What Is A Horizontal Translation

For the approach I now prefer to this topic, using transformation equations, please follow this link: Function Transformations: Translation

A part has been "translated" when it has been moved in a way that does not change its shape or rotate information technology in whatever style. A function can be translated either vertically, horizontally, or both. Other possible "transformations" of a function include dilation, reflection, and rotation.

Imagine a graph fatigued on tracing paper or a transparency, so placed over a separate set of axes. If you motility the graph left or correct in the direction of the horizontal axis, without rotating it, you are "translating" the graph horizontally. Motility the graph straight up or downwards in the direction of the vertical centrality, and you lot are translating the graph vertically.

In the text that follows, nosotros will explore how we know that the graph of a function similar

$g(x)=x^2-6x+10\\*~\\*~~~~~~~=(x-3)^2+1$

which is the bluish bend on the graph above, can exist described as a translation of the graph of the green bend above:

$f(x)=x^2 ~~~~\text{translated right by 3, and up by 1}$

Describing $g(x)$ as a translation of a simpler-looking (and more familiar) function like $f(x)$ makes it easier to understand and predict its behavior, and tin can make it easier to describe the beliefs of complex-looking functions. Before you lot dive into the explanations below, you may wish to play around a bit with the green sliders for "h" and "one thousand" in this Geogebra Applet to get a feel for what horizontal and vertical translations look like as they take place (the "a" slider dilates the office, every bit discussed in my Function Dilations post).

Vertical Translation

Consider the equation that describes the line that passes through the origin and has a gradient of 2:

$f(x)=2x$

What happens to the graph of this line if every value of $f(x)$ has 3 added to it? The office $g(x)$ is defined as the effect of $f(x)$ with three added to each event. If we then substitute the definition of $f(x)$ from higher up for $f(x)$ , we get:

$g(x)=f(x)+3\\*~\\*~~~~~~~=2x+3$

Since $f(x)$ produces the y-coordinate respective to x for every point on the original graph, adding 3 to each value movesevery point on the graph up by 3.

Adding "+3" to the definition of $f(x)$ causes the entire function to be "translated vertically" by a positive 3.

This procedure works for any function, and is usually thought through in the reverse lodge: when looking at a more complex role, do you come across a constant added or subtracted? If so, you tin can call up of it as a vertical translation of the rest of the office:

$g(x)=sin(x)+3\\*~\\*\text{is a vertical translation (by +3) of}\\*~\\*f(x)=sin(x)$

Another case:

$g(x)=\dfrac{1}{x}+k\\*~\\*~\\*\text{is a vertical translation (by k) of}\\*~\\*f(x)=\dfrac{1}{x}$

Horizontal Translation

Consider the same function described at the outset of the Vertical Translation section, which describes a line that passes through the origin with a slope of 2:

$f(x)~=~2(x)$

What happens to the graph of this equation if every "10" in the equation is replaced by a value that is 4 less? Nosotros can depict this algebraically by evaluating $f(x-4)$ instead of $f(x)$ , and let's phone call this new function $g(x)$ :

$g(x)~=~f(x-4)\\*~\\*~~~~~~~~=~2(x-4)$

At present let's compare the behaviors of $f(x)$ and $g(x)$ :

$f(1)~=~2~~~~g(5)~=~2\\*~\\*f(2)~=~4~~~~g(6)~=~4\\*~\\*f(3)~=~6~~~~g(7)~=~6$

$g(x)$ produces the same results as $f(x)$ , simply simply when its input values are four greater than the input to $f(x)$ . Comparing the graphs of the ii functions, the graph of $g(x)$ volition have the same shape as $f(x)$ , simply that shape has been shifted four units to the right along the x-axis.

A helpful way to think near the higher up (thanks to Michael Paul Goldenberg's 2016 comment below) is to think of the contained variable "x" as measuring time in seconds. Therefore, "x-4" is 4 seconds before and then "x", and evaluating $f(x-4)$ produces a result from 4 seconds before than time "ten". When we graph $f(x-4)$ , all of the results will appear to be 4 seconds later (to the right) than those on the graph of $f(x)$ .

The fact that substituting "x-4" for "x" produces a horizontal translation of +4 (not -4) is a source of errors when people get horizontal and vertical translation behaviors confused. Ane way to accost this is to apply a procedural approach whenever y'all run across a variable with a constant added or subtracted (oftentimes together in a set of parentheses). To detect the direction of the translation, set the transformation expression equal to goose egg and solve:

$(x-4)~=~0\\*~\\*x~=~+4$

The result volition e'er give your the magnitude and management of the translation (come across Keep Your Eye On The Variable). This process works for any role:

$g(x)~=~(x+3)^2\\*~\\*\text{if }f(x)~=~x^2\text{ then }\\*~\\*g(x)~=~f(x+3)$

and so set $(x+3)~=~0$ and solve for 10. The graph of $g(x)$ is the same as that of $f(x)$ translated horizontally past -3. Or for

$g(x)~=~(x+5)^2-3(x+5)+7\\*~\\*\text{if }f(x)~=~x^2-3x+7\text{ then }\\*~\\*g(x)~=~f(x+5)$

the graph of $g(x)$ is the same as that of $f(x)$ translated horizontally by -five. Annotation that $f(x+5)$ requireseveryinstance of "10" in $f(x)$ to have (ten+5) substituted for it. So a part like $g(x)$ volition merely be a horizontal translation of $f(x)$ if every instance of "10" has the same constant added or subtracted. The notation $g(x)=f(x+5)$ expresses this thought compactly and elegantly.

One final example:

$g(x)~=~\dfrac{1}{x+k}\\*~\\*~\\*\text{if }f(x)=\dfrac{1}{x}\text{ then }\\*~\\*~\\*g(x)=f(x+k)$

so the graph of $g(x)$ is the same as that of $f(x)$ translated horizontally past $-k$ .

Reconciling Horizontal And Vertical Translations

Let'due south re-examine why $f(x)+3$ translates a office in a positive vertical direction, yet $f(x+3)$ translates the function in a negative horizontal direction.

This credible difference in the style we analyze horizontal and vertical translations can exist reconciled past treating both independent and dependent variables in the same manner. If

$g(x)~=~(x+5)^2-3(x+5)+7$

and we subtract 7 from both sides, it becomes:

$(g(x)-7)~=~(x+5)^2-3(x+5)$

Since every example of $g(x)$ occurs equally a $g(x)-7$ , and every instance of "x" occurs as $(x+5)$ , you may treat both $g(x)$ and "x" as having been translated relative to a parent part, and you may analyze them both in exactly the same manner:
– what value of $g(x)$ makes $(g(x)-7)=0$ ? Positive 7. So the translation in the $g(x)$ management, forth the vertical axis, is positive 7.
– what value of "10" makes $(x+5)=0$ ? Negative five. So the translation in the "x" management, along the horizontal centrality, is negative 5.

Therefore, if we define $f(x)$ equally shown beneath, a $g(x)$ can be created which is translated horizontally by -5 and vertically by +7 when compared to $f(x)$ :

$f(x)=x^2-3x\\*~\\*(g(x)-7)=f(x+5)\\*~\\*g(x)~=~(x+5)^2-3(x+5)+7\\*~\\*g(x)~=~x^2+7x+17$

Equivalent Translations

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

$f(x)~=~(x-4)$

The above describes a horizontal translation past +4, just if we decrease 4 from both sides the equation becomes:

$f(x)-4~=~x$

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

Since both of the above are valid algebraic manipulations of the same equation, they must both have the same graph. Imagine the graph of $f(x)~=~x$ , which will be a line with a slope of one that passes through the origin. At present interpret the graph vertically past +four. This translation will besides cause the x-intercept to move… four to its left.

Equivalent translations do not always translate by the same altitude. If the slope of the line is non 1, we need to translate by different amounts:

$g(x)~=~2(x-2)\\*~\\*g(x)~=~2x-4\\*~\\*g(x)+4~=~2x$

The first representation of $g(x)$ to a higher place is a horizontal translation of $f(x)~=~2x$ by +2, while the terminal 1 is a vertical translation past -4. Yet, they both describe the same graph. We could be even trickier if we wished to:

$g(x)~=~2(x-2)\\*~\\*g(x)~=~2x-4\\*~\\*g(x)+2~=~2x-2\\*~\\*g(x)+2~=~2(x-1)$

So, we tin can choose to describe 1000(x) as either:
– f(10) translated horizontally by +2 (1st line)
– f(x) translated vertically past -4 (2nd line)
– f(ten) translated vertically by -2 and horizontally by +1 (4th line)

Simply every bit in that location are often multiple ways of describing something using English, a detail state of affairs can often exist described in more than one mathematically too.