What happens to $y=\ln x$ when it is translated and then dilated?

Question

I am struggling to understand my teacher's answer for the following question.

The curve $y=\ln x$ is translated to the left by $\pi$ units and then dilated horizontally by a scale factor of 3. What is the equation that describes the new curve?

I thought it was $y=\ln(x/3 +\pi)$ since the translation is applied first, which isn't typical order. But my teacher has the answer as $y=\ln\frac{x+\pi}3$, which would make sense to me if the order was dilation then translation, but I'm struggling to understand how that comes from translation then dilation.

Is my teacher correct, and if so, can someone explain?

@Henry, thank you! so the dilation is applied to pi as well as the x? — user1213971, Aug 29 '23 at 10:15
for clarification, would the result be any different if the dilation was performed before the translation? — user1213971, Aug 29 '23 at 10:16

Anne Bauval · Accepted Answer · 2023-08-29T10:23:14.123

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A curve $y=f(x)$ translated to the left by $\pi$ results in a curve $y=f(x+\pi).$

A curve $y=g(x)$ dilated horizontally by $3$ results in $y=g(x/3).$

Your teacher is therefore wrong and you are right.

If the dilation was performed before, it would be the opposite.

edited Aug 29 '23 at 10:23

answered Aug 29 '23 at 10:17

Anne Bauval

34,650

Im not sure!! I'm not downvoting it!!!! I really appreciate your answer it seems fine to me – user1213971 Aug 29 '23 at 10:20
$y = \ln(x)$ first becomes $y = \ln(x + \pi)$. Writing $g(x) = \ln(x + \pi)$, this is $y = g(x)$, so next this becomes $y = g(x/3) = \ln(x/3 + \pi)$. – Magdiragdag Aug 29 '23 at 10:21
I am not the downvoter but it looks to my eye like teacher is wrong and OP is right – M W Aug 29 '23 at 10:22
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Omg thank you! I correct immediately!! – Anne Bauval Aug 29 '23 at 10:22
actually, In my textbook it says that doing dilation first gives f(a(x+b), while doing translation first gives f(ax+b). That suggests to me that it should be ln(/3+) , but am I just misinterpreting the textbook? – user1213971 Aug 29 '23 at 10:24
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@user1213971 no you were correct all along. – M W Aug 29 '23 at 10:27
Oh for real?? cool! – user1213971 Aug 29 '23 at 10:27
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Thanks so much for your help guys!!! – user1213971 Aug 29 '23 at 10:28
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@user1213971 If you think this post answers your question/gives you the most insight, you may want to mark it as accepted it by clicking/tapping the check button. (This is to remove it from the unanswered section.) – soupless Aug 29 '23 at 10:30
Oh yeah perfect, sorry I am new, didn't know I could do that lol!! – user1213971 Aug 29 '23 at 10:31

What happens to $y=\ln x$ when it is translated and then dilated?

1 Answers1