Ch 2.5 Transformations of Functions.
Basic Functions and Their Graphs
Vertical Translations of Graphs
Consider a function defined byy=f(x). Letkrepresent a positive real number.
· The graph ofy=f(x) +kis the graph ofy=f(x) shiftedkunitsupward.
· The graph ofy=f(x) −kis the graph ofy=f(x) shiftedkunitsdownward.
Horizontal Translations of Graphs
Consider a function defined byy=f(x). Lethrepresent a positive real number.
· The graph ofy=f(x−h) is the graph ofy=f(x) shiftedhunits to theright.
· The graph ofy=f(x+h) is the graph ofy=f(x) shiftedhunits to theleft.
Reflections Across thex- andy-Axes
Consider a function defined byy=f(x).
· The graph ofy= −f(x) is the graph ofy=f(x) reflected across thex-axis.
· The graph ofy=f(−x) is the graph ofy=f(x) reflected across they-axis.
Vertical Shrinking and Stretching of Graphs
Consider a function defined byy=f(x). Letarepresent a positive real number.
· Ifa> 1, then the graph ofy=af(x) is the graph ofy=f(x) stretched vertically by a factor ofa.
· If 0 <a< 1, then the graph ofy=af(x) is the graph ofy=f(x) shrunk vertically by a factor ofa.
Note:For any point (x,y) on the graph ofy=f(x), the point (x,ay) is on the graph ofy=af(x).
Horizontal Shrinking and Stretching of Graphs
Consider a function defined byy=f(x). Letarepresent a positive real number.
· Ifa> 1, then the graph ofy=f(ax) is the graph ofy=f(x) shrunk horizontally by a factor of.
· If 0 <a< 1, then the graph ofy=f(ax) is the graph ofy=f(x) stretched horizontally by a factor of.
Note:For any point (x,y) on the graph ofy=f(x), the pointis on the graph ofy=f(ax).
Ex1. Use translations to graph the given functions.
a. b. gx=(x+3)2 c.
d. h(x) =x3+ 2 e. f.
Ex2. Use translations to graph the given functions.
The graph ofy=f(x) is shown. Graph
a. y=f(2x)
b.
Ex2. Use translations to graph the given functions.
The graph ofy=f(x) is given.
a. Graphy= −f(x).
b. Graphy=f(−x).
Ex2. Use translations to graph the given functions.
a. b. c. v(x) = −(x+ 2)2+ 1
d. e. f.