PreCalc 12/11
Unit 3- Transformations
Day 1 Notes
Shifting
Make a table of values and graph the line y = x.
Now make a table of values for y = x - 1
What do you notice about the graph of y = x - 1 when compared to the graph of y = x?
Equation
Graph
add to x
moves graph ____________
subtract from x
moves graph ____________
add to y
moves graph ____________
subtract from y
moves graph ____________
Matty 2014
PreCalc 12/11
Ex 1
Unit 3 Transformations
Day 1 Notes Page 2
Graph the following three functions on the same graph.
y x
y x 3
y  x 3
Note:
The notation
y  k  f (x) is often used instead of y  f (x)  k to emphasize that the parameter k
involves a translation in the y-direction only. For example, instead of y  x  3 , we could write
y  3 x .
Ex 2
Graph the following three functions on the same graph.
y x
Matty 2014
y  x3
y  x3
PreCalc 12/11
Unit 3 Transformations
Day 1 Notes Page 3
Changing Equations
Sometimes instead of changing the graph we can just change the equation.
Ex 1
Given the function y  f (x) , write the equation of the transformed function after each of
the following translations.
a) a vertical translation 4 units down.
b) a horizontal translation 5 units to the right.
c) a horizontal translation 3 units to the left and a vertical translation 6
units up.
Ex 2
In each case below, the given point is transformed into a second point by a certain
translation. Find the coordinates of the second point.
a) a horizontal translation 3 units to the left
(4 , –6)  (
,
)
b) a vertical translation 5 units down
(–3 , –5)  (
,
)
c) a horizontal translation 4 units to the right and a vertical translation 6 units up
(–7 , 2)  (
,
)
Ex 3
In each case below, a graph of y  f (x) is shown. Sketch the graph of the translated
function whose equation is given.
a)
y  f (x)  2
Matty 2014
b) y  f (x  3)
PreCalc 12/11
Unit 3 Transformations
Day 1 Notes Page 4
Reflections
The graph of y   f (x) is a reflection of the graph of y  f (x) in the_______________ .
The graph of y  f (x) is a reflection of the graph of y  f (x) in the_______________ .
Ex 1 Graph the indicated functions from the original y = f(x)
a) y   f (x)
Matty 2014
b) y  f (x)
PreCalc 12/11
Unit 3 Transformations
Day 1 Notes Page 5
Stretching / Compressing
Just like our translations were all backwards as far as what we expected, the expanding of a graph is the
same. Making an equation bigger has the opposite effect on the graph and makes the graph smaller .
Ex 1 Sketch the graph of y  3 x .
We can obtain the graph of y  3 x from the graph of y  x through two
transformations:
a) _____________________________________________________
b) ______________________________________________________
Ex 2
The grid below contains the graph of a function y  f (x) . Sketch the graph of
y  f ( 13 x) .
Matty 2014
PreCalc 12/11
Unit 3 Transformations
Day 1 Notes Page 6
Ex 3
The graph of a function y  f (x) is expanded vertically by a factor of 3, compressed
1
horizontally by a factor of 4 , and then reflected in the y–axis. If the equation of its
image is y  af (bx) , determine the values of a and b for the transformation.
To be consistent with the general transformation function y  af (b(x  p))  q , use the
following order when applying combinations of two or more transformations:
1) _________________________________________________________________
2) _________________________________________________________________
3) __________________________________________________________________
Ex 4
The point (5, –2) is on the graph of the function y  f (x) . Track (5, –2) through each of
the following five transformations to its final image.
Transformation
Original point of y  f (x)
Image of (5, –2)
(5 , –2)
reflected in the x-axis
(
,
)
compressed vertically by a factor of 1/2
(
,
)
expanded horizontally by a factor of 5
(
,
)
translated horizontally 4 units to the right
(
,
)
translated vertically 5 units down
(
,
)
Matty 2014
PreCalc 12/11
Unit 3 Transformations
Day 1 Notes Page 7
In summary, we can adjust all our graphs and all our equations in one of two ways:
a( y  d )  fb( x  c)
y  afb( x  c)  d
Matty 2014