Tuesday
• Evaluate these two functions
f ( x)  2 x  3x  4
f ( x) 
4
2
f ( x)  3x  5 x
f ( x) 
3
Function Characteristics
Even vs Odd
Symmetry
Concavity
Extreme
Objectives
• I can prove a function is even, odd, or
neither
• I can determine what type of symmetry a
function has from a graph
• I can find extreme of a function
(minimums/maximums)
• I can recognize concavity intervals based
on inflection points
Symmetry
• Symmetry means that one point on the
graph is exactly in the same position on the
other side of the symmetric line.
• Graphs can symmetric with respect to:
– x-axis
– y-axis
– A coordinate Point (Origin)
Section 1.2 : Figure 1.21,
Symmetry
Symmetric
wrt y-axis
Graphical Tests for Symmetry
• 1. A graph is symmetric wrt the x-axis, if
whenever (x, y) is on the graph, so is (x, -y)
• 2. A graph is symmetric wrt the y-axis, if
whenever (x, y) is on the graph, so is (-x, y)
• 3. A graph is symmetric wrt the origin, if
whenever (x, y) is on the graph, so is (-x, -y)
Symmetric about the y axis
FUNCTIONS
Symmetric about the origin
A function f is even if for each x in the domain of f,
f (– x) = f (x).
f (x) =
x2
y
Symmetric with
respect to the y-axis.
f (– x) = (– x)2 = x2 = f (x)
x
f (x) = x2 is an even function.
Even functions have y-axis Symmetry
8
7
6
5
4
3
2
1
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
-2
-3
-4
-5
-6
-7
So for an even function, for every point (x, y) on
the graph, the point (-x, y) is also on the graph.
A function f is odd if for each x in the domain of f,
f (– x) = – f (x).
f (x) = x3
f (– x) = (– x)3 = –x3 = – f (x)
y
Symmetric with
respect to the origin.
x
f (x) = x3 is an odd function.
Odd functions have origin Symmetry
8
7
6
5
4
3
2
1
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
-2
-3
-4
-5
-6
-7
So for an odd function, for every point (x, y) on the
graph, the point (-x, -y) is also on the graph.
x-axis Symmetry
We wouldn’t talk about a function with x-axis symmetry
because it wouldn’t BE a function.
8
7
6
5
4
3
2
1
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
-2
-3
-4
-5
-6
-7
A function is even if f( -x) = f(x) for every number x in
the domain.
So if you plug a –x into the function and you get the
original function back again it is even.
f x   5 x  2 x  1
4
2
Is this function even?
YES
f  x   5( x)  2( x)  1  5x  2 x  1
4
2
4
2
f x   2 x  x Is this function even?
NO
3
3
f  x   2( x)  ( x)  2 x  x
3
A function is odd if f( -x) = - f(x) for every number x in
the domain.
So if you plug a –x into the function and you get the
negative of the function back again (all terms change signs)
it is odd.
f x   5 x  2 x  1
4
2
Is this function odd?
NO
f  x   5( x)  2( x)  1  5x  2 x  1
4
2
4
2
f x   2 x  x Is this function odd? YES
3
3
f  x   2( x)  ( x)  2 x  x
3
If a function is not even or odd we just say neither
(meaning neither even nor odd)
Determine if the following functions are even, odd or
neither.
Not the original and all
3
terms didn’t change
signs, so NEITHER.
f x   5 x  1
f  x   5 x   1  5 x  1
3
3
f x   3x  x  2
4
2
Got f(x) back so
EVEN.
f  x   3( x)  ( x)  2  3x  x  2
4
2
4
2
Function Type Problems
Determine algebraically whether f(x) = –3x2 + 4 is even, odd, or neither.
f ( x)  3x 2  4
f ( x)  3( x) 2  4
= -3x 2  4
= f(x)
f(x) is an even function by definition.
Is this function symmetrical?
Practice Problem Seven
Determine algebraically whether f(x) = 2x3 - 4x is even, odd, or neither.
f ( x)  2 x 3  4 x
f ( x)  2( x)3  4( x)
= -2x 3  4 x
= -f(x)
f(x) is an odd function by definition.
Is this function symmetrical?
Practice Problem eight
Determine algebraically whether f(x) = 2x3 - 3x2 - 4x + 4 is even, odd, or neither.
f ( x)  2 x 3  3 x 2  4 x  4
f ( x)  2( x)3  3( x) 2  4( x)  4
= -2x 3  3 x 2  4 x  4
 f(x) or - f(x)
f(x) is neither odd or even.
Is this function symmetrical?
A function value f(a) is called a relative minimum of f
if there is an interval (x1, x2) that contains a such that
x1 < x < x2 implies f(a)  f(x).
y
Relative maximum
x
Relative minimum
A function value f(a) is called a relative maximum of f
if there is an interval (x1, x2) that contains a such that
x1 < x < x2 implies f(a)  f(x).
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Houghton Mifflin
19
f(-2) = 5
The value of c is
called a local
maximum of f.
increasing
here
When the graph of a
function is increasing to the
left of x = c and decreasing
to the right of x = c, then at
c the value of the function f
is largest (at least in the
area near there, hence
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20
“locally”).
Houghton Mifflin
8
7
6
5
4
3
2
1
decreasing
here
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
-2
-3
-4
-5
-6
-7
So 5 is called a local
maximum of the function
since for all x values
close to –2, 5 is the
maximum function value
(y value).
The value of c is
called a local
minimum of f.
When the graph of a
function is decreasing to the
left of x = c and increasing
to the right of x = c, then at
c the value of the function f
is smallest (at least in the
area near there, hence
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21
“locally”).
Houghton Mifflin
8
7
6
5
4
3
2
1
increasing
here
decreasing
here
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
-2
-3
-4
-5
-6
-7
f(4) = -1
So -1 is called a local
minimum of the function
since for all x values
close to 4, -1 is the
minimum function value
(y value).
Concavity
• A graph may be concave up or
concave down
• See graphs below:
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22
Concavity Examples
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23
Concavity Examples
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Houghton Mifflin
24
Inflection Points
• An inflection point on a graph is where the
graph changes concavity.
– It changes from concave up to concave
down
– Or it changes from concave down to up
25
Inflection Points
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Houghton Mifflin
26
Inflection Points
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27
Homework
WS 1-4
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