2024 ACJC Promo Q3
2024 ACJC Promo Q3
4 marks
Promo
The diagram below shows the graph of the curve \(y=\mathrm{f}\left( x \right)\), with turning point \(\left( \frac{k}{2},2 \right)\) and asymptotes with equations \(x=k\) and \(y=0\).
The curve undergoes the following sequence of transformations in succession
A: Translate \(k\) units in the negative \(x\)-direction
B: Reflection about the \(y\)-axis
C: Translate \(k\) units in the positive \(x\)-direction
- Sketch the curve after the transformations, indicating clearly the equations of the asymptotes and the coordinates of the turning point.[2]
- Find the equation of the curve after the transformations in the form \(y=\mathrm{f}\left( ax+\mathrm{b} \right)\) where \({a}\) and \({b}\) are constants to be determined, in terms of \(k\) where appropriate.[1]
- Another curve \(y=\mathrm{g}\left( x \right)\) is such that \(\mathrm{g}\left( x \right)=\mathrm{g}\left( 4-x \right)\) for all \(x\). State the equation of a line of symmetry of this curve.[1]
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Answer:(ii) \(\therefore a=-1\), \(b=2k\) (iii) From (i), we see that the curve \(y=\text{f}\hspace{0.1em}\left( -x+2k \right)\) is a reflection of the curve \(y=\text{f}\hspace{0.1em}\left( x \right)\) about the line \(x=k\). Hence if \(\text{g}\hspace{0.1em}\left( x \right)=\text{g}\hspace{0.1em}\left( 4-x \right)\) for all \(x\), then the curve \(y=\text{g}\hspace{0.1em}\left( x \right)\) is the same as when it is reflected about the line \(x=2\). Hence line of symmetry is \(x=2\).
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