September 09, 2014
Section 1.4
Shifting, Reflecting, and Stretching Graphs
Objective: In this lesson you learn how to identify and graph shifts, reflections, and
nonrigid transformations of functions.
Important Vocabulary
Vertical shift
A transformation of the graph of y = f(x), represented by h(x) = f(x) ± c, in which the
graph is shifted upward or downward c units respectively (c is a positive real number).
Horizontal shift
A transformation of the graph of y = f(x), represented by
h(x) = f(x ± c), in which the graph is shifted to the left or to the right c units respectively
(c is a positive real number).
Rigid transformations
Transformations in which the basic shape of the graph is unchanged.
Nonrigid transformations
Transformations that cause a distortion (a change in the original shape of the graph).
I. Summary of Graphs of Common Functions
Examples of each of the six most commonly used functions in algebra that you need to
know!
Constant Function
f(x)=c
Absolute Value Function
f(x)=|x|
Identity Function
f(x)=x
Square Root Function
f(x)=√x
September 09, 2014
Quadratic Function
f(x)=x
Cubic Function
2
f(x)=x
3
Graph y=x2+c for c=-2, 0, 2, 4 What happens?
Graph y=(x+c)2 for the same values. What do you notice?
II. Vertical and Horizontal Shifts
Let c be a positive real number. Complete the following representations of shifts in
the graph of :
1) Vertical shift c units upward:
h(x) = f(x) + c
2) Vertical shift c units downward:
h(x) = f(x) - c
3) Horizontal shift c units to the right:
h(x) = f(x - c)
4) Horizontal shift c units to the left:
h(x) = f(x + c)
Ex: Let f(x) = |x|. Write the equation for the function resulting from a vertical shift
of 3 units downward and a horizontal shift of 2 units to the right of the graph of f(x).
f(x) = | x - 2 | - 3
Ex: Given f(x)=x3+x, describe the shifts in the graph of f generated by the following
functions.
a) g(x)=(x+1)3+x+1
b) h(x)=(x-4)3+x
Horizontal shift 1 unit to the left
Horizontal shift 4 units right and vertical
shift 4 units up
September 09, 2014
III. Reflecting Graphs
Graph g(x)=-x2 and h(x)=(-x)2. Compare to f(x)=x2
What do you notice?
A reflection in the x-axis is a type of transformation of the graph of y = f(x)
represented by h(x) = - f(x) .
A reflection in the y-axis is a type of transformation of the graph of y = f(x)
represented by h(x) = f(- x) .
Ex: Let f(x)=|x|. Describe the graph of g(x)= -|x| in terms of f.
The graph of g is a reflection of the graph of f in the x-axis.
Ex: Describe the reflections in the graph of f(x)=x3+3 generated by...
a) g(x)=-x3+3
Reflected in the y-axis, g(x)=(-x)3+3
b) h(x)=-x3-3
Reflected in the x-axis, h(x)= -1(x3+3)
IV. Nonrigid Transformations
Three types of rigid transformations:
1) Horizontal shifts
again?
y
e
h
t
were
2) Vertical shifts
What
3) Reflections
Rigid transformations change only the position of the graph in the xy-plane.
Graph y=2x2 and y=0.5x2. Compare to y=x2.
Two types of nonrigid transformations:
1) Vertical stretch
2) Vertical shrink
A nonrigid transformation y=cf(x) of the graph of y=f(x) is a vertical stretch
if c > 1 or a vertical shrink if 0 < c < 1.
Note:
Sometimes two more types of nonrigid transformations:
1) Horizontal stretch
2) Horizontal shrink
A nonrigid transformation y=f(cx) of the graph of y=f(x) is a horizontal stretch if
see #48 p.49 (cubic)
c > 1 or horizontal shrink if 0 < c < 1.