Lesson Explainer: Combining Function Transformations Mathematics

In this explainer, we will learn how to identify function transformations that include a combination of translation, dilation, and reflection.

Let us begin by reviewing different types of function transformations. We recall that function transformations can be grouped as follows:

  • Horizontal: changes to the 𝑥-variable (e.g., 𝑥𝑥+3, 𝑥2𝑥),
  • Vertical: changes to the function (e.g., 𝑓(𝑥)𝑓(𝑥)3, 𝑓(𝑥)𝑓(𝑥)4).

The changes to the function 𝑓(𝑥) are applied directly to the expression 𝑓(𝑥), while the changes to the 𝑥-variable are applied to the variable 𝑥. For instance, if we begin with the expression 𝑥+4, the transformation 𝑓(𝑥)𝑓(𝑥)4 would result in the expression 14𝑥+4, which simplifies to 14𝑥1.

On the other hand, if we apply 𝑥𝑥4 to the same expression, this results in 𝑥4+4, which simplifies to 116𝑥+4.

Within each group of horizontal and vertical transformations, there are three different types of transformations:

  • Addition: for example, 𝑥𝑥3 is an addition of 3 and the 𝑥-variable;
  • Multiplication: for example, 𝑓(𝑥)𝑓(𝑥)7 is a multiplication of the function by the factor of 17;
  • Negation: there are 𝑥𝑥 and 𝑓(𝑥)𝑓(𝑥).

Each type of algebraic transformation listed above has geometric implications in the direction associated with the corresponding variable. The relationships between algebraic and geometric transformations are summarized in the following table.

AdditionShifts (horizontal and vertical)
MultiplicationDilations (horizontal and vertical)
NegationReflection in the axes

For horizontal transformations, the effects of addition and multiplication are the opposite of what we would expect. For example, the algebraic transformation 𝑥𝑥+3 results in the geometric transformation of shifting the graph of a function to the left by 3 units. Also, the multiplication 𝑥2𝑥 results in horizontal dilation by a factor of 12. On the other hand, algebraic transformations of the function result in the expected geometric transformations. For instance, 𝑓(𝑥)𝑓(𝑥)+3 results in a shift upward by 3 units, and 𝑓(𝑥)2𝑓(𝑥) results in a vertical dilation by a factor of 2. We can expand the table above to include all possible cases.

Definition: Function Transformations

Letting 𝑐>0 and 𝑑>1, we have the following.

Addition𝑥𝑥+𝑐Shifts to the left by 𝑐 units
𝑥𝑥𝑐Shifts to the right by 𝑐 units
𝑓(𝑥)𝑓(𝑥)+𝑐Shifts up by 𝑐 units
𝑓(𝑥)𝑓(𝑥)𝑐Shifts down by 𝑐 units
Multiplication𝑥𝑑𝑥Dilates horizontally with scale factor 1𝑑
𝑥𝑥𝑑Dilates horizontally with scale factor 𝑑
𝑓(𝑥)𝑑𝑓(𝑥)Dilates vertically with scale factor 𝑑
𝑓(𝑥)𝑓(𝑥)𝑑Dilates vertically with scale factor 1𝑑
Negation𝑥𝑥Reflects over the 𝑦-axis
𝑓(𝑥)𝑓(𝑥)Reflects over the 𝑥-axis

For transformations involving the function, the standard notation, for example, 𝑓(𝑥)𝑓(𝑥)+𝑐, may cause confusion if a problem already involves a function denoted 𝑓(𝑥). In these cases, we can denote functional transformations by a different letter: 𝑔(𝑥)𝑔(𝑥)+𝑐 or (𝑥)(𝑥)+𝑐.

In our first example, we will consider a combination of a reflection with horizontal and vertical shifts.

Example 1: Finding the Equation of a Graph after a Combination of Horizontal and Vertical Shifts

If the following graph is reflected in the 𝑥-axis and then shifted to the left by 1 unit and down by 3 units, what is the equation of the new graph?

Answer

We recall that the reflection in the 𝑥-axis corresponds to the algebraic transformation 𝑓(𝑥)𝑓(𝑥). Since the transformation in 𝑓(𝑥) is applied to the function directly, this changes the function |𝑥| to |𝑥|.

Next, we apply the shifts. Shifts are geometric transformations resulting from additions. An addition to the 𝑥-variable leads to a horizontal shift, and an addition to the function leads to a vertical shift.

From the resulting expression above, |𝑥|, this graph is shifted to the left by 1 unit and down by 3 units. We recall the following rules for transformation. For a positive constant 𝑐,

  • 𝑥𝑥+𝑐 results in a shift to the left by 𝑐 units,
  • 𝑓(𝑥)𝑓(𝑥)𝑐 results in a downward shift by 𝑐 units.

Since we want to shift the graph of our function to the left by 1 unit and downward by 3 units, we need both transformations 𝑥𝑥+1 and 𝑓(𝑥)𝑓(𝑥)3. Applying the transformation 𝑥𝑥+1 to 𝑦=|𝑥| leads to 𝑦=|𝑥+1|.

Next, we need to apply the transformation 𝑓(𝑥)𝑓(𝑥)3 to the expression above. This gives us the equation 𝑦=|𝑥+1|3.

Let us visualize this process using the graphs of these functions.

The figure above shows the sequence of this transformation. The round black arrow over the 𝑥-axis represents the reflection on the 𝑥-axis, which leads to 𝑦=|𝑥|. The green arrow is the shift to the left by 1 unit, which leads to the function 𝑦=|𝑥+1|. The purple arrow is the downward shift by 3 units, leading to the function 𝑦=|𝑥+1|3.

Hence, the equation of the new graph created by the given combination of transformations to 𝑦=|𝑥| is 𝑦=|𝑥+1|3.

In the next example, we will apply a combination of vertical dilation and a reflection to a given function.

Example 2: Graphing a Function Involving Combined Transformations of Graphs

The function 𝑓(𝑥)=(𝑥+2)+3 is stretched in the vertical direction by a scale factor of 2 then reflected in the 𝑦-axis. Write the equation of the transformed function 𝑔(𝑥).

Answer

We recall that a dilation in the vertical direction corresponds to a multiplication to the function, and a reflection over the 𝑦-axis comes from negating the 𝑥-variable. In our example, the vertical dilation by a scale factor of 2 comes from the algebraic transformation (𝑥)2(𝑥), and the reflection over the 𝑦-axis comes from the transformation 𝑥𝑥.

Let us first apply the transformation (𝑥)2(𝑥) to the given function 𝑓(𝑥). Hence, multiplying 𝑓(𝑥) by 2 leads to the expression 2(𝑥+2)+6.

Next, we apply the transformation 𝑥𝑥. We know that transformations in the 𝑥-variable are applied directly, so we obtain 2(𝑥+2)+6.

Let us visualize this process using the graphs of these functions.

The figure above shows the sequence of this transformation. The green arrow shows the vertical dilation with a scale factor 2, which leads to the function 2(𝑥+2)+6. The double-sided purple arrow represents the reflection over the 𝑦-axis, leading to the function 2(𝑥+2)+6.

Hence, the equation of the new transformed function 𝑔(𝑥) is 𝑦=2(𝑥+2)+6.

In the next example, we will obtain the combined expression of a function after four different transformations.

Example 3: Graphing a Function Involving Combined Transformations of Graphs

The graph of the function 𝑓(𝑥)=𝑥 is first reflected symmetrically over the 𝑦-axis, then shifted up by 2 units and right by 3 units, and finally horizontally stretched by 2 units to obtain the graph of the function 𝑔(𝑥). Write an equation for 𝑔(𝑥).

Answer

In this example, we need to apply four different geometric transformations: reflection over the 𝑦-axis, shift upward, shift to the right, and horizontal dilation with a scale factor of 2. We recall the algebraic transformations associated with these effects. If 𝑐>0 and 𝑑>1,

  • 𝑥𝑥 results in a reflection over the 𝑦-axis,
  • (𝑥)(𝑥)+2 results in a shift upward by 2 units,
  • 𝑥𝑥3 results in a shift to the right by 3 units,
  • 𝑥𝑥2 results in a horizontal dilation with a scale factor of 2.

Let us apply these transformations to 𝑓(𝑥) in the given order. Since we start by reflecting symmetrically over the 𝑦-axis, we first apply the algebraic transformation 𝑥𝑥 to our function 𝑓(𝑥)=𝑥. We know that the transformation in the 𝑥-variable is applied directly, so we can obtain this transformation by substituting 𝑥 into the expression 𝑥. This gives us the following. 𝑥.

The second transformation is shifting upward by 2 units, which corresponds to (𝑥)(𝑥)+2. We add 2 to the expression above to obtain the following. 𝑥+2.

Next, we need to shift the graph to the right by 3 units, which requires 𝑥𝑥3. Applying this transformation directly to the 𝑥-variable in the expression above, we obtain (𝑥3)+2, which simplifies to 𝑥+3+2.

Finally, we stretch the graph horizontally by 2, which requires 𝑥𝑥2. Applying this transformation directly to the 𝑥-variable in the expression above, we obtain the following. 𝑥2+3+2.

Hence, the equation for the resulting function 𝑔(𝑥) is 𝑔(𝑥)=𝑥2+3+2.

In the previous examples, we found algebraic expressions of functions after a combination of transformations were applied in a specified order. We now consider examples in the reverse direction. Given an algebraic expression of a function that is a result of a series of transformations, we can retrace the steps to identify all the transformations used to obtain the given expression.

Once we obtain the list of necessary transformations, we need to find their correct order. When we apply a collection of transformations in different orders, we may end up with different functions and graphs. For example, consider applying the following two transformations to the function 𝑓(𝑥)=|𝑥|:

  • shift to the left by 1 unit, given by 𝑥𝑥+1;
  • reflection over the 𝑦-axis, given by 𝑥𝑥.

First, let us apply the shift and then the reflection. The shift to the left by 1 unit is given by 𝑥𝑥+1, so this transformation leads to the graph 𝑦=|𝑥+1|. The reflection over the 𝑦-axis is given by 𝑥𝑥, so this leads to 𝑦=|𝑥+1|.

On the other hand, if we apply the reflection first, applying 𝑥𝑥 to |𝑥| results in |𝑥|, which is the same as |𝑥|. The reflection over the 𝑦-axis does not change the graph in this case, since it is symmetric about the 𝑦-axis. Then, applying the shift to the left by 1 unit, given by 𝑥𝑥+1, leads to the graph 𝑦=|𝑥+1|. This is different from 𝑦=|𝑥+1|, which we obtained by applying the same two transformations in a different order.

We can visualize this difference using the diagram.

Hence, after identifying a list of transformations used to obtain an expression, we must also identify an appropriate order of transformations. Let us consider an example in this direction.

Example 4: Combined Translations and Stretches of Graphs

This is the graph of the exponential function 𝑦=𝑓(𝑥).

Which of the following is the graph of 𝑦=4𝑓(2𝑥)?

Answer

To identify the graph given by 𝑦=4𝑓(2𝑥), we need to retrace the combination and the order of transformations needed to obtain the expression 4𝑓(2𝑥) from 𝑓(𝑥). Let us begin by observing key features from the resulting expression 4𝑓(2𝑥).

  • There is the number 4 outside the function. We can also write the given expression as 𝑓(2𝑥)+4, where we can observe the transformation 𝑔(𝑥)𝑔(𝑥)+4. This transformation results in the shift upward by 4 units.
  • There is a negative sign in front of the function 𝑓, indicating the transformation 𝑔(𝑥)𝑔(𝑥). This transformation results in a reflection over the 𝑥-axis.
  • 𝑥 in 𝑓(𝑥) has been replaced by 2𝑥. This indicates the transformation 𝑥2𝑥 corresponding to the horizontal dilation with a scale factor 12.

Hence, the algebraic transformations that were used to obtain 4𝑓(2𝑥) from 𝑓(𝑥) are 𝑔(𝑥)𝑔(𝑥)+4,𝑔(𝑥)𝑔(𝑥),𝑥2𝑥.

Let us apply these transformations in the stated order to see if we end with the given expression. Applying 𝑔(𝑥)𝑔(𝑥)+4 to the function 𝑓(𝑥) leads to 𝑓(𝑥)+4.

Applying 𝑔(𝑥)𝑔(𝑥) to this expression gives us (𝑓(𝑥)+4), which leads to 𝑓(𝑥)4.

Finally, applying 𝑥2𝑥 gives us 𝑓(2𝑥)4.

This order of transformations results in a different expression from 4𝑓(2𝑥), which tells us that this is a wrong order. But, from this sequence, we can observe that applying 𝑔(𝑥)𝑔(𝑥) before 𝑔(𝑥)𝑔(𝑥)+4 resulted in changing +4 outside the function to 4, leading to a different expression than the given one. Hence, we should apply 𝑔(𝑥)𝑔(𝑥) first and then apply 𝑔(𝑥)𝑔(𝑥)+4 to avoid changing the sign of the number 4.

Let us try the order given by 𝑔(𝑥)𝑔(𝑥),𝑔(𝑥)𝑔(𝑥)+4,𝑥2𝑥.

Applying 𝑔(𝑥)𝑔(𝑥) to 𝑓(𝑥) leads to 𝑓(𝑥).

Applying 𝑔(𝑥)𝑔(𝑥)+4 to this expression gives us 𝑓(𝑥)+4.

Finally, applying 𝑥2𝑥 gives us 𝑓(2𝑥)+4, which is the same as the given expression, 4𝑓(2𝑥). Hence, this is the correct order of transformations. Let us interpret this order of transformations geometrically. Since we applied 𝑔(𝑥)𝑔(𝑥) first, we begin by reflecting the graph of 𝑓(𝑥) over the 𝑥-axis.

The second transformation is 𝑔(𝑥)𝑔(𝑥)+4, so we shift the reflected graph up by 4.

The last transformation is 𝑥2𝑥, so we contract the graph horizontally with a scale factor 12.

Hence, the graph of 𝑦=4𝑓(2𝑥) is (b).

We can also reverse a given transformation. For instance, the shift to the right by 2 units, given by 𝑥𝑥2, is reversed by the shift to the left by 2 units, which is given by 𝑥𝑥+2. The reflection over the 𝑥-axis, given by 𝑓(𝑥)𝑓(𝑥), is reversed by the same transformation, since two reflections over the same axis bring back the original graph. The vertical dilation with a scale factor 5, given by 𝑓(𝑥)5𝑓(𝑥), is reversed by the vertical dilation with a scale factor 15, which is given by 𝑓(𝑥)15𝑓(𝑥).

To reverse a combination of transformations, we need to reverse each transformation starting from the last one. Let us consider an example where we identify the original function by reversing given order of transformations.

Example 5: Identifying the Original Function from a Transformed Graph

The graph represents a function 𝑔(𝑥) after a vertical positive shift by 3 units followed by a reflection on the 𝑥-axis. Which of the following represents the original function 𝑓(𝑥)?

  1. 𝑓(𝑥)=(𝑥+2)2
  2. 𝑓(𝑥)=3(𝑥1)6
  3. 𝑓(𝑥)=(𝑥1)+2
  4. 𝑓(𝑥)=(𝑥4)2
  5. 𝑓(𝑥)=(3𝑥1)+2

Answer

Since 𝑔(𝑥) comes from 𝑓(𝑥) after applying two given transformations, we can recover the original function 𝑓(𝑥) from 𝑔(𝑥) by reversing the transformations, beginning with the second one. This process will give 𝑓(𝑥) as an expression involving 𝑔(𝑥). We can finish the problem by finding the equation for 𝑔(𝑥).

The second transformation is the reflection on the 𝑥-axis, which can be reversed by the same transformation. The reflection on the 𝑥-axis comes from the algebraic transformation (𝑥)(𝑥). Applying this transformation to 𝑔(𝑥), we obtain 𝑔(𝑥).

Next, we reverse the first transformation, which is a vertical positive shift by 3 units. This transformation is reversed by a vertical negative (or downward) shift by 3 units, which corresponds algebraically to (𝑥)(𝑥)3. Applying this transformation to the expression above, we obtain 𝑔(𝑥)3.

Since we have reversed both transformations, this must be equal to our function. Thus, 𝑓(𝑥)=𝑔(𝑥)3.

Next, we need to find the equation for 𝑔(𝑥). From the given graph, we can see that 𝑔(𝑥) comes from the parent function 𝑦=𝑥. We see from the diagram below that a reflection over the 𝑦-axis makes the two graphs more similar.

Reflection over the 𝑦-axis comes from the transformation (𝑥)(𝑥), so this would lead to the equation 𝑦=𝑥 as indicated in the diagram. From the graph of 𝑦=𝑥, we can obtain the graph of 𝑔(𝑥) by shifting to the right by 1 unit, and then shifting down by 5 units.

The transformation 𝑥𝑥1 results in the shift to the right by 1 unit, and the transformation (𝑥)(𝑥)5 results in the shift down by 5 units. Applying these transformations in order, we obtain 𝑥𝑥1𝑦=(𝑥1),(𝑥)(𝑥)5𝑦=(𝑥1)5.leadstoleadsto

Hence, we obtain 𝑔(𝑥)=(𝑥1)5.

Finally, we can substitute this equation into the relation between 𝑓(𝑥) and 𝑔(𝑥) we obtained earlier: 𝑓(𝑥)=𝑔(𝑥)3=(𝑥1)53=(𝑥1)+53=(𝑥1)+2.

This leads to option C.

Sometimes, different combinations and orders of transformations may result in the same effect on the graph of a function. In our final example, we will observe this effect.

Example 6: Identifying Equivalent Combinations of Function Transformations

The red graph in the figure has equation 𝑦=𝑓(𝑥) and the green graph has equation 𝑦=𝑔(𝑥). Which of the following will not transform the graph of 𝑓(𝑥) onto the graph of 𝑔(𝑥)?

  1. A reflection in the 𝑥-axis.
  2. A horizontal shift of 4 units to the right followed by a reflection in the 𝑦-axis.
  3. A reflection in the 𝑦-axis followed by a horizontal shift of 4 units to the left.
  4. A vertical shift of 4 units down followed by a reflection in the 𝑥-axis.

Answer

We will apply each transformation to the red graph to see whether the resulting image overlaps with the green line. Since a line is fully characterized by two distinct points, it will be enough if the transformations move two distinct points on the red line onto the green line.

  1. We reflect the red line over the 𝑥-axis.
    In the diagram above, the round arrow around the 𝑥-axis represents the reflection over the 𝑥-axis. We can see that the reflection moves two distinct points on the red line onto the green line. Hence, this transformation brings the graph of 𝑓(𝑥) onto the graph of 𝑔(𝑥).
  2. We shift 4 units to the right and then reflect over the 𝑦-axis.
    In the diagram above, the straight arrow represents shifts, and the round arrow around the 𝑦-axis represents the reflection over the 𝑦-axis. We can see that the two transformations move two distinct points on the red line onto the green line. Hence, this option also brings the graph of 𝑓(𝑥) onto the graph of 𝑔(𝑥).
  3. We reflect over the 𝑦-axis and then shift to the left by 4 units.
    We can see that the two transformations move two distinct points on the red line onto the green line. Hence, this option also brings the graph of 𝑓(𝑥) onto the graph of 𝑔(𝑥).
  4. We shift down by 4 units and then reflect in the 𝑥-axis.
    Neither of the two selected points from the red line ends up on the green line. Hence, two transformations do not bring the graph of 𝑓(𝑥) onto the graph of 𝑔(𝑥).

This leads to option D.

Let us finish by recapping a few important concepts.

Key Points

  • Letting 𝑐>0 and 𝑑>1, we have the following.
    Addition𝑥𝑥+𝑐Shifts to the left by 𝑐 units
    𝑥𝑥𝑐Shifts to the right by 𝑐 units
    𝑓(𝑥)𝑓(𝑥)+𝑐Shifts up by 𝑐 units
    𝑓(𝑥)𝑓(𝑥)𝑐Shifts down by 𝑐 units
    Multiplication𝑥𝑑𝑥Dilates horizontally with scale factor 1𝑑
    𝑥𝑥𝑑Dilates horizontally with scale factor 𝑑
    𝑓(𝑥)𝑑𝑓(𝑥)Dilates vertically with scale factor 𝑑
    𝑓(𝑥)𝑓(𝑥)𝑑Dilates vertically with scale factor 1𝑑
    Negation𝑥𝑥Reflects over the 𝑦-axis
    𝑓(𝑥)𝑓(𝑥)Reflects over the 𝑥-axis
    𝑓(𝑥) in the notations for vertical transformations may be replaced by 𝑔(𝑥) or (𝑥) if the given problem already specifies the function 𝑓(𝑥).
  • A different order of transformations may result in a different outcome. When identifying the combination of transformations done to a function, we need to identify the order of the transformations.
  • A shift can be reversed by a shift to the opposite direction. A reflection can be reversed by the same reflection. A dilation can be reversed by a dilation of the reciprocal scale factor. When reversing a combination of transformations, we begin by reversing the last transformation.
  • Different combinations of transformations may result in an identical effect on a function.

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