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Section 5.2 Gist of Vertical and Horizontal Translations

To transform a function means to change it — either its input or its output. If the change is only in position (the graph looks the same, but in a different location) the change is called a translation.

In this chapter we learn how to translate a graph or a table of values. This will involve adding or subtracting values to the input and/or the output of a function's formula.

It will be important for us to distinguish between the input and the output of a function. For instance, we must understand that the function

\begin{equation*} f(x) = x^2 - 5x \end{equation*}

has an input \(x\) and an output \(f(x)\text{.}\)

If the input is decreased by \(3\text{,}\) then the output would be \(f(x−3)\text{.}\) If the output is increased by \(4\text{,}\) it would become \(f(x) + 4\text{.}\)

If you evaluate a function by using \(x\text{,}\) then the output will have \(x\)’s in it. But if you substitute \(x–3\) into the function, then the output will have \((x-3)\)’s in it.

One natural question to ask is “How does changing the input alter the graph of \(f(x)\text{?}\)” It turns out that adding a number to the input of a function will move the graph to the left or the right. This is called a horizontal translation (or horizontal shift).

In short:

  • Adding a positive number to the input will shift the graph to the left.

  • Adding a negative number to the input will shift the graph to the right.

In fact, whenever you do anything to the input of a function, the result is some kind of horizontal change to the graph.

On the other hand, if you make changes to the output of the function, it will affect the graph vertically. Adding a number to the output of a function will produce a vertical translation of the graph up or down.

In short:

  • Adding a positivenumber to the output will shift the graph up.

  • Adding a negativenumber to the output will shift the graph down.

Direction of the Translation.

It should be clear why adding a positive number to the output will move the graph up, and adding a negative number to the output will move the graph down. The output of a function is shown as the \(y\)-coordinate of a point on the graph, so adding to the output means adding to the \(y\)-coordinates. This produces a direct change in the \(y\)-coordinates, either upward or downward.

However, it may have surprised you to see that it was the opposite with horizontal shifts — adding a positive number to the input moved the graph to the left, and a negative number moved the graph to the right. To see why this should be, let us first think about a particular horizontal shift in the next example.

The function

\begin{equation*} f(t) = 32t \end{equation*}

represents the speed, measured in feet per second, of an object that was dropped from the top of a tall building at time \(t = 0\) seconds. The value

\begin{equation*} f(2) = 64 \end{equation*}

means that after falling for \(2\) seconds, the object will be traveling at \(64\) feet per second.

If another object was dropped \(1\) second after the timer was started, then its speed at \(2\) seconds would only be \(f(1) = 32\) feet per second. It experiences the same speeds that the first object had \(1\) second earlier.

That is, for any time \(t\text{,}\) the object that was dropped \(1\) second late would have a speed \(f(t - 1)\text{.}\) See the graphs below.

Figure 5.2.5. \(f(t) = 32t\text{:}\) Dropped at time \(t = 0\)
Figure 5.2.6. \(f(t-1) = 32(t-1)\text{:}\) Dropped at time \(t = 1\)

The first speed graph would be shifted to the right by \(1\) second to make the speed graph for the object that was dropped \(1\) second late.

In the next three examples, you will use a verbal description of a transformation in order to write a formula, and then use the formula for a transformation to provide a verbal description.

The function \(f(x) = x^2 + 2x\) is graphed below.

Figure 5.2.8. \(f(x) = x^2 + 2x\)

If you wanted to move this graph so it had the same shape, but it was \(3\) units to the right and \(2\) units down, what would be its formula?


To move the graph to the right by \(3\) units, you must subtract \(3\) from the input.

To move it down \(2\) units, you must subtract \(2\) from the output.

So the function should be:

\begin{equation*} f(x-3) - 2 = (x-3)^2 + 2(x-3) - 2 \end{equation*}

Remember to substitute \((x-3)\) in wherever \(x\) appears in the original formula, and then subtract \(2\) at the end.

The function \(g(x) = x - x^3\) is graphed below.

Figure 5.2.10. \(g(x) = x - x^3\)

A different function, \(h(x) = (x+1) - (x+1)^3 + 2\text{,}\) is a certain transformation of \(g(x)\text{.}\)

  1. Describe what transformations were done to \(g(x)\) to make \(h(x)\text{.}\)

  2. Then sketch a graph of \(h(x) = (x+1) - (x+1)^3 + 2\) by hand.


Notice that

\begin{align*} h(x) \amp= (x+1) - (x+1)^3 + 2\\ \amp= g(x+1)+2 \end{align*}

Using \((x+1)\) in place of \(x\) will shift the graph to the left by \(1\) unit, and adding \(2\) to the outside of the function will shift the graph up by \(2\) units.

Here is the graph:

Figure 5.2.11. \(h(x) = (x+1) - (x+1)^3 + 2\)

A straightforward way to sketch the transformed function is to take a known point from \(g(x)\text{,}\) such as \((0,0)\text{,}\) and move it left \(1\) and up \(2\text{.}\) Then sketch the rest of the graph around that new point.

Similarly, we should also be able to take the graph of a transformation and write its formula. In the next two exercises, you will be given the formula and graph of a function \(f(x)\text{,}\) and you will be shown a transformation of that graph. Observe the changes made to the graph of \(f(x)\) in order to write the formula for the function transformation.

Translations in context.

Of course, translations should do more for us than just provide a quick way to move graphs around. If a function describes something about a real object or situation, then knowing about changes to that object should help us alter the function to account for those changes.

Subsection 5.2.1 Conclusion

Test your understanding of this chapter with the following exercises.

Student Learning Outcome: List the transformations required to graph equations of the form \(y = f(x + h) + k\text{,}\) given the graph of \(y = f(x)\text{,}\) and use those transformations to sketch a graph.

Student Learning Outcome: Use transformations of a basic function to sketch graphs, model situations algebraically, and determine domain/range and asymptotes.

Exercises 5.2.2 Exercises