Translations, stretches, reflections, even and odd functions.

1. Practice what you've learned ...
The coordinates of a point, A, on the graph of y = ƒ(x) are (-2, -3). What are
the coordinates of it's image on each of the following graphs:
The image of point B after each transformation shown above is given
below as point C(n). Find the original coordinates of B.
C1 (2, 3) C2 (-3, 7) C3 (5, -4) C4 (-1, 6) C5 (-4, -2)

2. Consider the equation below. Which transformation do you think should be
applied first? second? third? fourth?
Given A(-2, -3) find the coordinates of its image under the transformation
given above.
The image of point B after the transformation shown above is (1, 4). Find the
original coordinates of B.

3. Given A(-2, -3) find the coordinates of its image under the transformation
given above.
The image of point B after the transformation shown above is (1, 4). Find the
original coordinates of B.

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

5. 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:
DICTIONARY
The role of parameter a:
a > 1 the graph of ƒ(x) is stretched vertically. Examples
0 < |a| < 1 the graph of ƒ(x) is compressed
vertically.
- the y-coordinates 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 x-coordinates are multiplied by .

9. Putting it all together ...
y = ƒ(x)
Try these examples ...
REMEMBER: stretches before translations

10. Reflections
DICTIONARY
Vertical Reflections Horizontal Reflections
Given any function ƒ(x): Given any function ƒ(x):
-ƒ(x) produces a reflection in the x-axis. ƒ(-x) produces a reflection in the y-axis.
y-coordinates are multiplied by (-1) x-coordinates are multiplied by (-1)
Inverses: the inverse of any function ƒ(x) is
(read as: quot;EFF INVERSEquot;)
WARNING:
undoes whatever ƒ did.

11. EVEN FUNCTIONS
Graphically: A function is quot;evenquot; if its graph is symmetrical about the y-axis.
These functions DICTIONARY
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

12. ODD FUNCTIONS DICTIONARY
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) = -(x³-x²)
-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