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:
- Addition—so we will think of as addition of .
- Multiplication—so we think of as multiplication by the factor .
- Negation—these 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:
|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.
We will use only positive numbers for now, so that we can talk unambiguously about “stretches.” The effects of going from to where are shown below: The effect is to “stretch” 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 “dilation” and “contraction” 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 direction—although 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.
The curve is the reflection of in the -axis. Curve is the image on reflection in the -axis.
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.
Because the curve is an “upward facing V”, 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:
Given the graph of a function .
|Shift left by 3 units|
|Shift right by 3 units|
|Shift up by 4 units|
|Shift down by 4 units|
Multiplication (Positive Constants Only)
|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|
|Reflection in the vertical axis|
|Reflection in the horizontal axis|