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End Behavior – non-infinite
x y
10
100
1,000
10,000
Fill in the following tables. Use the data you find to determine the end behavior of this rational function.
x y
10 2.0833
100 2.0098
1,000 2.0009
10,000 2.00009998
Right End Behavior
Right End Behavior: As x approaches ∞, y approaches 2
𝑔 (𝑥 )= 1𝑥+2
+2
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End Behavior – non-infiniteRecap:
Left End Behavior: as x approaches −∞, y
approaches 2
Right End Behavior: as x approaches ∞, y
approaches 2
OR
as x ∞, y
𝑔 (𝑥 )= 1𝑥+2
+2
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Which of the following have the same end behaviors?
End Behavior - Patterns
A
B
C
DAs ,
As As
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How are these functions similar?◦ They are all polynomial functions
Their equations are made up of the sum/difference of terms with integer exponents
◦ Their end behaviors always approach ∞ or -∞. A and D have even degrees
◦ A is a quadratic () and D is quartic () ◦ Even degree polynomial functions have the same
left and right end behaviors.◦ Meaning, either both ends go up (as ) or both ends
go down () .
End Behavior - Patterns
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B and C have odd degrees◦ B is a cubic () and D is quintic () ◦ Odd degree polynomial functions have opposite left
and right end behaviors. ◦ Meaning if the function goes down to the left (as
then it goes up to the right (as ) and vice versa.
End Behavior - Patterns
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The leading coefficient will determine whether the functions point up or down.◦ A negative leading coefficient will cause a reflection
over the x-axis.
Recap of the end behavior of polynomial functions
End Behavior - Patterns
Degree Leading Coefficient
Left End Behavioras
Right End Behavior
as ;
EvenPositive
Negative
OddPositive
Negative