tuesday evaluate these two functions function characteristics even vs odd symmetry concavity extreme

Tuesday

• Evaluate these two functions

4 2( ) 2 3 4

( )

f x x x

f x

3( ) 3 5

( )

f x x x

f x

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)

FUNCTIONSSymmetric about the y axis

Symmetric about the origin

A function f is even if for each x in the domain of f, f (– x) = f (x).

x

yf (x) = x2

f (– x) = (– x)2 = x2 = f (x)

f (x) = x2 is an even function.

Symmetric with respect to the y-axis.

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

7

123456

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.

Even functions have y-axis Symmetry

A function f is odd if for each x in the domain of f, f (– x) = – f (x).

x

y

f (x) = x3

f (– x) = (– x)3 = –x3 = – f (x)

f (x) = x3 is an odd function.

Symmetric with respect to the origin.

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

7

123456

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.

Odd functions have origin Symmetry

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

7

123456

8

-2-3-4-5-6-7

We wouldn’t talk about a function with x-axis symmetry because it wouldn’t BE a function.

x-axis Symmetry

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.

125 24 xxxf Is this function even?

1251)(2)(5 2424 xxxxxfYES

xxxf 32 Is this function even?

xxxxxf 33 2)()(2NO

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.

125 24 xxxf Is this function odd?

1251)(2)(5 2424 xxxxxfNO

xxxf 32 Is this function odd?

xxxxxf 33 2)()(2YES

If a function is not even or odd we just say neither (meaning neither even nor odd)

15 3 xxf

Determine if the following functions are even, odd or neither.

1515 33 xxxf

Not the original and all terms didn’t change signs, so NEITHER.

23 24 xxxf

232)()(3 2424 xxxxxf

Got f(x) back so EVEN.

Determine algebraically whether f(x) = –3x2 + 4 is even, odd, or neither.

Function Type Problems

2( ) 3 4f x x

2

2

( ) 3( ) 4

= -3x 4

= f(x)

f x x

f(x) is an even function by definition.

Is this function symmetrical?

Determine algebraically whether f(x) = 2x3 - 4x is even, odd, or neither.Practice Problem Seven

3( ) 2 4f x x x

Is this function symmetrical?

3

3

( ) 2( ) 4( )

= -2x 4

= -f(x)

f x x x

x

f(x) is an odd function by definition.

Practice Problem eightDetermine algebraically whether f(x) = 2x3 - 3x2 - 4x + 4 is even, odd, or neither.

3 2( ) 2 3 4 4f x x x x

Is this function symmetrical?

3 2

3 2

( ) 2( ) 3( ) 4( ) 4

= -2x 3 4 4

f(x) or -

f(x)

f x x x x

x x

f(x) is neither odd or even.

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19

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).

x

y

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).

Relative minimum

Relative maximum

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20

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

7

123456

8

-2-3-4-5-6-7

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 “locally”).

The value of c is called a local maximum of f.

increasing here

decreasing here

f(-2) = 5

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).

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21

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

7

123456

8

-2-3-4-5-6-7

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 “locally”).

The value of c is called a local minimum of f.

increasing here

decreasing here

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).

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Concavity• A graph may be concave up or

concave down• See graphs below:

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23

Concavity Examples

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24

Concavity Examples

25

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

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26

Inflection Points

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27

Inflection Points

Homework

WS 1-4

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