algebra 2 section 2-3
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EXTREMA AND END BEHAVIOR
SECTION 2-3
ESSENTIAL QUESTIONS
• How do you identify the end behavior of graphs?
• How do you identify extrema of functions?
VOCABULARY1. End Behavior:
2. Relative Maximum:
3. Relative Minimum:
VOCABULARY1. End Behavior: What occurs on a graph as x
approaches positive or negative infinity
2. Relative Maximum:
3. Relative Minimum:
VOCABULARY1. End Behavior: What occurs on a graph as x
approaches positive or negative infinity
2. Relative Maximum: When no other points nearby have a greater value for the y-coordinate
3. Relative Minimum:
VOCABULARY1. End Behavior: What occurs on a graph as x
approaches positive or negative infinity
2. Relative Maximum: When no other points nearby have a greater value for the y-coordinate
3. Relative Minimum: When no other points nearby have a lesser value for the y-coordinate
VOCABULARY4. Turning Points:
5. Extrema:
VOCABULARY4. Turning Points: Occur at the relative maxima or
minima; the point where the curve changes direction from up to down or vice versa
5. Extrema:
VOCABULARY4. Turning Points: Occur at the relative maxima or
minima; the point where the curve changes direction from up to down or vice versa
5. Extrema: The collective term for the relative maxima, relative minima, and turning points
EXAMPLE 1Describe the end behavior of each linear function.
x
ya.
EXAMPLE 1Describe the end behavior of each linear function.
x
ya.
As x→ +∞,f (x)→ +∞
EXAMPLE 1Describe the end behavior of each linear function.
x
ya.
As x→ +∞,f (x)→ +∞As x→ −∞,f (x)→ −∞
y
x
EXAMPLE 1Describe the end behavior of each linear function.
b.
y
x
EXAMPLE 1Describe the end behavior of each linear function.
b.
As x→ +∞,g(x)→ −4
y
x
EXAMPLE 1Describe the end behavior of each linear function.
b.
As x→ +∞,g(x)→ −4As x→ −∞,g(x)→ −4
EXAMPLE 2Describe the end behavior of each nonlinear function.
a.
EXAMPLE 2Describe the end behavior of each nonlinear function.
a.
As x→ +∞,h(x)→ +∞
EXAMPLE 2Describe the end behavior of each nonlinear function.
a.
As x→ +∞,h(x)→ +∞As x→ −∞,h(x)→ −∞
EXAMPLE 2Describe the end behavior of each nonlinear function.
b.
EXAMPLE 2Describe the end behavior of each nonlinear function.
b.
As x→ +∞, j(x)→ −∞
EXAMPLE 2Describe the end behavior of each nonlinear function.
b.
As x→ +∞, j(x)→ −∞As x→ −∞, j(x)→ −∞
EXAMPLE 3The table and graph below are a function with extrema. Estimate the zeros. Then estimate the
coordinates at which relative maxima and minima occur.x y-3 9-2 0-1 -10 01 -32 -163 -45
EXAMPLE 3The table and graph below are a function with extrema. Estimate the zeros. Then estimate the
coordinates at which relative maxima and minima occur.x y-3 9-2 0-1 -10 01 -32 -163 -45
Zeros: x = -2, x = 0
EXAMPLE 3The table and graph below are a function with extrema. Estimate the zeros. Then estimate the
coordinates at which relative maxima and minima occur.x y-3 9-2 0-1 -10 01 -32 -163 -45
Zeros: x = -2, x = 0Relative Maximum: near x = 0
EXAMPLE 3The table and graph below are a function with extrema. Estimate the zeros. Then estimate the
coordinates at which relative maxima and minima occur.x y-3 9-2 0-1 -10 01 -32 -163 -45
Zeros: x = -2, x = 0Relative Maximum: near x = 0Relative Minimum: near x = -1
EXAMPLE 4The table and graph represent the balance on
Maggie Brann’s savings account over a year. Use the table and graph to estimate the extrema for
the function. Then explain the extrema in the context of the situation.
EXAMPLE 4Month Balance ($)
1 11002 12003 13004 13005 14006 12007 13008 14009 900
10 100011 110012 1200
Acco
unt B
alan
ce ($
)0
200400600800
100012001400
Month0 1 2 3 4 5 6 7 8 9 10 11 12
EXAMPLE 4Month Balance ($)
1 11002 12003 13004 13005 14006 12007 13008 14009 900
10 100011 110012 1200
Acco
unt B
alan
ce ($
)0
200400600800
100012001400
Month0 1 2 3 4 5 6 7 8 9 10 11 12
Relative maxima: In May and August, the balance was $1400
EXAMPLE 4Month Balance ($)
1 11002 12003 13004 13005 14006 12007 13008 14009 900
10 100011 110012 1200
Acco
unt B
alan
ce ($
)0
200400600800
100012001400
Month0 1 2 3 4 5 6 7 8 9 10 11 12
Relative maxima: In May and August, the balance was $1400These are the months where the balance was highest in relation to the months before and after.
EXAMPLE 4Month Balance ($)
1 11002 12003 13004 13005 14006 12007 13008 14009 900
10 100011 110012 1200
Acco
unt B
alan
ce ($
)0
200400600800
100012001400
Month0 1 2 3 4 5 6 7 8 9 10 11 12
EXAMPLE 4Month Balance ($)
1 11002 12003 13004 13005 14006 12007 13008 14009 900
10 100011 110012 1200
Acco
unt B
alan
ce ($
)0
200400600800
100012001400
Month0 1 2 3 4 5 6 7 8 9 10 11 12
Relative minima: In June, the balance was $1200, September, the balance was $900
EXAMPLE 4Month Balance ($)
1 11002 12003 13004 13005 14006 12007 13008 14009 900
10 100011 110012 1200
Acco
unt B
alan
ce ($
)0
200400600800
100012001400
Month0 1 2 3 4 5 6 7 8 9 10 11 12
Relative minima: In June, the balance was $1200, September, the balance was $900These are the months where the balance was lowest in relation to the months before and after.
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