Function Transformations: Translation

The red curve above is a “transformation” of the green one. It has been “translated” (or shifted) four units to the right. A translation is a change in position resulting from addition or subtraction, one that does not rotate or change the size or shape in any way.

Transformations are often easiest to analyze by focusing on how the location of specific  points on the curve have changed. In the image above, the point A_1 on the green curve “corresponds” to point A_2 on the red curve. By this we mean that the transformation has moved point A_1 to A_2.

Horizontal Translations

In looking at the coordinates of the two corresponding points identified in the graph above, you can see that the y-coordinate has not changed and the x-coordinate has increased by 4. If you were to examine any other pair of corresponding points, you would see the exact same difference. This can be described algebraically by the equation:

\begin{cases}x_1+4~=~x_2~~~~~(1)\\*y_1~=~y_2\end{cases}

where x_1 is the starting x-coordinate from a point on the green curve, and x_2 is the translated x-coordinate from the corresponding point on the red curve. The y coordinates do not change, and thus are set equal to each other. Note that every translation equation will involve the addition or subtraction of a quantity to the original coordinate.

The green curve above is the graph of the equation:

y_1=(x_1)^2~~~~~~~(2)

To find the equation of the translated red curve, merge the effects of the green curve equation (2) with the translation equation (1). Since equation (2) is based on the variable x_1, we’ll need to start by solving equation (1) for x_1:

x_1+4~=~x_2~~~~~(1)\\*~\\*x_1~=~x_2-4

and our system of transformation equations, now ready for substitution, becomes

\begin{cases}x_1~=~x_2-4\\*y_1~=~y_2\end{cases}

Substituting this system into (2) to produces an equation for y_2:

y_1=(x_1)^2~~~~~~(2)~~Green\\*~\\*y_2=(x_2-4)^2~~~~~~~~Red

Notice that the process of solving for x_1 before substituting causes the horizontal translation amount to move to the other side of the equation, changing sign in the process. A translation to the right, by positive four, becomes (x_2-4) in the transformed equation.

When translation amounts are shown next to the variable they affect (as opposed to the other side of the equation), “keep your eye on the variable”. Ask yourself “what value of x_2 makes the expression in parentheses equal 0?”. The answer to this question for this example is +4, which is the amount and direction of the translation in the x_2 (horizontal) direction.

Vertical Translations

Translations-Y

In the same way we approached horizontal translations above, look at the coordinates of the corresponding points on the green and red graphs above. The x-coordinate has not changed and the y-coordinate has increased by two. If you were to examine any other pair of corresponding points, you would see the exact same difference. This can be described algebraically by the system of transformation equations:

\begin{cases}x_1~=~x_2\\*y_1+2~=~y_2~~~~~(3)\end{cases}

where y_1 is the starting y-coordinate from a point on the green curve, and y_2 is the translated y-coordinate from the corresponding point on the red curve. The x coordinates do not change, and thus are set equal to one another.

The green curve is the graph of the equation:

y_1~=~(x_1)^2~~~~~~~(2)

To find the equation of the translated curve, merge the two equations above by substituting for the variable they have in common. To do so, we’ll need to solve equation (3) for y_1:

y_1+2~=~y_2~~~~~(3)\\*~\\*y_1~=~y_2-2

and the system of transformation equations becomes:

\begin{cases}x_1~=~x_2\\*y_1~=~y_2-2\end{cases}

then substitute the result into (2) to produce an equation for y_2:

(y_1)~=~(x_1)^2~~~~~~(2)~~Green\\*~\\*(y_2-2)~=~(x_2)^2~~~~~~~~Red

Notice that the process of solving for y_1 before substituting moves the vertical translation amount to the other side of the equation, so that a translation by positive two (up), becomes (y_2-2) in the transformed equation. If this equation is then solved for y_2, we get:

(y_2-2)~=~(x_2)^2~~~~~~~~Red\\*~\\*y_2~=~(x_2)^2+2

The last equation above can be interpreted as the original result of the function, x^2, with two added to every result. Thus, every y-value from the original function must be two higher. To summarize, vertical translation amounts appear:

  • with the “wrong” sign when on the vertical variable side of the equation
  • with the “right” sign when on the horizontal variable side of the equation

When analyzing vertical translations, “keep your eye on the variable”. If the translation amount is next to the vertical variable, ask yourself “what value of y makes the expression equal 0?”.  On the other hand, if the translation amount is on the horizontal variable side of the equation, its value correctly reflects the direction and amount of the vertical translation.

Multiple Translations

Translations-XY

The image above shows the result of two transformations. The green curve has been translated both horizontally and vertically to produce the red curve. Looking at the coordinates of the corresponding point A, you can see that it has been moved right four units, and up one unit. Algebraically, this can be described by the system:

\begin{cases}~~x_1+4~=~x_2\\*~~y_1+1~=~y_2\end{cases}

Looking at the two corresponding points from the graph above, (1,1) is the orginal point represented by (x_1,y_1) in the system of translation equations above, and (5,2) is the translated point represented by (x_2, y_2) in the system of translation equations above.

Solving both equations for the “original” (green) curve variables produces

\begin{cases}~~x_1~=~x_2-4\\*~~y_1~=~y_2-1\end{cases}

and substituting both into the equation of the green curve leads to

~~~~~~~~y_1~=~(x_1)^2~~~~~~~(2)~~Green\\*~\\*(y_2-1)~=~(x_2-4)^2~~~~~~~~~Red\\*~~~~~~~~~~~~~or\\*~~~~~~~~~y_2~=~~(x_2-4)^2+1~~~Red

Note that translation amounts appear:

  • with the “wrong” sign when next to the variable they affect
  • with the “right” sign when on the other side of the equation from the variable they affect

Translations in More Complex Equations

When using substitution, all instances of the variable being substituted for must be replaced. The same applies when translating a function. For example, if we were translating a function with multiple x terms, such as

y_1~=~(x_1)^2+3(x_1)+5

by a horizontal amount

x_1+4~=~x_2\\*~\\*x_1~=~x_2-4

then every instance of x_1 must have the translation equation substituted in its place

y_2~=~(x_2-4)^2+3(x_2-4)+5

So, when analyzing one function to see if it a translation of another, check that every instance of each variable has had the same transformation equation applied before asserting that the function is a transformation!

In this example,

y_2~=~(x_2-4)^2+3(x_2+1)+5

y2 does not describe a horizontal translation of y1, because the “x” term did not have the same transformation equation substituted for it as the “x squared” term did.

Equivalent Translations

In mathematics, it is often (but not always) possible to produce the same end result in different ways. When working with linear equations and using the approach described above, you may have wondered how to handle a situation such as:

y~=~x-4

The above describes a horizontal translation by +4, but if we add 4 to both sides the equation becomes:

y+4~=~x

which describes a vertical translation by -4. Are they both valid interpretations?

Since both of the above are algebraic manipulations of the same equation, they must both have the same graph.

Translations-Either

Examine the graph of y~=~x in green above, which is a line with a slope of one that passes through the origin. Now, think about what happens to the x-intercept of this graph as you translate it vertically by -4 to produce the red line. The x-intercept moves… four to its right. So, both interpretations above are valid.

Equivalent translations do not always translate by the same distance. If the slope of the line is not 1, we will need to translate by different amounts in each direction to achieve the same end result:

y~=~2(x-3)\\*~\\*y~=~2x-6\\*~\\*y+6~=~2x

The first equation above can be thought of as y = 2x translated horizontally by +3, while the last one is a vertical translation by -6. Yet, they both describe the same graph. We could make things even more complex by not moving all of  the translation amount to the other side of the equation:

y~=~2(x-3)\\*~\\*y~=~2x-6\\*~\\*y+4~=~2x-2\\*~\\*y+4~=~2(x-1)

So, we can choose to describe this function as y=2x translated either:
– horizontally by +3 (1st line), or
– vertically by -4 and horizontally by +1 (4th line)

Just as there are often multiple ways of describing a specific situation using words, math can often also describe something in multiple ways.

Published by

Whit Ford

Math Tutor in Yarmouth, Maine

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