Finding the image of $y=2x+5$ under the transformation $w=(1+i)z-2$

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I am trying to find the image of the line $y=2x+5$ under the transformation $w=(1+i)z-2$.




I'm uncertain of how to proceed in questions that do not involve the mapping $w=frac1z$. I thought that if I could rewrite the mapping as $w=sqrt2e^fracpi4iz-2$, this would indicate that the straight line is rotated anticlockwise about the origin by an angle of $fracpi4$, is dilated by $sqrt2$ and translated by 2 units to the right, but I am not confident in these statements.



Is there a general method on how to solve this question? A hint would be very helpful :)




complex-analysis functions




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    I am trying to find the image of the line $y=2x+5$ under the transformation $w=(1+i)z-2$.




    I'm uncertain of how to proceed in questions that do not involve the mapping $w=frac1z$. I thought that if I could rewrite the mapping as $w=sqrt2e^fracpi4iz-2$, this would indicate that the straight line is rotated anticlockwise about the origin by an angle of $fracpi4$, is dilated by $sqrt2$ and translated by 2 units to the right, but I am not confident in these statements.



    Is there a general method on how to solve this question? A hint would be very helpful :)




    complex-analysis functions




    share|cite|improve this question





















      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite












      I am trying to find the image of the line $y=2x+5$ under the transformation $w=(1+i)z-2$.




      I'm uncertain of how to proceed in questions that do not involve the mapping $w=frac1z$. I thought that if I could rewrite the mapping as $w=sqrt2e^fracpi4iz-2$, this would indicate that the straight line is rotated anticlockwise about the origin by an angle of $fracpi4$, is dilated by $sqrt2$ and translated by 2 units to the right, but I am not confident in these statements.



      Is there a general method on how to solve this question? A hint would be very helpful :)




      complex-analysis functions




      share|cite|improve this question












      I am trying to find the image of the line $y=2x+5$ under the transformation $w=(1+i)z-2$.




      I'm uncertain of how to proceed in questions that do not involve the mapping $w=frac1z$. I thought that if I could rewrite the mapping as $w=sqrt2e^fracpi4iz-2$, this would indicate that the straight line is rotated anticlockwise about the origin by an angle of $fracpi4$, is dilated by $sqrt2$ and translated by 2 units to the right, but I am not confident in these statements.



      Is there a general method on how to solve this question? A hint would be very helpful :)





      complex-analysis functions





      share|cite|improve this question










      share|cite|improve this question




      share|cite|improve this question









      asked Aug 1 at 9:36









      Bell

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          You're overcomplicating it.



          Set $z=t+(2t+5)i$. Plug into your transformation. Evaluate. You get a parametric equation for the image of the line.






          share|cite|improve this answer





















          • Can this method be extended to other transformations? Or only in the case where $w$ is linear?
            – Bell
            Aug 1 at 10:13










          • @Bell: It will always give you a parametric equation for whatever the image is, but the shape of the parametric equation is not necessarily nice when your $w$ isn't.
            – Henning Makholm
            Aug 1 at 10:16










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          1 Answer
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          active

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          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes








          up vote
          2
          down vote



          accepted










          You're overcomplicating it.



          Set $z=t+(2t+5)i$. Plug into your transformation. Evaluate. You get a parametric equation for the image of the line.






          share|cite|improve this answer





















          • Can this method be extended to other transformations? Or only in the case where $w$ is linear?
            – Bell
            Aug 1 at 10:13










          • @Bell: It will always give you a parametric equation for whatever the image is, but the shape of the parametric equation is not necessarily nice when your $w$ isn't.
            – Henning Makholm
            Aug 1 at 10:16














          up vote
          2
          down vote



          accepted










          You're overcomplicating it.



          Set $z=t+(2t+5)i$. Plug into your transformation. Evaluate. You get a parametric equation for the image of the line.






          share|cite|improve this answer





















          • Can this method be extended to other transformations? Or only in the case where $w$ is linear?
            – Bell
            Aug 1 at 10:13










          • @Bell: It will always give you a parametric equation for whatever the image is, but the shape of the parametric equation is not necessarily nice when your $w$ isn't.
            – Henning Makholm
            Aug 1 at 10:16












          up vote
          2
          down vote



          accepted







          up vote
          2
          down vote



          accepted






          You're overcomplicating it.



          Set $z=t+(2t+5)i$. Plug into your transformation. Evaluate. You get a parametric equation for the image of the line.






          share|cite|improve this answer













          You're overcomplicating it.



          Set $z=t+(2t+5)i$. Plug into your transformation. Evaluate. You get a parametric equation for the image of the line.







          share|cite|improve this answer













          share|cite|improve this answer



          share|cite|improve this answer











          answered Aug 1 at 9:40









          Henning Makholm

          225k16290516




          225k16290516











          • Can this method be extended to other transformations? Or only in the case where $w$ is linear?
            – Bell
            Aug 1 at 10:13










          • @Bell: It will always give you a parametric equation for whatever the image is, but the shape of the parametric equation is not necessarily nice when your $w$ isn't.
            – Henning Makholm
            Aug 1 at 10:16
















          • Can this method be extended to other transformations? Or only in the case where $w$ is linear?
            – Bell
            Aug 1 at 10:13










          • @Bell: It will always give you a parametric equation for whatever the image is, but the shape of the parametric equation is not necessarily nice when your $w$ isn't.
            – Henning Makholm
            Aug 1 at 10:16















          Can this method be extended to other transformations? Or only in the case where $w$ is linear?
          – Bell
          Aug 1 at 10:13




          Can this method be extended to other transformations? Or only in the case where $w$ is linear?
          – Bell
          Aug 1 at 10:13












          @Bell: It will always give you a parametric equation for whatever the image is, but the shape of the parametric equation is not necessarily nice when your $w$ isn't.
          – Henning Makholm
          Aug 1 at 10:16




          @Bell: It will always give you a parametric equation for whatever the image is, but the shape of the parametric equation is not necessarily nice when your $w$ isn't.
          – Henning Makholm
          Aug 1 at 10:16












           

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