Chapter Sixteen — Graphing Functions
The Humongous Book of Algebra Problems
The domain and range of k(x) both consist of all real numbers except for 0.
The graph is symmetric about the origin, as the reciprocal of a number x and its
opposite –x are opposites: k(–x) = –k(x).
Graphing Functions Using Transformations
Move, stretch, squish, and ip graphs
16.31 Transform the graph of f(x) = x
to graph the function g(x) = x
The only difference between functions f(x) and g(x) is a constant, –5. Adding
or subtracting a constant c from a function affects its graph—all points on the
graph are shifted c units up or down, respectfully, on the coordinate plane.
In this problem, the graph of g(x) is exactly 5 units below the graph of f(x), as
illustrated by Figure 16-19.
Figure 16-19: The graphs of f(x) = x
and g(x) = x
– 5 are equivalent, except that g(x)
is ﬁve units below f(x).
16.32 Transform the graph of f(x) = x
to graph the function h(x) = (x – 3)
Whereas only x is squared in function f(x), the expression x – 3 is squared
in h(x). When a constant c is added to, or subtracted from, the input x of a
function, the corresponding graph is shifted –c units to the right or left, as
illustrated by Figure 16-20.
have 0 in the
denominator, so the
domain must exclude
0. There’s no number
you can divide into 1 to
get a quotient of 0, so
you have to exclude
0 from the range as
If you subtract 3
from the input of a
function, it moves the
graph 3 units to the
RIGHT. Adding 3 to
the input would move
it 3 units to the LEFT.
That’s the opposite of
what you might think—
negative numbers move
things right and positive
numbers move things