Ch 3 lessons

Chapter 3: Transformations of Graphs and Data
Purpose: Sometimes in
statistics, the result of a study is in
a form that is not usable, so we
can transform it into a useful form.
An example is the last quiz we
took - using 1985 gave us an
overload in the calculator, so we
used only 85.

3.1 I can adjust the window to view critical points on
graphs

Definitions:
Transformation: a one-to-one correspondence between sets of points.
Types of transformations:
Translation: Shift left/right/up/down
Scale change: Dilations in regards to the x or y axis.

Parent Functions: non-transformed functions.

Greatest integer (floor function)

Asymptote - a line that the graph of a function
approaches but never crosses.

Points of discontinuity - "gaps" in the graph where it
appears to "jump" to a new location.

Parent functions you are expected to know are on p. 162
Examples:
Graph y = 3x2 usint the following windows

Find the equation of the asymtotes of the following function:
f(x) = 4
x + 1
-2

3.2 I can translate functions along the x and y
axis.

ex. A translation of the
function f(x) = x2 three units
to the right.

Definition: A translation in the plane is a transformation that
maps each point (x, y) onto (x + h, y + k).
ex. Come up with a transformation rule for translating a
function down 3 units and left 5.
T(x,y) = ( , )
T(4,4) = ( , )

Translating functions vertically
ex. Translate f(x) = x2 up 3 units
In general: Let g(x) represent the transformed function.
Vertical translation of up/down k units
g(x) = f(x) +/- ____

Translating functions horizontally
Consider the function f(x) = x2
Adjust the x value.
Graph g(x) = (x + 3)2
What happened?

Translating functions horizontally
In general: g(x) = f(x + k) is a translation
to the left k units.
ex. Consider h(x) = √(x - 5)
How does this compare to the parent function?

ex. Translate f(x) = x3 down 4 units and to the right 5.
p. 170 – 172 #1-6, 8-13

3.3 I can understand and apply symmetries of graphs to functions
There are 2 types of symmetry that we will discuss:
reflective and rotational
Any type of symmetry implies that one part of a graph is congruent to another
- that makes it easier to draw.

Line of symmetry: A figure
has a line of symmetry if
it can be mapped onto
itself by a reflection over
the line of symmetry.

Symmetry with respect to the y-axis.
y = x2 Prove below that y = x2 is symmetric over the y-axis.
In other words, show that
(x,y) = (-x,y)
Which other parent functions have symmetry over the y axis?

Symmetry with respect to the x-axis.
x = y2
Make a table of values:
What has to be true for it to be symmetric over
x?
(x,y) = (x,-y)

Symmetry with respect to the origin
(x,y) rotated 1800 about the origin gives (-x, -y)
y = x3
Make a table of values:
Which other parent functions have symmetry with respect to the origin
(rotational)?

examples of power functions:
Power functions (f(x) = xn) where n is even are considered even functions.

Definition: A function f is an even function if and only if for all values of x in its domain f(-
x) = f(x).
These functions are always symmetrical
about the y-axis.

Definition: A function f is an odd function if and only if for all values of x in its domain f(-
x) = -f(x).
These functions are always symmetrical
about the origin.

ex. Consider the function f(x) = (x + 3)2 - 10
Find any lines of symmetry or points of symmetry.

ex. Consider the function F with f(x) = 1/(x - 5)2 - 4
Find any lines/points of symmetry. Find any asymptotes and give equations of them.

Clarification of even and odd functions:
Even functions MUST fit into the pattern f(-x) = f(x)
Therefore, The highest power has to be an even number AND
the function must be symmetric to the y axis.
For example:
f(x) = x2 + 2 is even because when you look at the table, -3
and 3 have the same y value. This is true for ANY x value
and its opposite.
g(x) = (x - 3)2 + 5 is not even. However, it has a vertical line
of symmetry.
In summary, a function with the highest power being even
MAY be an even function and it MAY have a vertical line of
symmetry. You have to look at the table and graph to see.

Odd functions:
Much like even functions, odd functions have to fit a
pattern. The pattern is f(-x) = -f(x). That means that a point
with both coordinates being positive will map onto a point
where both are negative and vice versa. For example, in an
odd function, the point (3,5) will map onto (-3,-5). This will
ONLY happen if the symmetry is about the origin.
For example, f(x) = x5 is odd. Let's look at what it looks like
in the calculator.
graph table
However, g(x) = (x + 3)3 has rotational symmetry, but it is not
considered an odd function.

ex. Prove that the graph of y = x3 + 2x5 is symmetric to the origin.
First consider the point (a,b) on the graph.
Substitute a for x and b for y.
Do a little mathmagic and show that (-a, -b) is on the graph. (hint: first multiply both sides by -1)
p. 183-185 #1, 3-4, 8-10, 13-16

Extra practice for symmetries
A = (3, 4)
Plot B so that it is symmetrical to
A over the x - axis.
B = ( , )
Symmetric over x-axis rule: (x, y) ( , )

A = (3, 4)
Plot C so that it is symmetrical to
A over the y - axis.
C = ( , )
Symmetric over y-axis rule: (x, y) ( , )

A = (3, 4)
D = ( , )
Symmetric about the origin rule: (x, y) ( , )

Extra practice for even/odd functions
Characteristics of even functions:
Characteristics of odd functions:

State whether the functions are even, odd, or neither.

State whether the functions are even, odd, or neither.
f(x) = 2x + 4

3.4 I can dilate functions along the x and y axis.
p. 186 in class activity

Vertical scale factor
af(x) results in a vertical dilation
if 0 < a < 1 the vertical dilation will squish the graph.
y = x2 y = .5x2

if a > 1 the vertical dilation will stretch the graph.
y = x2 y = 2x2

if a < 0 the vertical dilation will flip the whole graph over the x-axis.
y = x2 y = -x2

Horizontal scale factor
f(ax) results in a horizontal dilation.
If 0 < a < 1, then the graph will stretch along the x-axis.
y = x3
y = (x/3)3

If a > 1, then the graph will shrink along the x- axis.
y = x3
y = (3x)3

If a < 0, then the graph will flip over the x-axis.
y = x3
y = (-x)3

3.5 I can find compositions of functions algebraically

Consider f(x) = x3, g(x) = 2x
f(3) =
g(f(3)) =
g(f(x)) is a composite function.
g = g(f(x))
o f "g of f of x"

Consider f(x) = x2 + x,
g(x) = x - 5
f(4) =
g(f(-2)) =

Consider f(x) = x2 + x,
g(x) = x - 5
g(f(x)) =
f(g(x)) =

Does g(f(x)) = f(g(x))?
Using the functions in the previous example, find f(g(2)) and g(f(2))
p. 205-207 #5, 8, 9a, 10-12

3.6 I can find inverse functions algebraically

Inverse of a function: Switching the coordinates in the set
of ordered pairs defined by a function.

ex. Let f = {(-3,9), (-2,4), (-1,1), (0,0), (1,1), (2,4), (3,9)}
a. Find the inverse of the function.
b. Is the inverse a function?

ex. Consider f(x) = x2 Find f-1(x) by switching the x
and y values.
Is the inverse a function?

Notice: Taking the inverse of a function reflects
it over the y=x line.

Horizontal line test for inverses:
The inverse of a function f is itself a function if and only if no horizontal line intersects the
graph of f in more than one point.
Is the inverse a function? Is the inverse a function?

ex. Verify that and
, then f(f-1(x)) = x
for all x not equal to 2 and f-1(f(x)) = x for all x not equal to 0.
Step 1: Find f(f-1(x))
Step 2: Find f-1(f(x))
Step 3: See if the answers to step 1 and 2 are equal.

Inverse function theorem: Any 2 functions f and g are inverse functions if and only if f(g(x))
= x for all x in the domain of g, and g(f(x)) = x for all x in the domain of f.
The previous example shows this since both steps
turned out to be x.

ex. Us the inverse function theorem to show that f and g, with
f(x) = 3x + 4 and g(x) = (1/3)x - 4, are not inverses.
p. 212-214 #5-7, 11, 14, 15, 16
Step 1: Find f(g(x))
Step 2: Find g(f(x))
If either one of these is not equal to x, they are not inverses.

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