Lesson Explainer: Function Transformations: Translations
Mathematics
In this explainer, we will learn how to identify function transformations involving horizontal and vertical shifts.
It may be best to split transformations into groups as follows:
changes to the -variable
changes to the -variable
In practice, the changes to the -variable are applied directly to the
expression . So, instead of
, the (algebraic) effect of
is to produce
The second dimension on which we split the examination of operations is type, of which we
have three:
Additionβso we will think of
as addition of .
Multiplicationβso we
think of as multiplication by the factor
.
Negationβthese are just
and .
The use of negation means that we will restrict all multiplication to positive factors only.
The reason for this is that we have the following dictionary between the algebraic operations
on variables and the geometric effects on graphs:
Algebra
Geometry
Addition
Shifts (horizontal and vertical)
Multiplication
Stretches (horizontal and vertical)
Negation
Reflection in the axes
Consider the function whose graph is shown.
The graph has -intercepts at , and 2. The marked dot
is the point which is the inflection
point on the curve: a rotation through
about this point maps the curve
back on to itself.
We look at the operations in turn.
Addition ()
The graph of the function is
produced by an upward shift of by 2 units.
A downward shift by would correspond to
.
Addition ()
Notice that the curve
is given by a shift of curve
to the right, not the left, while the original
curve is shifted 2 units to the left to produce
. The reason is that for
to be on the curve , we
need to be on the graph of
. So, the curve
is to the left of
, by 2 units.
Multiplication
() We will use only positive numbers for now, so that we
can talk unambiguously about βstretches.β The effects of going from
to where
are shown below:
The effect is
to βstretchβ the curve in the vertical direction by
a factor of . Notice that the -axis is fixed by this
transformation and the difference between a stretching factor and
. You will often see the words βdilationβ and βcontractionβ used
to distinguish these two types of stretches.
Multiplication () Going from the equation to
changes The graphs of these
functions are related by a stretch in the horizontal directionβalthough not in the way you
may expect.
The geometric effect of is a stretch in the horizontal direction by a factor of
(not ). The reasons are similar to the
counterintuitive effect of and
above.
Negation The curve is the
reflection of in the -axis. Curve
is the image on reflection in the
-axis.
Combinations
Sometimes, you will come across combinations of transformations. For example, how does the
graph of relate to the graph of
? One way to approach an example like this is to
separate the transformation into changes in and changes in
. Starting with , we have
applied the transformations The combination of transformations that takes combines a stretch in the horizontal direction by
a factor of a with a shift to the right by
a unit. The transformation, which takes
, is a vertical stretch by a factor of 3. The figures
demonstrate this series of transformations for the graph of a particular function
.
When negatives
are involved, the appropriate reflections must be used. For example, the graph of
starts with a transformation
, which is a reflection in -axis. The next
transformation is , which is a shift by
1 unit to the left. Following this, we make a series of
transformations as follows: These correspond to
the following series of geometric transformations on the curve: reflection in
-axis vertical stretch by a factor of a
vertical shift by 4 units upwards.
Example 1: Using Transformations of Functions to Find the Equation of a Function from a
Graph
The following is a transformation of the graph of .
What is the function it represents? Write your answer in a form related to the
transformation.
Answer
Because the curve is an βupward facing Vβ, the
transformation must involve . Looking at the slope of the lines,
we see that these are 1 and . So, there has been no stretching (in
either direction). We then observe that we can get this curve by two shifts of
. First, a horizontal one by 1 unit left:
. Then, a vertical one by 4 units. This gives us the
answer:
Key Points
Given the graph of a function .
Addition
Operation
Transformed Equation
Geometrically
Shift left by 3 units
Shift right by 3 units
Shift up by 4 units
Shift down by 4 units
Multiplication (Positive Constants Only)
Operation
Transformed Equation
Geometrically
Horizontal stretch by a factor of
Horizontal stretch by a factor of 3
Vertical stretch by a factor of 4
Vertical stretch by a factor of
Negation
Operation
Transformed Equation
Geometrically
Reflection in the vertical axis
Reflection in the horizontal axis
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