Lesson Explainer: Transformation of Trigonometric Functions | Nagwa Lesson Explainer: Transformation of Trigonometric Functions | Nagwa

Lesson Explainer: Transformation of Trigonometric Functions Mathematics • Second Year of Secondary School

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.

Let’s 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 [1,1]. Both functions are periodic, as demonstrated in the diagram, with a period of 360 or 2𝜋 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 𝑥- and 𝑦-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 𝑓(𝑥). A transformation maps 𝑓(𝑥) to 𝑓(𝑥)2. Determine the coordinates of 𝐴 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 𝑦-intercept of 1, is periodic with period 2𝜋, and has roots at 𝑥=90, 90, and 180, we can conclude that it must be the cosine function, that is, 𝑓(𝑥)=𝑥.cos

For this example, we just need to identify the effect that the transformation that maps 𝑓(𝑥) onto 𝑓(𝑥)2 has on a single point. We can see that the point 𝐴 in the graph has coordinates (180,1), which corresponds to the fact that cos(180)=1.

Hence, we can find the new coordinate by finding the value of 𝑓(𝑥)2 at the point 𝑥=180: 𝑓(180)2=(180)2=12=3.cos

Therefore, the transformation maps (180,1) to (180,3).

Let us further consider the implications of the transformation used in the previous example. We saw that for a specific value of 𝑥, the output of 𝑓(𝑥)2 is 2 less than the output of 𝑓(𝑥).

Let us consider how the transformation affects the outputs of the function 𝑓(𝑥) at some other values of 𝑥.

𝑥04590180
𝑓(𝑥)=(𝑥)cos12201
𝑓(𝑥)2=𝑥2cos122223

For each point, we can see that the value of 𝑓(𝑥)2 is 2 less than the value of 𝑓(𝑥), which is as expected. If we plotted this for all possible values of 𝑥, we would have the following.

We have highlighted the transformation of point 𝐴 from the previous example to a new point 𝐵. 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 𝑓(𝑥) is mapped to 𝑓(𝑥)+𝑑 for a constant 𝑑, this is equivalent to a translation by (0,𝑑) in the graph (i.e., it is shifted upward by 𝑑). If 𝑑 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 𝑥 before substituting it into the function 𝑓(𝑥)=𝑥cos, for example, 𝑓(𝑥90). Let us write the effect this has on the outputs for some values of 𝑥 in a table.

𝑥090180270360
𝑓(𝑥)=(𝑥)cos10101
𝑓(𝑥90)=(𝑥90)cos01010

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 (180,1) (which represents cos(180)=1) has been shifted to (270,1) (which represents cos(27090)=1). 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 90. That is to say, by subtracting 90 from 𝑥 directly, the output has been moved by 90 in the opposite direction.

This result, too, can be generalized. A function 𝑓(𝑥) mapped to 𝑓(𝑥+𝑐), for a constant 𝑐, has its graph translated by (𝑐,0) (i.e., it is shifted to the left by 𝑐). If 𝑐 is negative, as we have just shown, this results in a shift to the right.

Definition: Function Transformations

Consider the function 𝑓(𝑥), for real constants 𝑎, 𝑏, 𝑐, and 𝑑:

  • 𝑎𝑓(𝑥) represents a vertical stretch by a scale factor 𝑎.
  • 𝑓(𝑏𝑥) represents a horizontal stretch by a scale factor 1𝑏 for 𝑏0.
  • 𝑓(𝑥+𝑐) represents a translation by (𝑐,0).
  • 𝑓(𝑥)+𝑑 represents a translation by (0,𝑑).

We might notice that a vertical stretch by a scale factor 𝑎, where 𝑎<0, can be alternatively represented as a reflection in the 𝑥-axis followed by a vertical stretch by a scale factor |𝑎|. Similarly, a horizontal stretch by a scale factor 𝑎𝑏, where 𝑏<0, can be represented as a reflection in the 𝑦-axis followed by a horizontal stretch by a scale factor 𝑎|𝑏|. 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 𝑓(𝑥). A transformation maps 𝑓(𝑥) to 𝑓(2𝑥). Determine the coordinates of 𝐴 following this transformation.

Answer

Remember, a function 𝑓(𝑥) is mapped onto 𝑓(𝑏𝑥) after a horizontal stretch by a scale factor 1𝑏. Since the transformation in this question maps 𝑓(𝑥) to 𝑓(2𝑥), we define 𝑏=2. This represents a horizontal stretch by a scale factor of 12, as shown in the following diagram.

Since the graph has been stretched horizontally by a scale factor of 12, the 𝑥-coordinate of the image of point 𝐴 will be 12×180=90 and the 𝑦-coordinate of the image will remain unchanged.

The coordinates of the image of point 𝐴 are (90,1).

It is worth noting that we can check this answer by substituting 𝑥=90 into 𝑓(2𝑥): 𝑓(2𝑥)=𝑓(2×90)=𝑓(180).

We observe from the graph of 𝑓(𝑥) that 𝑓(180)=1. This is the 𝑦-coordinate of the image of 𝐴.

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 𝑦=(𝑥)+1cos?

Answer

Remember, the graph of the cosine function is as shown.

In order to identify the correct graph, we use the fact that 𝑓(𝑥) is mapped onto 𝑓(𝑥)+𝑑 by a translation 𝑑 units in the vertical direction, or by the vector (0,𝑑). This means that 𝑦=(𝑥)cos is mapped onto 𝑦=(𝑥)+1cos by a translation one unit upward. After this translation, the 𝑦-intercept will have coordinates (0,2), and the points of intersection of the curve with the 𝑥-axis will lie at 180+360𝑛, for integer values of 𝑛.

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 2(𝑥)+1cos and 2((𝑥)+1)cos. The graph of both functions are some transformation of the graph of 𝑦=𝑥cos. Figure 1 shows the graph of 𝑦=(𝑥)cos and 𝑦=2(𝑥)+1cos, where 𝑦=2(𝑥)+1cos is obtained by performing a vertical stretch by a scale factor of 2 and then a translation by (0,1). Figure 2 shows the graph of 𝑦=(𝑥)cos and 𝑦=2((𝑥)+1)cos, where the blue plot is obtained by a translation (0,1)  then a vertical stretch.

To avoid errors, we should follow the order given below.

How To: Sequencing Transformations of Functions

𝑓(𝑥) maps onto 𝑎𝑓(𝑏(𝑥+𝑐))+𝑑 in the following order:

  1. A vertical stretch by a scale factor 𝑎, where 𝑎<0, resulting in a reflection in the 𝑥-axis
  2. A horizontal stretch by a scale factor 1𝑏, where 𝑏<0, resulting in a reflection in the 𝑦-axis
  3. A horizontal translation given by (𝑐,0)
  4. A vertical translation given by (0,𝑑)

For instance, let’s identify the series of transformations that map 𝑦=(𝑥)sin onto 𝑦=(90𝑥)sin. We rewrite sin(90𝑥) as sin((𝑥90)) and use the sequencing of transformations. We see that sin(𝑥) undergoes two separate transformations to map it onto sin((𝑥90)): a horizontal stretch by a scale factor of 11=1 followed by a horizontal translation by (90,0).

The graph of 𝑦=(𝑥)sin is shown in figure 1. A horizontal stretch by a scale factor of 1 results in the graph shown in figure 2.

Finally, a horizontal translation by (90,0) gives the graph shown below.

Let’s 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 𝑓(𝑥). A transformation maps 𝑓(𝑥) to 4𝑓(3𝑥45)+1. Determine the coordinates of 𝐴 following this transformation.

Answer

Remember, 𝑓(𝑥) maps onto 𝑎𝑓(𝑏(𝑥+𝑐))+𝑑 in the following order:

  1. A vertical stretch by a scale factor 𝑎, where 𝑎<0, resulting in a reflection in the 𝑥-axis
  2. A horizontal stretch by a scale factor 1𝑏, where 𝑏<0, resulting in a reflection in the 𝑦-axis
  3. A horizontal translation given by (𝑐,0)
  4. A vertical translation given by (0,𝑑)

To identify the transformation that maps 𝑓(𝑥) to 4𝑓(3𝑥45)+1, we rewrite 4𝑓(3𝑥45)+1 as 4𝑓(3(𝑥15))+1 and let 𝑎=4, 𝑏=3, 𝑐=15, and 𝑑=1. Then, 𝑓(𝑥) undergoes the following:

  1. A vertical stretch by a scale factor of 4
  2. A horizontal stretch by a scale factor of 13
  3. A horizontal translation given by ((15),0)=(15,0)
  4. A vertical translation given by (0,1)

We can apply each step to point 𝐴 with coordinates (180,1):

  1. A vertical stretch by a scale factor of 4 maps (180,1) onto (180,1×4)=(180,4).
  2. A horizontal stretch by a scale factor of 13 maps (180,4) onto 180×13,4=(60,4).
  3. A horizontal translation given by (15,0) maps (60,4) onto (60+15,4)=(75,4).
  4. A vertical translation given by (0,1) maps (75,4) onto (75,4+1)=(75,3).

Hence, the coordinates of the image of 𝐴 are (75,3).

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 𝑦=𝑥41sin?

Answer

We recall that 𝑓(𝑥) maps onto 𝑎𝑓(𝑏(𝑥+𝑐))+𝑑 in the following order:

  1. A vertical stretch by a scale factor 𝑎, where 𝑎<0, resulting in a reflection in the 𝑥-axis
  2. A horizontal stretch by a scale factor 1𝑏, where 𝑏<0, resulting in a reflection in the 𝑦-axis
  3. A horizontal translation given by (𝑐,0)
  4. A vertical translation given by (0,𝑑)

If we define 𝑓(𝑥)=(𝑥)sin, then we can define the image after some series of transformations as 𝑔(𝑥)=𝑥41=14𝑥1sinsin.

Then, we can let 𝑏=14 and 𝑑=1, so the function 𝑓(𝑥) maps onto 𝑔(𝑥) by a horizontal stretch by a scale factor of 1=4 followed by a vertical translation given by (0,1).

The graph of 𝑦=𝑥sin is shown in figure 1, and a horizontal stretch of sin(𝑥) by a scale factor of 4 is shown in figure 2. We observe that point 𝐴 with coordinates (90,1) maps onto 𝐴 with coordinates (360,1).

Finally, this graph is translated one unit down as shown in figure 3. Point 𝐴 maps onto 𝐴 with coordinates (360,0).

This is option B.

Let us finish by recapping some key concepts from this explainer.

Key Points

  • For a function 𝑓(𝑥) and real constants 𝑎, 𝑏, 𝑐, and 𝑑,
    • 𝑎𝑓(𝑥) represents a vertical stretch by a scale factor 𝑎, where 𝑎<0 will result in a reflection in the 𝑥-axis;
    • 𝑓(𝑏𝑥) represents a horizontal stretch by a scale factor 1𝑏 for 𝑏0, where 𝑏<0 will result in a reflection in the 𝑦-axis;
    • 𝑓(𝑥+𝑐) represents a translation by (𝑐,0);
    • 𝑓(𝑥)+𝑑 represents a translation by (0,𝑑).
  • If we consider a series of transformations that maps 𝑓(𝑥) onto 𝑎𝑓(𝑏(𝑥+𝑐))+𝑑, they should be applied in the following order:
    • First, a vertical stretch by a scale factor 𝑎;
    • Second, a horizontal stretch by a scale factor 1𝑏;
    • Third, a horizontal translation given by (𝑐,0);
    • Finally, a vertical translation given by (0,𝑑).

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