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I am working through a chapter on circles, tangents and parabolas and changes in origin. I am answering the questions at the end of a section on changes in origin. This question says sketch the following pairs of related curves:

$(a) y= x^3, y= x^3 - 3x^2+3x+1$

$(b) xy=1, xy=2x+y-1$

$(c) y^2=-12x, y^2= 12(y-x)$

This puzzles me. The preceding end of section questions have involved equations of parabolas in the form of $(y-k)^2=4a(x-h)$

These equations don't fit into that pattern. I can see that they all involve transitions and I could laboriously plot $y= x^3 - 3x^2+3x+1$ - or just plug it into Desmos - but I am thinking I must be missing the point.

So my question is: Are these questions simply asking me to plot the curves or is there some deeper significance which I am missing?

Steblo
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    $x^3-3x^2+3x+1=(x-1)^3+2$ – Teddy38 Jul 07 '21 at 11:15
  • Ah, so I am looking to refactor these equations so that the shifts in x and y become apparent? – Steblo Jul 07 '21 at 11:17
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    I think the question asks you to: (1) sketch the first curve of each pair, and then (2) use certain transforms to "convert" the sketch of the first curve into a sketch of the second curve, without doing much extra work. – WhatsUp Jul 07 '21 at 11:18

2 Answers2

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Since $x^3-3x^2+3x+1=(x-1)^3+2$, the second curve is a translation of the first one.

José Carlos Santos
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Part a) has already been done. For part b), use Simon's Favourite Factoring Trick to get $xy - 2x - y + 1 = 0 \Rightarrow (x - 1)(y - 2) + \ ? = \ ?$

For part c), complete the square for $y^2 - 12y = -12x$.

Toby Mak
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