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## Section5.2Gist 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.

###### Exercise5.2.1
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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.

###### Exercise5.2.2
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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.

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##### 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.

###### Example5.2.4

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 travelling 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.

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.

###### Example5.2.8

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

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?

Solution

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.

###### Example5.2.10

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

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.

Solution

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:

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.

###### Exercise5.2.13
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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.

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##### 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.

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