Lesson Explainer: Function Transformations: Translations | Nagwa Lesson Explainer: Function Transformations: Translations | Nagwa

Lesson Explainer: Function Transformations: Translations Mathematics • Second Year of Secondary School

In this explainer, we will learn how to identify function transformations involving horizontal and vertical shifts.

A translation in geometry is a rigid motion of a plane where we shift each point on the plane in a given direction and distance. More specifically, a translation can be algebraically represented by the transformation where each point with coordinates (𝑥,𝑦) is moved to a new point with coordinates (𝑥+𝑎,𝑦+𝑏) for some fixed constants 𝑎 and 𝑏. We can apply the same geometric concept to the graph of a function to define a translation of a function.

In the figure above, we can see that the point (𝑥,𝑦) on the solid green graph is translated to a new point (𝑥+𝑎,𝑦+𝑏). When every point on the solid green graph is translated to a corresponding new point using the same rule, we can obtain the dashed graph. The dashed graph is a translation of the solid green graph. We can see that the shape of the graph is not affected by this translation, which is a property of a rigid motion. While the two graphs have the same shape, they are not identical since the coordinates on the two graphs are not identical. This means that the dashed graph represents a different function than the one represented by the solid green graph. In other words, the geometric translation is associated with a change in the algebraic expression of the function, known as a function transformation.

In this explainer, we want to understand what type of function transformation results in a translation in the graph of the function. To examine the translation of a graph algebraically, it is beneficial to restrict translations to be strictly horizontal or vertical. Any translation on the plane can be stated as a combination of a horizontal and a vertical translation. For the translation depicted in the diagram above, we can realize this rigid motion as two rigid motions described by the horizontal shift 𝑥𝑥+𝑎 and the vertical shift 𝑦𝑦+𝑏.

In the graph of a function, the horizontal direction is related to the input variable, which is often represented by the variable 𝑥, and the vertical direction is related to the function 𝑓(𝑥) directly. Hence, by restricting a translation to be horizontal or vertical, we are restricting the function transformation to be purely in the 𝑥-variable or the function 𝑓(𝑥).

Let us begin by considering function transformations for horizontal shifts.

Definition: Function Transformations for Horizontal Shifts

For any 𝑐>0,

  • the function transformation 𝑥𝑥𝑐 represents a shift to the right by 𝑐 units,
  • the function transformation 𝑥𝑥+𝑐 represents a shift to the left by 𝑐 units.

The notation 𝑥𝑥𝑐, for instance, means that we replace the variable 𝑥 in the function with the expression 𝑥𝑐. Translating the graph of the function 𝑓(𝑥) to the right by 𝑐 units would lead to the graph of the function 𝑓(𝑥𝑐). It is easy to confuse the direction of a horizontal shift since the positive direction of the 𝑥-variable is to the right, but the associated function transformation requires subtracting by 𝑐, rather than adding. We should always keep in mind that a function transformation in the 𝑥-variable has the opposite of the intuitive effect geometrically. This is also true for transformations associated with dilation, which is not covered in this explainer.

To understand why this function transformation results in a horizontal translation in the graph, we consider the following input–output table of the function 𝑓(𝑥).

𝑥21012
𝑓(𝑥)𝑓(2)𝑓(1)𝑓(0)𝑓(1)𝑓(2)
𝑓(𝑥3)𝑓(5)𝑓(4)𝑓(3)𝑓(2)𝑓(1)
𝑓(𝑥+2)𝑓(0)𝑓(1)𝑓(2)𝑓(3)𝑓(4)

We can note that the row of outputs for 𝑓(𝑥3) can be obtained by shifting the row of outputs for 𝑓(𝑥) to the right by 3 units. This means that the graph of 𝑓(𝑥3) is obtained by shifting the graph of 𝑓(𝑥) to the right by 3. Similarly, the row of outputs for 𝑓(𝑥+2) is obtained from the row for 𝑓(𝑥) to the left by 2 units. Hence, the graph of 𝑓(𝑥+2) results from a horizontal shift to the left by 2 units.

In our first example, we will determine the algebraic expression of a function resulting from a horizontal translation.

Example 1: Identifying and Expressing Translations That Are Expressed Graphically

The red graph in the figure has the equation 𝑦=𝑓(𝑥) and the blue graph has the equation 𝑦=𝑔(𝑥). Express 𝑔(𝑥) as a transformation of 𝑓(𝑥).

Answer

In the given figure, we can see that the red and blue graphs have the same shape, which means that the blue graph can be obtained from the red graph by a rigid motion of the plane. More specifically, the blue graph is a translation of the red graph.

To determine the direction and the distance of this translation, we can examine the change in coordinates of corresponding points on both graphs. We can see that a local minimum of the red graph with coordinates (1,0) is translated to the corresponding point on the blue graph with coordinates (2,0).

This tells us that the rigid motion from the red graph to the blue graph is a translation to the right by 3 units. We recall that, for any 𝑐>0, the function transformation 𝑥𝑥𝑐 represents a shift to the right by 𝑐 units. This means that the function transformation 𝑥𝑥3 gives us the translation from the red graph to the blue graph. Applying this function transformation to 𝑓(𝑥), we get the function 𝑓(𝑥3).

Hence, 𝑔(𝑥)=𝑓(𝑥3).

In the previous example, we determined the algebraic expression of a function resulting from a horizontal translation. Using the same method, we can also identify a graph of a function after a function transformation, as we will see in the next example.

Example 2: Identifying Transformations of Graphs

The figure shows the graph of 𝑦=𝑓(𝑥).

Which of the following is the graph of 𝑦=𝑓(𝑥+1)?

Answer

In this example, we need to determine the graph of 𝑓(𝑥+1) based on the given graph of 𝑓(𝑥). The change in algebraic expressions from 𝑓(𝑥) to 𝑓(𝑥+1) can be represented by the function transformation 𝑥𝑥+1.

We recall that, for any 𝑐>0, the function transformation 𝑥𝑥+𝑐 represents a shift to the left by 𝑐 units. This means that 𝑥𝑥+1 represents a shift to the left by 1 unit. Hence, we can find the graph of 𝑦=𝑓(𝑥+1) by translating the given graph to the left by 1 unit. In particular, we can use the coordinates of a local maximum (1,0) in the original graph. Translating this point to the left by 1 unit leads to the coordinates (2,0), which gives us a reference point for the new graph.

Therefore, the answer is option C.

In previous examples, we considered function transformations associated with horizontal shifts. Let us now consider function transformations for vertical shifts. We remember that a vertical shift is associated with a function transformation in the function 𝑓(𝑥) directly rather than the 𝑥 variable.

Definition: Function Transformations for Vertical Shifts

For any 𝑐>0,

  • the function transformation 𝑓(𝑥)𝑓(𝑥)+𝑐 represents a shift up by 𝑐 units,
  • the function transformation 𝑓(𝑥)𝑓(𝑥)𝑐 represents a shift down by 𝑐 units.

The notation 𝑓(𝑥)𝑓(𝑥)+𝑐 means that we add 𝑐 directly to the given expression for the function. Hence, translating the graph of a function 𝑓(𝑥) upward by 𝑐 units leads to the graph of the function 𝑓(𝑥)+𝑐. Unlike the transformation in the 𝑥-variable, the geometric effect of function transformation in the function 𝑓(𝑥) is what we would expect. The upward direction is the positive direction in the function 𝑓(𝑥), and an upward translation of the graph is associated with the function transformation where we add 𝑐 to the function 𝑓(𝑥).

We can understand this function transformation directly since adding to or subtracting 𝑐 from the function 𝑓(𝑥) changes the 𝑦-coordinate of each point on the graph by the added or subtracted amount. For instance, the 𝑦-coordinate of the graph of the function 𝑓(𝑥)+2 would be precisely 2 more than the corresponding point on the graph of the function 𝑓(𝑥). Hence, the graph of 𝑓(𝑥)+2 can be obtained by shifting each point on the graph of 𝑓(𝑥) upward by 2 units.

In the next example, we will identify a corresponding point in the graph of a function resulting from this function transformation.

Example 3: Identifying the Coordinates of Points Following a Transformation

The figure shows the graph of 𝑦=𝑓(𝑥) and the point 𝐴. The point 𝐴 is a local maximum. Identify the corresponding local maximum for the transformation 𝑦=𝑓(𝑥)2.

Answer

In this example, we want to identify the coordinates of the corresponding local maximum for the function 𝑦=𝑓(𝑥)2. We recall that, for any 𝑐>0, the function transformation 𝑓(𝑥)𝑓(𝑥)𝑐 represents a shift down by 𝑐 units. The function 𝑓(𝑥)2 can be obtained from 𝑓(𝑥) by the function transformation 𝑓(𝑥)𝑓(𝑥)2, which means that the graph of 𝑦=𝑓(𝑥)2 is obtained by translating the graph of 𝑓(𝑥) down by 2 units.

We know that a translation is a rigid motion that does not change the shape of a graph. Hence, the corresponding point for the local maximum is obtained also by shifting the given point (2,1) down by 2 units. This means that the 𝑥-coordinate of this point will remain the same, while the 𝑦-coordinate will change to 12=1.

Hence, the corresponding local maximum is (2,1).

So far, we have considered function transformations associated with horizontal and vertical shifts. When we combine horizontal and vertical shifts, we can produce a translation of a graph in any direction and distance. Earlier in the explainer, we mentioned that the translation in the plane (𝑥,𝑦)(𝑥+𝑎,𝑦+𝑏) can be split into the horizontal shift 𝑥𝑥+𝑎 and the vertical shift 𝑦𝑦+𝑏. The order in which we apply the vertical and horizontal shifts does not matter, since either order will lead to the same result in the end.

In the next example, we will find the coordinates of a corresponding point on the graph of a function obtained by translating the graph of a given function both horizontally and vertically.

Example 4: Identifying the Coordinates of Points Following a Transformation

The figure shows the graph of 𝑦=𝑓(𝑥) and the point 𝐴. The point 𝐴 is a local maximum. Identify the corresponding local maximum for the transformation 𝑦=𝑓(𝑥1)+4.

Answer

In this example, we want to identify the coordinates of the corresponding local maximum for the function 𝑦=𝑓(𝑥1)+4. We can obtain the function 𝑓(𝑥1)+4 by first replacing 𝑥 by 𝑥1 and then adding 4 to the resulting expression. This means that we need to apply two function transformations, 𝑥𝑥1 and 𝑓(𝑥)𝑓(𝑥)+4. We recall that, for any 𝑐>0,

  • 𝑥𝑥𝑐 represents a shift to the right by 𝑐 units,
  • 𝑓(𝑥)𝑓(𝑥)+𝑐 represents a shift up by 𝑐 units.

Hence, for our example, 𝑥𝑥1 represents a shift to the right by 1 unit and 𝑓(𝑥)𝑓(𝑥)+4 represents a shift up by 4 units. We remember that the order in which we apply vertical and horizontal translations does not matter since either order will lead to the same result. We can obtain the graph of 𝑦=𝑓(𝑥1)+4 by first shifting the graph of 𝑓(𝑥) to the right by 1 unit and then up by 4 units.

We know that a translation is a rigid motion that does not change the shape of a graph. Hence, the corresponding point for the local maximum is obtained also by shifting the given point (2,1) to the right by 1 unit and then up by 4 units. This means that the 𝑥-coordinate of this point is 2+1=3, while the 𝑦-coordinate is 1+4=5.

Hence, the corresponding local maximum is (3,5).

If we are given a function along with a graph of the translation of that function, we can find the expression of the translated function by applying the rules for function transformations to the original function. In the next example, we will compute the algebraic expression for the function whose graph is given as a translation of a cubic function.

Example 5: Finding the Equation of a Function Following a Translation That Is Represented Graphically

Graph A shows the curve 𝑦=𝑥+2𝑥, which has a point of inflection at the origin. Determine the equation of graph B, given that it is a translation of graph A.

Answer

We are given that graph B is a translation of graph A. To determine the direction and the distance of this translation, we can examine the change in coordinates of corresponding points on both graphs. We are given that a point of inflection of graph A is at the origin. In graph B, we see that this point of inflection is located at (1,4). This means that the point at (0,0) is translated to the point (1,4).

Recall that function transformations are associated with horizontal and vertical translations of the graph. Hence, we need to find the horizontal and vertical translations that collectively shift a point from coordinates (0,0) to (1,4). This can be achieved by first translating to the right by 1 unit and then translating down by 4 units.

We recall the function transformations associated with such transformations: for any 𝑐>0,

  • 𝑥𝑥𝑐 represents a shift to the right by 𝑐 units,
  • 𝑓(𝑥)𝑓(𝑥)𝑐 represents a shift down by 𝑐 units.

Since we need a shift to the right by 1 unit and a shift down by 4 units, we need to apply the transformations 𝑥𝑥1 and 𝑓(𝑥)𝑓(𝑥)4 to the function whose graph is given in A. We remember that the order of horizontal and vertical translations does not matter, since either order will lead to the same result.

Let us first apply the transformation 𝑥𝑥1 to the function 𝑥+2𝑥. Applying this function transformation means that we are replacing each 𝑥-variable in the function by the expression 𝑥1. This gives us (𝑥1)+2(𝑥1).

We recall that we can expand a cube of a binomial expression by (𝑎𝑏)=𝑎3𝑎𝑏+3𝑎𝑏𝑏. Applying this formula and simplifying, we have (𝑥1)+2(𝑥1)=𝑥3𝑥+3𝑥1+2𝑥2=𝑥3𝑥+5𝑥3.

Next, let us apply the transformation 𝑓(𝑥)𝑓(𝑥)4 to this function. Applying this function transformation means that we need to subtract 4 from the given function. This leads to 𝑥3𝑥+5𝑥34=𝑥3𝑥+5𝑥7.

Hence, the equation of graph B is 𝑦=𝑥3𝑥+5𝑥7.

In our final example, we will find the value of an unknown shift based on given algebraic expressions for both the original and the resulting functions.

Example 6: Finding the Value of an Unknown Using the Equation of the Function Before and After a Translation

The function 𝑓(𝑥)=3𝑥+9 is translated +2 units in the 𝑦-direction and 𝑐 units in the 𝑥-direction to form the function 𝑔(𝑥)=3𝑥+2. Find the value of 𝑐.

Answer

In this example, we need to find the value of an unknown horizontal (𝑥-direction) shift based on the given vertical shift (𝑦-direction) and the algebraic expressions for both the original and resulting functions. We begin by recalling the function transformations associated with these translations: for any 𝑐>0,

  • 𝑓(𝑥)𝑓(𝑥)+𝑐 represents a shift up by 𝑐 units,
  • 𝑥𝑥𝑐 represents a shift to the right by 𝑐 units.

This tells us that we can shift 𝑓(𝑥) up by 2 units by applying the function transformation 𝑓(𝑥)𝑓(𝑥)+2. Applying this function transformation means that we need to add 2 to the function 𝑓(𝑥). This leads to (3𝑥+9)+2=3𝑥+11.

Next, we need to shift this function 𝑐 units in the 𝑥-direction, which means that we need to apply the transformation 𝑥𝑥𝑐. Applying this transformation means that we need to replace the 𝑥-varaible in the function with the expression 𝑥𝑐. This leads to 3(𝑥𝑐)+11=3𝑥3𝑐+11.

After both function transfomations, the graph of this function is the same as the graph of 𝑔(𝑥). Hence, we can equate the two expressions to write 3𝑥3𝑐+11=3𝑥+2.

The common term 3𝑥 on both sides of the equation cancels out, leading to 3𝑐+11=2.

Rearranging this equation so that 𝑐 is the subject, we find 𝑐=3.

Let us finish by recapping a few important concepts from this explainer.

Key Points

  • Horizontal and vertical shifts in the graph of a function are associated with the function transformations in the 𝑥-variable or function 𝑓(𝑥) respectively.
  • For any 𝑐>0,
    • the function transformation 𝑥𝑥𝑐 represents a shift to the right by 𝑐 units,
    • the function transformation 𝑥𝑥+𝑐 represents a shift to the left by 𝑐 units,
    • the function transformation 𝑓(𝑥)𝑓(𝑥)+𝑐 represents a shift up by 𝑐 units,
    • the function transformation 𝑓(𝑥)𝑓(𝑥)𝑐 represents a shift down by 𝑐 units.
  • The order of horizontal or vertical translations does not matter since either order will result in the same graph.

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