Horizontal And Vertical Translations

• Aug 11th 2010, 01:00 PM
Gordon
Horizontal And Vertical Translations
http://library.thinkquest.org/20991/...rizandvert.gif
When you alter a graph, you transform it. If you transform a graph without changing its shape, you translate it. Vertical and horizontal transformations are translations. When y = f(x) + d, shift (translate) the graph of y = f(x) vertically (upward if d > 0, downward if d < 0).

Example: 1. Problem: Translate y = x2 upward by 1. Solution: You have been asked to shift the graph upward 1. Rewrite the equation to do this, and then graph. y = x2 + 1 The figure below is a graph of the solution.
http://library.thinkquest.org/20991/media/alg2_vt.gif

When y = f(x + c), translate the graph of y = f(x) horizontally (left if c > 0, right if c < 0).

Example: 2. Problem: Sketch the graph of y = |x + 2|. Solution: First, graph y = |x|, then shift it to the left 2 places. The figure below is a graph of the solution.http://library.thinkquest.org/20991/media/alg2_ht.gif

I saw this online and i was wondering how can you tell if it's a Horizontal Translation or Vertical one? If so, what are usually these translation used for?
• Aug 11th 2010, 01:15 PM
Ackbeet
Suppose you have a function $\displaystyle y=f(x).$ Vertical translations look like $\displaystyle y=f(x)\pm c$: they correspond to a shifting of the range of a function, or its output. Horizontal translations look like $\displaystyle y=f(x\pm c)$: they correspond to a shifting of the domain of a function, or its input.

There are a few functions where you can't tell the difference between the two translations: straight lines, for example: $\displaystyle y=x+c$ could be viewed either way. Or, if you had a slope of $\displaystyle 2,$ then the line $\displaystyle y=2x+4$ could have a positive vertical translation of $\displaystyle 4$ from the line $\displaystyle y=2x$, or it could be viewed as $\displaystyle y=2(x+2),$ in which case it's a horizontal translation of $\displaystyle 2$ to the left.

I'll give you one application of translations: modeling switches in electrical circuits. The Heaviside step function is defined as

$\displaystyle u(x)=\begin{cases}1&x\ge 0\\0&x<0\end{cases}.$

I can think of a switch turning on when I start a clock timer at $\displaystyle t=0$ as the function $\displaystyle u(t)$. But now, suppose I want to flip a switch for one second, and then switch it back off? I could model that as a horizontally translated difference of step functions: $\displaystyle u(t)-u(t-1).$ We might want to model that if we're trying to solve the differential equation governing the behavior of the circuit.
• Aug 12th 2010, 06:49 PM
Gordon
Thank you very much and it was detailed =D
• Aug 13th 2010, 02:54 AM
Ackbeet
You're welcome. Have a good one!