Translations and stretches. There's more here but we didn't get to it.

1. What transformations
do you see?
Elefant slide by flickr
user Jan the manson

2. Translations
What do the graphs of
these functions look like?

3. In general we write ...
Now you try ...
Electric Slide by
flickr user kretyen

4. The equation of the black function is given.
Write the equations of the red, blue, and green functions.

5. Translations y = ƒ(x ­ b) + a
The role of parameter a:
b > 0 the graph shifts right b units. Examples
­ the x­coordinates are increased
b units.
b < 0 the graph shifts left b units.
­ the x­coordinates are decreased
b units.
WARNING: watch the sign of a
The role of parameter b:
a > 0 the graph shifts up a units.
­ the y­coordinates are increased
a units.
a < 0 the graph shifts down a units.
­ the y­coordinates are decreased
a units.

6. Stretches ...
and some
reflections.
Ballet stretch

7. Stretches ... (and a wee bit about reflections)
Let's start with a circle ...
Let's look at some graphs ... We'll head over to fooplot.com ...

8. Stretches and Compressions:
The role of parameter a:
a > 1 the graph of ƒ(x) is stretched Examples
vertically.
0 < |a| < 1 the graph of ƒ(x) is
compressed vertically.
­ the y­coordinates of ƒ are multiplied
by a.
The role of parameter b:
b > 1 the graph of ƒ(x) is compressed
horizontally.
(Everything quot;speeds upquot;)
0<|b|<1 the graph of ƒ(x) is stretched
horizontally. (Everything quot;slows downquot;)
­ the x­coordinates are multiplied
by .

9. Given y = ƒ(x) sketch the graph of:

10. Putting it all together ...
Try these examples ... y = ƒ(x)
Really! You're
Don't Look
asking for it ...
Behind Here!
REMEMBER: stretches
before translations

11. The graph of g(x) is formed by sliding the graph of ƒ(x) 4 units to the left. If
ƒ(x) = sin(x ­ 2) + 5, write an equation in terms of sine to represent g(x).

12. Given , write the equation that translates the graph of ƒ(x) three
units to the left.

13. Write g as a function of ƒ, and ƒ as a function of g.
ƒ
g

14. Homework tonight is
exercise 7.
Studying. As you
should probably be
doing right now.

15. Reflections
Th r e
Vertical Reflections
es
s t
Given any function ƒ(x):
e a t c
­ƒ(x) produces a reflection in the x­axis.
The y­coordinates of ƒ are multiplied by (­1).
re h e
sim s
Horizontal Reflections
ila
Given any function ƒ(x):
r to
ƒ(­x) produces a reflection in the y­axis.
The x­coordinates of ƒ are multiplied by (­1).
Inverses: the inverse of any function ƒ(x) is this isn't
(read as: quot;EFF INVERSEquot;)
WARNING:
undoes whatever ƒ did.

16. EVEN FUNCTIONS
Graphically: A function is quot;evenquot; if its graph is symmetrical about the y­axis.
These functions
are even...
These are
not ...
Symbolically (Algebraically)
a function is quot;evenquot; IFF (if and only if) ƒ(­x) = ƒ(x)
Examples: Are these functions even?
1. f(x) = x² 2. g(x) = x² + 2x
f(­x) = (­x)² g(­x) = (­x)² + 2(­x)
f(­x) = x² g(­x) = x² ­ 2x
since f(­x)=f(x) since g(­x) is not equal to g(x)
f is an even function g is not an even function

17. ODD FUNCTIONS
Graphically: A function is quot;oddquot; if its graph is symmetrical about the origin.
These
functions
These are
are odd ... not ...
Symbolically (Algebraically)
a function is quot;oddquot; IFF (if and only if) ƒ(­x) = ­ƒ(x)
Examples: 1. ƒ(x) = x³ ­ x 2. g(x) = x³­ x²
ƒ(­x) = (­x)³ ­ (­x) g(­x) = (­x)³ ­ (­x)²
ƒ(x) = ­x³ + x g(x) = ­x³ ­ x²
­ƒ(x) = ­(x³ ­ x) ­g(x) = ­(x³­x²)
­ƒ(x) = ­x³ + x ­g(x) = ­x³+ x²
since ƒ(­x)= ­ƒ(x) since g(­x) is not equal to ­g(x)
ƒ is an odd function g is not an odd function

18. Baby Play
or
All About
Inverse
Functions
duck wrangling by toyfoto

19. Inverses ...
The concept ...
Numerically speaking ...

20. Inverses ...
Algebraically speaking ...
Conceptually analyzing the function ...

21. Inverses ...
Graphically speaking ...

22. Attachments
Picture clipping.pictClipping
Studying. As you should probably be doing right now.
Double Spiral
http://fooplot.com/