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Math Help - Horizontal And Vertical Translations

  1. #1
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    Horizontal And Vertical Translations


    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.


    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.

    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?
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  2. #2
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    Suppose you have a function y=f(x). Vertical translations look like y=f(x)\pm c: they correspond to a shifting of the range of a function, or its output. Horizontal translations look like 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: y=x+c could be viewed either way. Or, if you had a slope of 2, then the line y=2x+4 could have a positive vertical translation of 4 from the line y=2x, or it could be viewed as y=2(x+2), in which case it's a horizontal translation of 2 to the left.

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

    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 t=0 as the function 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: 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.
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  3. #3
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    Thank you very much and it was detailed =D
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  4. #4
    A Plied Mathematician
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    You're welcome. Have a good one!
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