Rational function curve sketching

8 A curve has equation \(y = \frac { x ^ { 2 } - 4 } { ( 3 x - 2 ) ^ { 2 } }\).

1. Find the equations of the asymptotes.
2. Describe the behaviour of the curve for large positive and large negative values of \(x\), justifying your description.
3. Sketch the curve.
4. Solve the inequality \(\frac { x ^ { 2 } - 4 } { ( 3 x - 2 ) ^ { 2 } } \geqslant - 1\).

5 A curve has equation \(y = \frac { x ^ { 3 } - k ^ { 3 } } { x ^ { 2 } - 4 }\), where \(k\) is a positive constant and \(k \neq 2\).

1. Find the equations of the three asymptotes.
2. Use your graphical calculator to obtain rough sketches of the curve in the two separate cases \(k < 2\) and \(k > 2\).
3. In the case \(k < 2\), your sketch may not show clearly the shape of the curve near \(x = 0\). Use calculus to show that the curve has a minimum point when \(x = 0\).
4. In the case \(k > 2\), your sketch may not show clearly how the curve approaches its asymptote as \(x \rightarrow + \infty\). Show algebraically that the curve crosses this asymptote.
5. Use the results of parts (iii) and (iv) to produce more accurate sketches of the curve in the two separate cases \(k < 2\) and \(k > 2\). These sketches should indicate where the curve crosses the axes, and should show clearly how the curve approaches its asymptotes. The presence of stationary points should be clearly shown, but there is no need to find their coordinates. RECOGNISING ACHIEVEMENT \section*{OXFORD CAMBRIDGE AND RSA EXAMINATIONS} \section*{MEI STRUCTURED MATHEMATICS} Further Methods for Advanced Mathematics (FP2)
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5 A curve has equation \(y = \frac { x ^ { 3 } - k ^ { 3 } } { x ^ { 2 } - 4 }\), where \(k\) is a positive constant and \(k \neq 2\).

1. Find the equations of the three asymptotes.
2. Use your graphical calculator to obtain rough sketches of the curve in the two separate cases \(k < 2\) and \(k > 2\).
3. In the case \(k < 2\), your sketch may not show clearly the shape of the curve near \(x = 0\). Use calculus to show that the curve has a minimum point when \(x = 0\).
4. In the case \(k > 2\), your sketch may not show clearly how the curve approaches its asymptote as \(x \rightarrow + \infty\). Show algebraically that the curve crosses this asymptote.
5. Use the results of parts (iii) and (iv) to produce more accurate sketches of the curve in the two separate cases \(k < 2\) and \(k > 2\). These sketches should indicate where the curve crosses the axes, and should show clearly how the curve approaches its asymptotes. The presence of stationary points should be clearly shown, but there is no need to find their coordinates.

7 Fig. 7 shows part of the curve with equation \(y = \frac { x + 5 } { ( 2 x - 5 ) ( 3 x + 8 ) }\). \begin{figure}[h] \end{figure}

1. Write down the coordinates of the two points where the curve crosses the axes.
2. Write down the equations of the two vertical asymptotes and the one horizontal asymptote.
3. Determine how the curve approaches the horizontal asymptote for large positive and large negative values of \(x\).
4. On the copy of Fig. 7, sketch the rest of the curve.
5. Solve the inequality \(\frac { x + 5 } { ( 2 x - 5 ) ( 3 x + 8 ) } < 0\).

7 A curve has equation \(y = \frac { ( x + 1 ) ( 2 x - 1 ) } { x ^ { 2 } - 3 }\).

1. Find the coordinates of the points where the curve crosses the axes.
2. Write down the equations of the three asymptotes.
3. Determine whether the curve approaches the horizontal asymptote from above or from below for
   (A) large positive values of \(x\),
   (B) large negative values of \(x\).
4. Sketch the curve.
5. Solve the inequality \(\frac { ( x + 1 ) ( 2 x - 1 ) } { x ^ { 2 } - 3 } < 2\).

7 Fig. 7 shows a sketch of \(y = \frac { x - 4 } { ( x - 5 ) ( x - 8 ) }\). \begin{figure}[h] \end{figure}

1. Write down the equations of the three asymptotes and the coordinates of the points where the curve crosses the axes. Hence write down the solution of the inequality \(\frac { x - 4 } { ( x - 5 ) ( x - 8 ) } > 0\).
2. The equation \(\frac { x - 4 } { ( x - 5 ) ( x - 8 ) } = k\) has no real solutions. Show that \(- 1 < k < - \frac { 1 } { 9 }\). Relate this result to the graph of \(y = \frac { x - 4 } { ( x - 5 ) ( x - 8 ) }\).

7 A curve has equation \(y = \frac { x ^ { 2 } - 25 } { ( x - 3 ) ( x + 4 ) ( 3 x + 2 ) }\).

1. Write down the coordinates of the points where the curve crosses the axes.
2. Write down the equations of the asymptotes.
3. Determine how the curve approaches the horizontal asymptote for large positive values of \(x\), and for large negative values of \(x\).
4. Sketch the curve.

7 A curve has equation \(\mathrm { y } = \frac { ( 3 \mathrm { x } + 2 ) ( \mathrm { x } - 3 ) } { ( \mathrm { x } - 2 ) ( \mathrm { x } + 1 ) }\).

1. Write down the equations of the three asymptotes and the coordinates of the points where the curve crosses the axes.
2. Sketch the curve, justifying how it approaches the horizontal asymptote.
3. Find the set of values of \(x\) for which \(y \geqslant 3\).

7 The equation of a curve is \(y = \frac { x ^ { 2 } + 1 } { ( x + 1 ) ( x - 7 ) }\).

1. Write down the equations of the asymptotes.
2. Find the coordinates of the stationary points on the curve.
3. Find the coordinates of the point where the curve meets one of its asymptotes.
4. Sketch the curve.

7 It is given that \(\mathrm { f } ( x ) = \frac { x ^ { 2 } - 25 } { ( x - 1 ) ( x + 2 ) }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Write down the equations of the asymptotes of the curve \(y = \mathrm { f } ( x )\).
3. Find the value of \(x\) where the graph of \(y = \mathrm { f } ( x )\) cuts the horizontal asymptote.
4. Sketch the graph of \(y ^ { 2 } = \mathrm { f } ( x )\).

7 The curve \(C\) has equation $$y = \frac { 2 x ^ { 2 } + 5 x - 1 } { x + 2 }$$ Find the equations of the asymptotes of \(C\). Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } > 2\) at all points on \(C\). Sketch C.

9 The curve \(C\) has equation $$y = \frac { 3 x - 9 } { ( x - 2 ) ( x + 1 ) }$$

1. Find the equations of the asymptotes of \(C\). \includegraphics[max width=\textwidth, alt={}, center]{68e31138-756a-433a-bf42-0fdfadad091e-14_61_1566_513_328}
2. Show that there is no point on \(C\) for which \(\frac { 1 } { 3 } < y < 3\).
3. Find the coordinates of the turning points of \(C\).
4. Sketch \(C\).

3 The curve \(C\) has equation \(y = \frac { x + 1 } { x ^ { 2 } - 4 }\).

1. Show that the gradient of \(C\) is always negative.
2. Sketch \(C\), showing all significant features.

6 The curve \(S\) has equation \(y = \frac { x ^ { 2 } + 1 } { ( x + 1 ) ^ { 2 } }\).

1. Write down the equations of the asymptotes of \(S\).
2. Determine \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the coordinates of any turning points of \(S\).
3. Sketch \(S\).

Answer only one of the following two alternatives. EITHER The curve \(C\) has equation \(y = \frac{5(x-1)(x+2)}{(x-2)(x+3)}\).

1. Express \(y\) in the form \(P + \frac{Q}{x-2} + \frac{R}{x+3}\). [3]
2. Show that \(\frac{dy}{dx} = 0\) for exactly one value of \(x\) and find the corresponding value of \(y\). [4]
3. Write down the equations of all the asymptotes of \(C\). [3]
4. Find the set of values of \(k\) for which the line \(y = k\) does not intersect \(C\). [4]

OR A curve has equation \(y = \frac{5}{3}x^{\frac{3}{2}}\), for \(x \geq 0\). The arc of the curve joining the origin to the point where \(x = 3\) is denoted by \(R\).

1. Find the length of \(R\). [4]
2. Find the \(y\)-coordinate of the centroid of the region bounded by the \(x\)-axis, the line \(x = 3\) and \(R\). [5]
3. Show that the area of the surface generated when \(R\) is rotated through one revolution about the \(y\)-axis is \(\frac{232\pi}{15}\). [5]

The curve \(C\) has equation $$y = \frac{x^2}{x + \lambda},$$ where \(\lambda\) is a non-zero constant. Obtain the equations of the asymptotes of \(C\). [3] In separate diagrams, sketch \(C\) for the cases where

1. \(\lambda > 0\),
2. \(\lambda < 0\).

[4]

The curve \(C\) has equation \(y = \frac{2x^2 + kx}{x + 1}\), where \(k\) is a constant. Find the set of values of \(k\) for which \(C\) has no stationary points. [5] For the case \(k = 4\), find the equations of the asymptotes of \(C\) and sketch \(C\), indicating the coordinates of the points where \(C\) intersects the coordinate axes. [6]