We can summarize the different transformations and their related effects on the graph of a function in the following table.
Transformations of functions
Transformation of
Effect on the graph of
Vertical shift up
units
Vertical shift down
units
Shift left by
units
Shift right by
units
Vertical stretch if
vertical compression if
Horizontal stretch if
horizontal compression if
Reflection about the
-axis
Reflection about the
-axis
Transforming a function
For each of the following functions, a. and b., sketch a graph by using a sequence of transformations of a well-known function.
Starting with the graph of
shift
units to the left, reflect about the
-axis, and then shift down 3 units.
The function
can be viewed as a sequence of three transformations of the function
Starting with the graph of
reflect about the
-axis, stretch the graph vertically by a factor of 3, and move up 1 unit.
The function
can be viewed as a sequence of three transformations of the function
The power function
is an even function if
is even and
and it is an odd function if
is odd.
The root function
has the domain
if
is even and the domain
if
is odd. If
is odd, then
is an odd function.
The domain of the rational function
where
and
are polynomial functions, is the set of
such that
Functions that involve the basic operations of addition, subtraction, multiplication, division, and powers are algebraic functions. All other functions are transcendental. Trigonometric, exponential, and logarithmic functions are examples of transcendental functions.
A polynomial function
with degree
satisfies
as
The sign of the output as
depends on the sign of the leading coefficient only and on whether
is even or odd.
Vertical and horizontal shifts, vertical and horizontal scalings, and reflections about the
- and
-axes are examples of transformations of functions.
Key equations
Point-slope equation of a line
Slope-intercept form of a line
Standard form of a line
Polynomial function
For the following exercises, for each pair of points, a. find the slope of the line passing through the points and b. indicate whether the line is increasing, decreasing, horizontal, or vertical.