5 Steps to a 5 AP Calculus AB 2019 - William Ma

MA 3972-MA-Book May 8, 2018 13:46

Review of Precalculus 75

Shifting, Reflecting, and Stretching Graphs
Vertical and Horizontal Shifts
Giveny= f(x) anda >0, the graph ofy= f(x)+ais a vertical shift of the graph of
y=f(x)byaunits upward. Andy=f(x)−ais a vertical shift of the graph ofy=f(x)by
aunits downward. (See Figure 5.4-13.)

x

y

f(x)

y = f(x) + a

y = f(x) − a

−a

a

Figure 5.4-13

y = f(x −a) is a horizontal shift of the graph of y = f(x)bya units to the right.
y = f(x+a) is a horizontal shift of the graph ofy= f(x)byaunits to the left. (See
Figure 5.4-14.)

x

y

y = f(x + a) y = f(x) y = f(x − a)

−aa 0

Figure 5.4-14

Reflections About thex-Axis,y-Axis, and the Origin
Giveny= f(x), then the graph ofy=−f(x) is a reflection of the graph ofy= f(x) about
thex-axis. (See Figure 5.4-15.)

x

y y = f(x)

y = −f(x)

a

−a

0

Figure 5.4-15