In this explainer, we will learn how to translate or stretch the trigonometric function and find the rule of a trigonometric function given the transformation.
Letβs recall some of the key features of the graphs of the main trigonometric functions: the sine and cosine functions.
Graphs of Trigonometric Functions
The Sine Function
When the sine of an angle is plotted against that angle, the result is a sine curve.
The Cosine Function
When the cosine of an angle is plotted against that angle, the result is as shown.
The domain of the sine and cosine functions is the set of real numbers, while their range is . Both functions are periodic, as demonstrated in the diagram, with a period of or radians.
We will need to be able to recognize the graphs of these functions, so we must familiarize ourselves with their key features, such as the locations of any turning points, the - and -intercepts, and the equations of any asymptotes, before considering how to interpret their transformations.
For our first example, let us practice identifying the graph of the cosine function by recalling the properties that it has.
Example 1: Identifying the Image of a Point on a Trigonometric Graph Following a Transformation
A figure shows the graph of . A transformation maps to . Determine the coordinates of following this transformation.
Answer
To begin with, although not strictly required, let us first determine the trigonometric function represented by the graph. Since the graph has a -intercept of 1, is periodic with period , and has roots at , , and , we can conclude that it must be the cosine function, that is,
For this example, we just need to identify the effect that the transformation that maps onto has on a single point. We can see that the point in the graph has coordinates , which corresponds to the fact that .
Hence, we can find the new coordinate by finding the value of at the point :
Therefore, the transformation maps to .
Let us further consider the implications of the transformation used in the previous example. We saw that for a specific value of , the output of is 2 less than the output of .
Let us consider how the transformation affects the outputs of the function at some other values of .
| 0 | |||||
| 1 | 0 | ||||
For each point, we can see that the value of is 2 less than the value of , which is as expected. If we plotted this for all possible values of , we would have the following.
We have highlighted the transformation of point from the previous example to a new point . Graphically, this corresponds to the point shifting vertically downward by 2. In fact, we can see that the entire graph has been shifted downward by 2 in this way.
This is actually a result that we can generalize. If a function is mapped to for a constant , this is equivalent to a translation by in the graph (i.e., it is shifted upward by ). If is negative, as we just saw, then this results in a downward shift in the graph.
Now, let us consider what might happen if we were to add or subtract a constant from the value of Β before substituting it into the function , for example, . Let us write the effect this has on the outputs for some values of in a table.
| 0 | ||||||
| 1 | 0 | 0 | 1 | |||
| 0 | 1 | 0 | 0 |
It may not be immediately obvious, but the outputs in the bottom row have all been shifted to the right. As an example, the point (which represents ) has been shifted to (which represents ). Let us plot the two graphs below, including the shift of this point.
As we can see, the graph has been shifted to the right by . That is to say, by subtracting from directly, the output has been moved by in the opposite direction.
This result, too, can be generalized. A function mapped to , for a constant , has its graph translated by (i.e., it is shifted to the left by ). If is negative, as we have just shown, this results in a shift to the right.
Definition: Function Transformations
Consider the function , for real constants , , , and :
- represents a vertical stretch by a scale factor .
- represents a horizontal stretch by a scale factor for .
- represents a translation by .
- represents a translation by .
We might notice that a vertical stretch by a scale factor , where , can be alternatively represented as a reflection in the followed by a vertical stretch by a scale factor . Similarly, a horizontal stretch by a scale factor , where , can be represented as a reflection in the followed by a horizontal stretch by a scale factor . These interpretations are interchangeable, but we will use the former notation in this explainer.
In our next example, we will demonstrate how to find the coordinates of a point after a transformation using these definitions.
Example 2: Identifying the Image of a Point on a Trigonometric Graph Following a Transformation
The figure shows the graph of . A transformation maps to . Determine the coordinates of following this transformation.
Answer
Remember, a function is mapped onto after a horizontal stretch by a scale factor . Since the transformation in this question maps to , we define . This represents a horizontal stretch by a scale factor of , as shown in the following diagram.
Since the graph has been stretched horizontally by a scale factor of , the -coordinate of the image of point will be and the -coordinate of the image will remain unchanged.
The coordinates of the image of point are .
It is worth noting that we can check this answer by substituting into :
We observe from the graph of that . This is the -coordinate of the image of .
In our third example, we will apply these definitions to help us recognize the graph of a transformed function.
Example 3: Identifying the Graph of a Trigonometric Function after a Transformation
Which of the following is the graph of ?
Answer
Remember, the graph of the cosine function is as shown.
In order to identify the correct graph, we use the fact that is mapped onto by a translation units in the vertical direction, or by the vector . This means that is mapped onto by a translation one unit upward. After this translation, the -intercept will have coordinates , and the points of intersection of the curve with the will lie at , for integer values of .
This is option D.
There will be occasions when a function is mapped onto another function by a series of transformations. In this case, there is a limited number of situations where the order in which these are performed is unimportant. Generally, the order matters if the transformations act in the same direction (in other words, two transformations that have a horizontal effect).
For instance, consider the functions defined by and . The graph of both functions are some transformation of the graph of . Figure 1 shows the graph of and , where is obtained by performing a vertical stretch by a scale factor of 2 and then a translation by . Figure 2 shows the graph of and , where the blue plot is obtained by a translation Β then a vertical stretch.
To avoid errors, we should follow the order given below.
How To: Sequencing Transformations of Functions
maps onto in the following order:
- A vertical stretch by a scale factor , where , resulting in a reflection in the
- A horizontal stretch by a scale factor , where , resulting in a reflection in the
- A horizontal translation given by
- A vertical translation given by
For instance, letβs identify the series of transformations that map onto . We rewrite as and use the sequencing of transformations. We see that undergoes two separate transformations to map it onto : a horizontal stretch by a scale factor of followed by a horizontal translation by .
The graph of is shown in figure 1. A horizontal stretch by a scale factor of results in the graph shown in figure 2.
Finally, a horizontal translation by gives the graph shown below.
Letβs demonstrate how to apply this process to find the image of a point on a curve.
Example 4: Identifying the Image of a Point on a Trigonometric Graph Following Multiple Transformations
The figure shows the graph of . A transformation maps to . Determine the coordinates of following this transformation.
Answer
Remember, maps onto in the following order:
- A vertical stretch by a scale factor , where , resulting in a reflection in the
- A horizontal stretch by a scale factor , where , resulting in a reflection in the
- A horizontal translation given by
- A vertical translation given by
To identify the transformation that maps to , we rewrite as and let , , , and . Then, undergoes the following:
- A vertical stretch by a scale factor of 4
- A horizontal stretch by a scale factor of
- A horizontal translation given by
- A vertical translation given by
We can apply each step to point with coordinates :
- A vertical stretch by a scale factor of 4 maps onto .
- A horizontal stretch by a scale factor of maps onto .
- A horizontal translation given by maps onto .
- A vertical translation given by maps onto .
Hence, the coordinates of the image of are .
In our final example, we will demonstrate how to apply this process to find the graph of the image of a function.
Example 5: Identifying the Graph of a Trigonometric Function after Two Transformations
Which of the following is the graph of ?
Answer
We recall that maps onto in the following order:
- A vertical stretch by a scale factor , where , resulting in a reflection in the
- A horizontal stretch by a scale factor , where , resulting in a reflection in the
- A horizontal translation given by
- A vertical translation given by
If we define , then we can define the image after some series of transformations as .
Then, we can let and , so the function maps onto by a horizontal stretch by a scale factor of followed by a vertical translation given by .
The graph of is shown in figure 1, and a horizontal stretch of by a scale factor of 4 is shown in figure 2. We observe that point with coordinates maps onto with coordinates .
Finally, this graph is translated one unit down as shown in figure 3. Point maps onto with coordinates .
This is option B.
Let us finish by recapping some key concepts from this explainer.
Key Points
- For a function and real constants
, , , and
,
- represents a vertical stretch by a scale factor , where will result in a reflection in the ;
- represents a horizontal stretch by a scale factor for , where will result in a reflection in the ;
- represents a translation by ;
- represents a translation by .
- If we consider a series of transformations that maps
onto
, they should be applied
in the following order:
- First, a vertical stretch by a scale factor ;
- Second, a horizontal stretch by a scale factor ;
- Third, a horizontal translation given by ;
- Finally, a vertical translation given by .
