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 three main trigonometric functions, the sine, cosine, and tangent function.

### 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, whilst their range is . Both functions are periodic, as demonstrated in the diagram, with a period of or radians.

**The Tangent Function**

The graph of the tangent function is shown.

Since , it follows that is undefined at values of where . In fact, the vertical asymptotes for the graph of the tangent function occur at or , for any integer value . The domain of the tangent function is therefore , or in radians, and the range is all real numbers. The tangent function has 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.

In our first example, we will demonstrate a numerical interpretation of a function transformation which will allow us to generalize translations to trigonometric functions.

### Example 1: 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

We might first recognize the graph of this function; it is the graph the tangent function where is given in degrees. We can therefore state,

In order to identify the transformation that maps onto , letβs consider how the transformation affects the outputs of the function at various values of .

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For any value of in the domain of , the output of is 3 less than the output of . We can interpret this as a translation of the original curve 3 units down, or by .

Since point has coordinates , after a translation by , its image will have coordinates .

In the previous example, we saw that a function is mapped onto after a translation by . We can generalize this as follows:

A function is mapped onto , for real constants , after a translation by .

Now, letβs consider what might happen if we were to add a constant to the value of Β *before* substituting it into the function , say .

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The function is mapped onto by a translation. This time, the vector that describes this translation is . We can generalize this result as follows:

A function is mapped onto , for real constants , after a translation by .

Whilst we can consider a numerical interpretation of the main function transformations, in practice it makes sense to learn their general form. Letβs summarize the key transformations:

### Definition: Function Transformations

Consider the function , for real constants , , , and ,

- represents a translation by ,
- represents a translation by ,
- represents a vertical stretch by a scale factor ,
- represents a horizontal stretch by a scale factor for .

We might notice that a vertical stretch by a scale factor , where , can be alternatively represented as a reflection in the -axis followed by a vertical stretch by a scale factor of . Similarly, a horizontal stretch by a scale factor , where , can be represented as a reflection in the -axis followed by a horizontal stretch by a scale factor of . 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

Recall, a function is mapped onto after a horizontal stretch by scale factor . Since the transformation in this question maps to , we define , so this represents a horizontal stretch by scale factor 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 whilst 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 recognise 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

Recall, 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. This means that is mapped onto by a translation one unit upwards. After this translation, the -intercept will have coordinates , and the points of intersection of the curve with the -axis will lie at for integer values of .

This is option D:

There will be occasions where a function is mapped onto another function by a series of several transformations. In this case, there are 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 given order:

### How To: Sequencing Transformations of Functions

maps onto in the following order:

- A vertical stretch by a scale factor where will result in a reflection in the -axis,
- A horizontal stretch by a scale factor , where will result in a reflection in the -axis,
- 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

Recall, maps onto in the following order:

- A vertical stretch by a scale factor where will result in a reflection in the -axis,
- A horizontal stretch by a scale factor , where will result in a reflection in the -axis,
- 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 4,
- A horizontal stretch by a scale factor ,
- 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 4 maps onto ,
- A horizontal stretch by a scale factor 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 will result in a reflection in the -axis,
- A horizontal stretch by a scale factor , where will result in a reflection in the -axis,
- 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 , 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 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 function , for real constants , , , and ,
- represents a translation by ,
- represents a translation by ,
- represents a vertical stretch by a scale factor ,
- represents a horizontal stretch by a scale factor .

- A series of transformations can be applied in the following order. To see how maps onto :
- A vertical stretch by a scale factor where will result in a reflection in the -axis,
- A horizontal stretch by a scale factor , where will result in a reflection in the -axis,
- A horizontal translation given by ,
- A vertical translation given by .