function characteristics – end behavior
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Function Characteristics – End Behavior. - PowerPoint PPT PresentationTRANSCRIPT
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.
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
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
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
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
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?
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 ”
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.
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 ”.
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
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
End Behavior – visual approach
End behavior: As Let’s visually confirm this by ‘zooming out’ on the
graph.
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
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
These functions level off as they go to the left and/or right. The y-values do not necessarily approach ±∞.
End Behavior – non-infinite
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
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 ∞
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
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
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
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
Which of the following have the same end behaviors?
End Behavior - Patterns
A
B
C
DAs ,
As As
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
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
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