mba admissions in india
TRANSCRIPT
Essential Questions
1)What is the difference between an odd and even function?
2)How do you perform transformations on polynomial functions?
admission.edhole.com
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3
Even and Odd Functions (graphically)
If the graph of a function is symmetric with respect to the y-axis, then it’s even.
If the graph of a function is symmetric with respect to the origin, then it’s odd.
admission.edhole.com
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4
Even and Odd Functions (algebraically)
A function is even if f(-x) = f(x)
A function is odd if f(-x) = -f(x)
If you plug in -x and get the original function, then it’s even.
If you plug in -x and get the opposite function, then it’s odd.
admission.edhole.com
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5
Let’s simplify it a little…
We are going to plug in a number to simplify things. We will usually use 1 and -1 to compare, but there is an exception to the rule….we will see soon!
admission.edhole.com
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6
f x x( ) =Even, Odd or Neither?Ex. 1
f x x( ) =Graphically Algebraically
(1) 1 1f = =( 1) 1 1f − = − =
They are the same, so it is.....
admission.edhole.com
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7
f x x x( ) = −3Even, Odd or Neither?
Ex. 2
3( )f x x x= −Graphically Algebraically
3(2) (2) (2)f = − =6=
6= −3( 2) ( 2) ( 2)f − = − − −
They are opposite, so…
What happens if we plug in 1?
admission.edhole.com
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8
f x x( ) = +2 1Even, Odd or Neither?
2( ) 1f x x= +Graphically Algebraically
2(1) (1) 1f = +
2=2( 1) ( 1) 1f − = − +
Ex. 3
2=They are the same, so.....
admission.edhole.com
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9
3( ) 1f x x= −Even, Odd or Neither?
3( ) 1f x x= −Graphically Algebraically
3(1) (1) 1f = −
0=
2= −
3( 1) ( 1) 1f − = − −
They are not = or opposite, so...
Ex. 4
admission.edhole.com
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10
Let’s go to the Task….
admission.edhole.com
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11
What happens when What happens when we change the we change the
equations of these equations of these parent functions?parent functions?
admission.edhole.com
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12
14)9()( 2 −+= xxf
3)2()( −+= xxf
Left 9 , Down 14
Left 2 , Down 3
admission.edhole.com
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13
-f(x)
f(-x)
Reflection in the x-axis
Reflection in the y-axis
What did the negative on the outside do?
What do you think the negative on the inside will do?
Study tip: If the sign is on the outside it has “x”-scaped
Study tip: If the sign is on the inside, say “y” am I in here?
admission.edhole.com
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14
Write the Equation to this Graph
2)3( 2 +−= xy
admission.edhole.com
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 15
Write the Equation to this Graph
1)2( 3 −+= xy
admission.edhole.com
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 16
Write the Equation to this Graph
11 +−−= xy
admission.edhole.com
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 17
Write the Equation to this Graph
3 3( ) ( ) 2 or ( ) ( ) 2f x x f x x= − + = − +admission.edhole.com
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 18
Example: Sketch the graph of f (x) = – (x + 2)4 .
This is a shift of the graph of y = – x 4 two units to the left.
This, in turn, is the reflection of the graph of y = x 4 in the x-axis.
x
y
y = x4
y = – x4f (x) = – (x + 2)4
admission.edhole.com
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 19
Compare:3 3 31
( ) ( ) 4 ( )4
f x x and g x x and h x x= = =
admission.edhole.com
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 20
Compare…
What does the “a” do?
2 2( ) to ( ) 3f x x f x x= =
Compare… 2 21( ) to ( )
2f x x f x x= =
• What does the “a” do?
Vertical stretch
Vertical shrink
admission.edhole.com
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 21
Nonrigid Transformations
h(x) = c f(x) c >1
0 < c < 1
Vertical stretch
Vertical shrink
Closer to y-axis
Closer to x-axis
admission.edhole.com
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 22
Polynomial functions of the form f (x) = x n, n ≥ 1 are called
power functions.
If n is even, their graphs resemble the graph of
f (x) = x2.
If n is odd, their graphs resemble the graph of
f (x) = x3.
x
y
x
y
f (x) = x2
f (x) = x5f (x) = x4
f (x) = x3
admission.edhole.com