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2.9 G) Multiple Transformations
2.9 G) Multiple Transformations
Sometimes in the exam you will be given a function and asked to perform multiple transformations on them. These transformations shouldn’t be too tricky, but they can sometimes be confusing.
When you are working with multiple transformations, it is best to split them up into different transformations and perform each one separately. You should start with the transformations on the inside of the bracket for the function, then the sign of the function and then finally any numbers that are added or subtracted to the function (outside of the brackets of the function).
When you are working with multiple transformations, it is best to split them up into different transformations and perform each one separately. You should start with the transformations on the inside of the bracket for the function, then the sign of the function and then finally any numbers that are added or subtracted to the function (outside of the brackets of the function).
Example 1
For example, suppose that we had the function f(x), which is shown below.
For example, suppose that we had the function f(x), which is shown below.
The minimum for the function f(x) is (1, -4).
Sketch the function below clearing showing the coordinates of the new minimum.
Sketch the function below clearing showing the coordinates of the new minimum.
There are two transformations in the function above. It is always best to start with the function bracket. Therefore, the first transformation that I will undertake is the f(x + 3). This is going to be a translation parallel to the x axis (horizontal). As the number inside the bracket is being added, it results in the curve moving 3 to the left. A sketch of this is shown below.
This results in the new minimum of the function being (-2, -4).
We now move onto the next translation, which is the + 1 at the end. This is a translation that is parallel to the y axis (vertical). As this is positive, it results in all of the points on the curve moving up by 1.
We now move onto the next translation, which is the + 1 at the end. This is a translation that is parallel to the y axis (vertical). As this is positive, it results in all of the points on the curve moving up by 1.
The new minimum of the curve is (-2, -3).
For the above example, it would not have matter if we would have performed the translation parallel to the x or y axis first. However, I would suggest always working with the inside of the bracket for the function, then the sign of the function (see example 2) and then finally any numbers that are added or subtracted to the function.
For the above example, it would not have matter if we would have performed the translation parallel to the x or y axis first. However, I would suggest always working with the inside of the bracket for the function, then the sign of the function (see example 2) and then finally any numbers that are added or subtracted to the function.
Example 2
The graph below shows a sketch of the function g(x).
The graph below shows a sketch of the function g(x).
The maximum of the function is (5, 4).
Sketch the function below clearing showing the coordinates of the new turning point.
Sketch the function below clearing showing the coordinates of the new turning point.
I am going to answer this question by splitting the overall transformation into 3 different transformations. The first transformation that I am going to look at is the inside of the bracket; g(x – 3). A number added or subtracted inside the bracket results in a translation that is parallel to the x axis; the curve will move horizontally. A negative number inside the bracket results in the curve moving towards the right. Therefore, this translation results in the curve moving 3 to the right.
The coordinates of the turning point are (8, 4).
We now move on to the next transformations, which will be the negative on the outside of the function. A negative outside the function results in a reflection in the x axis. This results in the signs for the y coordinates all changing (which can be viewed as multiplying all of the y coordinates by -1). The x coordinates for all of the points on the curve remain the same. Our function looks like what is shown below.
We now move on to the next transformations, which will be the negative on the outside of the function. A negative outside the function results in a reflection in the x axis. This results in the signs for the y coordinates all changing (which can be viewed as multiplying all of the y coordinates by -1). The x coordinates for all of the points on the curve remain the same. Our function looks like what is shown below.
The coordinates of the turning point are now (8, -4).
We are now onto the final transformation, which is the taking of 2 at the end of the function. This results in a translation that is parallel to the y axis. As the value is negative (-2), the curve moves down by 2. We take 2 off of all of the y coordinates and the x coordinates remain the same.
We are now onto the final transformation, which is the taking of 2 at the end of the function. This results in a translation that is parallel to the y axis. As the value is negative (-2), the curve moves down by 2. We take 2 off of all of the y coordinates and the x coordinates remain the same.
Therefore, the new coordinates for the turning point are (8, -6).
The order of the transformations for this question would have made a difference for the answer. Therefore, you should always do the inside of the bracket first, then the sign of the function and then finally any numbers that are added or subtracted to the function (outside the bracket of the function).
The order of the transformations for this question would have made a difference for the answer. Therefore, you should always do the inside of the bracket first, then the sign of the function and then finally any numbers that are added or subtracted to the function (outside the bracket of the function).
End Note
A good way to check your understanding of all of these transformations is to use a graph plotting software such as Desmos (click here to be taken to the Desmos website). You can create a function and make prediction about what will happen when you translate them (e.g. f(x) + 5, or f(-x), or -f(x) etc). You can use the software to check whether your predictions are correct.
A good way to check your understanding of all of these transformations is to use a graph plotting software such as Desmos (click here to be taken to the Desmos website). You can create a function and make prediction about what will happen when you translate them (e.g. f(x) + 5, or f(-x), or -f(x) etc). You can use the software to check whether your predictions are correct.