Translations and Reflections of Graphs
Dr. Howard
As we continue our discussion of graphs of
functions, you should familiarize yourself with the graphs of some common
functions. Using Scientific Notebook, sketch the graphs of the following
functions:
f(x) = x1/2
f(x) = |x|
f(x) = x
f(x) = x2
f(x) = x3
f(x) = x4
This assignment is designed to lead the student to discover shifting
and reflection properties of graphs of functions.
Example:
The graphs of the functions f(x) = x2
and g(x) = x2 + 1 are depicted
below.
The graph of y = f(x) appears in black, and the graph of y
= g(x) appears in red. The
red graph can be obtained by shifting the black graph vertically by one
unit. This is sometimes referred to as a "vertical translation" of the
graph.
Assignment:
-
Sketch the graph of f(x) = x2 using the color black.
a). In the same picture, include graphs of the following
functions. You should choose a different
color for each of the functions. Your plot must use the x-interval
[ -4, 4 ] and the y-interval [ -3, 5 ].
-
g(x) = (x - 2)2 + 1
-
h(x) = (x - 2)2 - 1
-
p(x) = (x + 2)2 + 1, and
-
q(x) = (x - 2)2 - 2.
b). Describe how one may obtain each graph by shifting the
graph of f(x).
c). Write a formula for a function, r(x), whose
graph is obtained by shifting the graph of f(x) 3 unit to
the left and 5 units upward.
-
Sketch the graph of f(x) = |x| in black and the graph
of g(x) = |x-1| in red,
in the same picture, using the x-interval [ -5, 5 ] and the y-interval
[ -1, 5 ]. Describe how the graph of g(x) compares to the
graph of f(x).
-
Using the x-interval [ -8, 8 ] and setting Sample = 500, sketch
the graph of f(x) = x1/2 in black.
a). To that graph, add the graphs of the following functions.
-
g(x) = (x-1)1/2,
-
h(x) = (-x)1/2, and
-
r(x) = (1-x)1/2.
b). Describe how each graph is related to the graph of
f(x).