Transformations I
Shifts
Math 141
Group
Members:__________________________________________________
A transformation of a function changes the function in a predictable
way. Follow the instructions below, and write your own rules for the
following types of transformations.
For full credit, write all descriptions in complete sentences.
1. Consider how the graph of f x   c compares with the graph of
f x  .
2
Look at what happens to the function f x   x when you add a
constant c to f x  . Graph the function f x   x in Y1 using the
standard WINDOW. Look at the graph now!
2
Then graph the function g x  f x  3 or x  3 in Y2 and
hx  f x  5 or x 2  5 in Y3 all together in the same window on your
calculator. Then, draw the graphs here.
2
1
2. The changes you saw on page 1 are called vertical shifts.
a) Note that you are adding the constant –5 to the output of the
function in h(x). How does adding the constant change the graph? Be
specific!
Adding -5 shifts the graph 5 units down from the original.
b) Generalize and write a rule that you can remember, in your own
words, that tells how the graph of f x   c is different from the graph
of f x  . Use the word “shift” in your rule, and remember that a
distance must always be a positive number.
The y-value of every coordinate on the graph moves c units vertically,
up if c is positive and down if c is negative.
2
3. Now consider how the graph of f x  c  compares with the graph of
f x  . Look at what happens to the function f x   x when you add a
constant c to x . Graph f x   x in Y1. (Use MATH ► NUM and [1]
for abs to get absolute value.) Look at the graph! Then graph
g x   f x  6  x  6 in Y2 and hx   x  4 in Y3 in the standard
WINDOW on your calculator.
g x , you are adding the constant 6 to the input of f x  to
change the graph, and in h x  you are adding the constant –4.
Note: In
These transformations are called horizontal shifts.
a) Note that you are adding the constants 6 and –4 to the input of the
function. How does adding the constant 6 change the graph of the
original function?
Adding 6 shifts the graph 6 units to the left.
b) How does adding the constant –4 change the graph of the original
function?
Adding -4 shifts the graph 4 units to the right.
c) Generalize and write a rule that you can remember, in your own
words, that tells how the graph of f x  c is different from the graph
of
f x  .
Use the word “shift” in your rule.
The graph of the function horizontally shifts the absolute value of c
units. The graph shifts to the right when it’s negative, to the left
when it’s positive.
3
4. Compare the graph of the function  f x  with the graph of the
function f x  .
2
Look at what happens to f x    x  2 x  4 when you make it
negative. Graph f x    x  2 x  4 in the standard WINDOW.
2
Look at the graph now! Then graph g x    f x   x  2 x  4 in
the same window.
This change is called a reflection in the x-axis.
2
a) Look at TABLE. The inputs (for both functions) are in the X column,
and the outputs are in the corresponding Y columns. What do you
notice about the outputs? What do you notice about the distance of
each function from the x-axis?
The output values switch their signs, unless the output is zero.
The distance from the x-axis remains the same.
b) This transformation is called a reflection or a flip. What line was the
graph reflected over?
The x-axis, or axis of symmetry, is y = 0.
4