Transformations
We are going to look at
some transformation
rules today:
Lets take a function
f  x  and change it to f   x 
What happens to the graph of the function w
Ex : f  x   x
f  x    x
hen we do this?
Lets check the graphs of
each function to see
what happens!
f x   x
When you negate the x,
you are reflecting the
graph in the y-axis!
g x    x
Lets take a function
f  x  and change it to  f  x 
What happens to the graph of the function w
Ex : f  x   x
2
When you negate the
function, or y, you are
reflecting the graph in
the x-axis!
g x    x
f x   x
hen we do this?
2
2
This means we
are negating the
function, not x.
So we just put a
negative in front
of the function.
Page 15
When you negate the x,
you are reflecting the
graph in the y-axis!
f x  
x
g x  
x
Page 15
When you negate the
function, or y, you are
reflecting the graph in
the x-axis!
g x    x
f x   x
5
Page 16
Since the x is negated, then we would reflect the function over the y-axis.
Since there is a negative in front of the function, then we reflect it over the
x-axis. This is also the same as a reflection through the origin!
Page 16
When we reflect over the
x-axis, we negate the
function, or y.
The function is negated,
or y, so we reflect over
the x-axis.
Homework
•TEQ
Lets take a function
f  x  and change it to f  ax 
What happens to the graph of the function w
Ex : f  x   sin( x )
f  2 x   sin  2 x 
hen we do this?
Lets check the graphs of
each function to see
what happens!
When you multiply the x
by a, it is a horizontal
dilation of 1/a.
D
H,
1
a
f  x   sin  x 
g  x   sin  2 x 
D
H,
1
2
1 
Lets take a function f  x  and change it to f  x 
a 
What happens to the graph of the function w hen we do this?
Ex : f  x   sin( x )
1 
1 
f  x   sin  x 
2 
2 
Lets check the graphs of
each function to see
what happens!
f  x   sin  x 
D H ,2
When you multiply the x
by 1/a, it is a horizontal
dilation of a.
D H ,a
1 
g  x   sin  x 
2 
Lets take a function
f  x  and change it to a f  x 
What happens to the graph of the function w
Ex : f  x   sin( x )
When you multiply the
function by a, just put a
in front of the function.
f  x   sin  x 
hen we do this?
When you multiply f(x)
by a, it is a vertical
dilation of a.
DV ,a
g  x   2 sin  x 
DV , 2
Lets take a function
f  x  and change it to
1
a
What happens to the graph of the function w
Ex : f  x   sin( x )
When you multiply the
function by 1/a, just put 1/a
in front of the function.
f x 
hen we do this?
When you multiply f(x)
by 1/a, it is a vertical
dilation of 1/a.
D
V,
1
a
f  x   sin  x 
g x  
D
V,
1
2
1
2
sin  x 
Page 18
This is a horizontal
D H ,2
dilation
of 2.
Page 18
This is a vertical
DV , 5
dilation
of 5.
Page 18
The y-intercept of the original
function is -6. When you do a
vertical dilation, you multiply
the y-value.
0 ,  6 
0 ,  12 
I multiplied the y-value
by 2, and got the new
y-intercept.
Page 18
We are multiplying
the x-value by 2,
which is a
horizontal dilation
of ½. So we
multiply all of the
x-values by ½.
We are multiplying
the function by 2,
which is a vertical
dilation of 2. So
we multiply all of
the y-values by 2.
Homework
•Page 24
#3,6,7,9,10