Transcript 1.6 Transformation of Functions

Functions
Our objectives:
• Recognize “Parent Functions”
– Graphically & Algebraically
– Please take notes and ALWAYS ask questions 
Today’s Agenda
1. Do
2. CW Note-taking
guide on Parent
Functions
Today’s Objectives:
Students Will Be Able
To…
Define domain and range
Recognize Parent
Functions
Homework:
Do NOW:
. Define domain and range in your notebook.
10 mins
The following basic graphs will be used
extensively in this section. It is important
to be able to sketch these from memory.
Constant Function
f(x) = a
Linear function
f(x) = x
quadratic function
f ( x)  x
2
cubic function
f ( x)  x
3
Polynomial Function
• http://zonalandeducation.com/mmts/functio
nInstitute/polynomialFunctions/graphs/polyn
omialFunctionGraphs.html
•
•
•
•
•
*zero degree
*first Degree
*second degree
*third degree
Fourth degree
Exponential Function
x
f(x) = a
Logarithmic Function
f(x)=log x
a
square root function
f ( x)  x
cube root function
f ( x)  x
3
absolute value function
f ( x)  x
Rational Function
( x  1)(x  2)
f(x) =
2
( x  1)(x  2)
Reciprocal Function
1
f(x) =
x
Inverse Function
Piece-wise Function
Piece-wise Function
•
We will now see how certain
transformations (operations) of a
function change its graph. This will
give us a better idea of how to
quickly sketch the graph of certain
functions. The transformations are
(1) translations, (2) reflections, and
(3) stretching.
Vertical Translation
Vertical Translation
For b > 0,
the graph of y = f(x) + b is
the graph of y = f(x)
shifted up b units;
f ( x)  x
2
f ( x)  x2  3
the graph of y = f(x)  b is
the graph of y = f(x)
shifted down b units.
f ( x)  x2  2
Horizontal Translation
f ( x)  x2
Horizontal Translation
For d > 0,
the graph of y = f(x  d) is
the graph of y = f(x)
shifted right d units;
the graph of y = f(x + d) is
the graph of y = f(x)
shifted left d units.
y   x  2
2
y   x  2
2
• Vertical shifts
– Moves the graph up or
down
– Impacts only the “y”
values of the function
– No changes are made to
the “x” values
• Horizontal shifts
– Moves the graph left or
right
– Impacts only the “x”
values of the function
– No changes are made to
the “y” values
The values that translate the graph of a
function will occur as a number added or
subtracted either inside or outside a
function.
y  f (x  d )  b
Numbers added or subtracted inside
translate left or right, while numbers
added or subtracted outside translate up
or down.
Recognizing the shift from the equation,
examples of shifting the function
f(x) =
2
x
• Vertical shift of 3 units up
f ( x)  x , h( x)  x  3
2
2
• Horizontal shift of 3 units left (HINT: x’s go the
opposite direction that you might believe.)
f ( x)  x , g ( x)  ( x  3)
2
2
Points represented by (x , y) on the graph
of f(x) become
x
d , y  b for the function f ( x  d )  b
If the point (6, -3) is on the graph of f(x),
find the corresponding point on the
graph of f(x+3) + 2
(3,1)
Combining a vertical & horizontal shift
• Example of function
that is shifted down 4
units and right 6 units
from the original
function.
f ( x)  x ,
g ( x)  x  6  4
Reflections
• The graph of f(x) is the reflection of the graph of
f(x) across the x-axis.
• The graph of f(x) is the reflection of the graph of
f(x) across the y-axis.
• If a point (x, y) is on the graph of f(x), then
(x, y) is on the graph of f(x), and
• (x, y) is on the graph of f(x).
Reflecting
• Across x-axis (y becomes negative, -f(x))
• Across y-axis (x becomes negative, f(-x))
Vertical Stretching and Shrinking
The graph of af(x) can be obtained from the graph
of f(x) by
stretching vertically for |a| > 1, or
shrinking vertically for 0 < |a| < 1.
For a < 0, the graph is also reflected across the xaxis.
(The y-coordinates of the graph of y = af(x) can be
obtained by multiplying the y-coordinates of y = f(x)
by a.)
VERTICAL STRETCH (SHRINK)
• y’s do what we think
they should: If you see
3(f(x)), all y’s are
MULTIPLIED by 3 (it’s
now 3 times as high or
low!)
f ( x)  x  4
2
f ( x)  3 x  4
2
1 2
f ( x)  x  4
2
Horizontal stretch & shrink
• We’re MULTIPLYING by
an integer (not 1 or 0).
• x’s do the opposite of
what we think they
should. (If you see 3x in
the equation where it
used to be an x, you
DIVIDE all x’s by 3, thus
it’s compressed
horizontally.)
g ( x)  (3 x)  4
2
f ( x)  x  4
2
1 2
f ( x)  ( x)  4
3