College Algebra 9th txtbk

190 CHAPTER 3 FUNCTIONS
EXAMPLE 1 Vertical and Horizontal Shifts
(A) How are the graphs of y x 2 2 and y x 2 3 related to the graph of y x 2 ?
Confirm your answer by graphing all three functions in the same coordinate system.
(B) How are the graphs of y (x 2) 2 and y (x 3) 2 related to the graph of y x 2 ?
Confirm your answer by graphing all three functions in the same coordinate system.
SOLUTIONS (A) Note that the output of y x 2 2 is always exactly two more than the output of
y x 2 . Consequently, the graph of y x 2 2 is the same as the graph of y x 2
2 shifted upward two units, and the graph of
is the same as the graph of
y x 2 shifted downward three units. Figure 3 confirms these conclusions. (It
appears that the graph of y f(x) k is the graph of y f(x) shifted up if k is
positive and down if k is negative.)
y
5
5
y x 2 2
y x 2
x
5
y x 2 3
5
Z Figure 3 Vertical shifts.
(B) Note that the output of y (x 2) 2 is zero for x 2, while the output of y x 2
is zero for x 0. This suggests that the graph of y (x 2) 2 is the same as the
graph of y x 2 shifted to the left two units, and the graph of y (x 3) 2 is the
same as the graph of y x 2 shifted to the right three units. Figure 4 confirms these
conclusions. It appears that the graph of y f(x h) is the graph of y f(x) shifted
right if h is negative and left if h is positive.
y x 2 y y (x 3) 2
5
y (x 2) 2
5
5
x
Z Figure 4 Horizontal shifts.
MATCHED PROBLEM 1
(A) How are the graphs of y 1x 3 and y 1x 1 related to the graph of
y 1x? Confirm your answer by graphing all three functions in the same coordinate
system.
(B) How are the graphs of y 1x 3 and y 1x 1 related to the graph of
y 1x? Confirm your answer by graphing all three functions in the same coordinate
system.
To summarize our experiences in Explore-Discuss 1 and Example 1: We can graph
y f(x) k by vertically shifting the graph of y f(x) upward k units if k is positive