2024 VJC Promo Q3
2024 VJC Promo Q3
6 marks
The diagram shows the curve \(y = \mathrm{f}(x)\) which cuts the axes at \(A(0, -a)\) and \(B(b, 0)\), where \(a\) and \(b\) are positive constants.
- State, if it is possible to do so, the coordinates of the points where the following curves cut the axes.
- \(y = \mathrm{f}^{-1}(x)\)[1]
- \(y = \mathrm{f}(x - 2)\)[1]
- \(y = \mathrm{f}(3x)\)[1]
- Sketch the curve \(y = \frac{1}{\mathrm{f}(x)}\), stating the equations of any asymptotes and the coordinates of the point where \(y = \frac{1}{\mathrm{f}(x)}\) crosses the axes.[3]
Solution:
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Answer:
(a)(i) x-axis \((-a,0)\), y-axis \((0,b)\). (ii) x-axis \((b+2,0)\); y-intercept not numerically determined. (iii) x-axis \(\left(\frac b3,0\right)\), y-axis \((0,-a)\).
(b) Asymptotes \(x=b,y=0\), y-intercept \(\left(0,-\frac1a\right)\), no x-intercept; sketch shown.
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