1.02n Sketch curves: simple equations including polynomials

7

1. By sketching the curves \(y = x ( 2 x + 5 )\) and \(y = \cos ^ { - 1 } x\) (where \(y\) is in radians) in a single diagram, show that the equation \(x ( 2 x + 5 ) = \cos ^ { - 1 } x\) has exactly one real root.
2. Use the iterative formula $$x _ { n + 1 } = \frac { \cos ^ { - 1 } x _ { n } } { 2 x _ { n } + 5 } \text { with } x _ { 1 } = 0.25$$ to find the root correct to 3 significant figures. Show the result of each iteration correct to at least 4 significant figures.
3. Two new curves are obtained by transforming each of the curves \(y = x ( 2 x + 5 )\) and \(y = \cos ^ { - 1 } x\) by the pair of transformations:
   reflection in the \(x\)-axis followed by reflection in the \(y\)-axis.
   State an equation of each of the new curves and determine the coordinates of their point of intersection, giving each coordinate correct to 3 significant figures.

7

1. Show algebraically that the function \(\mathrm { f } ( x ) = \frac { 2 x } { 1 - x ^ { 2 } }\) is odd. Fig. 7 shows the curve \(y = \mathrm { f } ( x )\) for \(0 \leqslant x \leqslant 4\), together with the asymptote \(x = 1\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{28ce1bcc-e9d5-4ae6-98c0-67b5b8c50bc6-4_730_817_431_607} \captionsetup{labelformat=empty} \caption{Fig. 7}
   \end{figure}
2. Use the copy of Fig. 7 to complete the curve for \(- 4 \leqslant x \leqslant 4\).

9 A curve has parametric equations \(x = 1 - \cos t , y = \sin t \sin 2 t\), for \(0 \leqslant t \leqslant \pi\).

1. Find the coordinates of the points where the curve meets the \(x\)-axis.
2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 \cos 2 t + 2 \cos ^ { 2 } t\). Hence find, in an exact form, the coordinates of the stationary points.
3. Find the cartesian equation of the curve. Give your answer in the form \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x )\) is a polynomial.
4. Sketch the curve.

1

1. Use the trapezium rule with four strips to estimate \(\int _ { - 2 } ^ { 2 } \sqrt { 1 + \mathrm { e } ^ { x } } \mathrm {~d} x\), showing your working. Fig. 1 shows a sketch of \(y = \sqrt { 1 + \mathrm { e } ^ { x } }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f657e167-e6f8-4df2-901b-067c32835877-02_535_1074_571_532} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure}
2. Suppose that the trapezium rule is used with more strips than in part (i) to estimate \(\int _ { - 2 } ^ { 2 } \sqrt { 1 + \mathrm { e } ^ { x } } \mathrm {~d} x\). State, with a reason but no further calculation, whether this would give a larger or smaller estimate.

8 Fig. 8 shows part of the graph of \(y = \frac { x ^ { 2 } - 3 } { ( x - 4 ) ( x + 2 ) }\). Two sections of the graph have been omitted. \begin{figure}[h] \end{figure}

1. Write down the coordinates of the points where the curve crosses the axes.
2. Write down the equations of the two vertical asymptotes and the one horizontal asymptote.
3. Copy Fig. 8 and draw in the two missing sections.
4. Solve the inequality \(\frac { x ^ { 2 } - 3 } { ( x - 4 ) ( x + 2 ) } \leqslant 0\).

7 A curve has equation \(y = \frac { 5 x - 9 } { ( 2 x - 3 ) ( 2 x + 7 ) }\).

1. Write down the equations of the two vertical asymptotes and the one horizontal asymptote.
2. Describe the behaviour of the curve for large positive and large negative values of \(x\), justifying your answers.
3. Sketch the curve.
4. Solve the inequality \(\frac { 5 x - 9 } { ( 2 x - 3 ) ( 2 x + 7 ) } \leqslant 0\).

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 \(y = \frac { x - 4 } { ( x - 5 ) ( x - 8 ) }\).

3

1. Sketch the graph of \(y = \frac { 2 } { x + 4 }\).
2. Solve the inequality $$\frac { 2 } { x + 4 } \leqslant x + 3$$ showing your working clearly.

7 A curve has equation \(y = \frac { ( x + 2 ) ( 3 x - 5 ) } { ( 2 x + 1 ) ( x - 1 ) }\).

1. Write down 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 below for
   (A) large positive values of \(x\),
   (B) large negative values of \(x\).
4. Sketch the curve.

7 Fig. 7 shows an incomplete sketch of \(y = \frac { ( 2 x - 1 ) ( x + 3 ) } { ( x - 3 ) ( x - 2 ) }\). \begin{figure}[h] \end{figure}

1. Find the coordinates of the points where the curve cuts the axes.
2. Write down the equations of the three asymptotes.
3. Determine whether the curve approaches the horizontal asymptote from above or below for large positive values of \(x\), justifying your answer. Copy and complete the sketch.
4. Solve the inequality \(\frac { ( 2 x - 1 ) ( x + 3 ) } { ( x - 3 ) ( x - 2 ) } < 2\).

7 A curve has equation \(y = \frac { ( x + 9 ) ( 3 x - 8 ) } { x ^ { 2 } - 4 }\).

1. Write down 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 below for
   (A) large positive values of \(x\),
   (B) large negative values of \(x\).
4. Sketch the curve.

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 Fig. 7 shows an incomplete sketch of \(y = \frac { c x ^ { 2 } } { ( b x - 1 ) ( x + a ) }\) where \(a , b\) and \(c\) are integers. The asymptotes of the curve are also shown. \begin{figure}[h] \end{figure}

1. Determine the values of \(a , b\) and \(c\). Use these values of \(a , b\) and \(c\) throughout the rest of the question.
2. Determine how the curve approaches the horizontal asymptote for large positive values of \(x\), and for large negative values of \(x\), justifying your answer. On the copy of Fig. 7, sketch the rest of the curve.
3. Find the \(x\) coordinates of the points on the curve where \(y = 1\). Write down the solution to the inequality \(\frac { c x ^ { 2 } } { ( b x - 1 ) ( x + a ) } < 1\).

7 A curve has equation \(y = \frac { x ^ { 2 } - 5 } { ( x + 3 ) ( x - 2 ) ( a x - 1 ) }\), where \(a\) is a constant.

1. Find the coordinates of the points where the curve crosses the \(x\)-axis and the \(y\)-axis.
2. You are given that the curve has a vertical asymptote at \(x = \frac { 1 } { 2 }\). Write down the value of \(a\) and the equations of the other asymptotes.
3. Sketch the curve.
4. Find the set of values of \(x\) for which \(y > 0\).

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 \(y \geqslant 3\).

8 A curve has equation \(y = \frac { 3 x ^ { 2 } - 9 } { x ^ { 2 } + 3 x - 4 }\).

1. Find the equations of the two vertical asymptotes and the one horizontal asymptote of this curve.
2. State, with justification, how the curve approaches the horizontal asymptote for large positive and large negative values of \(x\).
3. Sketch the curve.
4. Solve the inequality \(\frac { 3 x ^ { 2 } - 9 } { x ^ { 2 } + 3 x - 4 } \geqslant 0\).

3 The function f is defined by \(\mathrm { f } ( x ) = \frac { 5 a x } { x ^ { 2 } + a ^ { 2 } }\), for \(x \in \mathbb { R }\) and \(a > 0\).

1. For the curve with equation \(y = \mathrm { f } ( x )\),
   1. write down the equation of the asymptote,
   2. find the range of values that \(y\) can take.
   3. For the curve with equation \(y ^ { 2 } = \mathrm { f } ( x )\), write down
      (a) the equation of the line of symmetry,
      (b) the maximum and minimum values of \(y\),
   4. the set of values of \(x\) for which the curve is defined.

2 The equation of a curve is \(y = \frac { x ^ { 2 } - 3 } { x - 1 }\).

1. Find the equations of the asymptotes of the curve.
2. Write down the coordinates of the points where the curve cuts the axes.
3. Show that the curve has no stationary points.
4. Sketch the curve and the asymptotes.

4 \includegraphics[max width=\textwidth, alt={}, center]{074597e7-5bb1-4249-9cfa-784974a6fd2b-2_947_1305_986_420} The diagram shows the curve with equation $$y = \frac { a x + b } { x + c }$$ where \(a , b\) and \(c\) are constants.

1. Given that the asymptotes of the curve are \(x = - 1\) and \(y = - 2\) and that the curve passes through \(( 3,0 )\), find the values of \(a , b\) and \(c\).
2. Sketch the curve with equation $$y ^ { 2 } = \frac { a x + b } { x + c }$$ for the values of \(a , b\) and \(c\) found in part (i). State the coordinates of any points where the curve crosses the axes, and give the equations of any asymptotes.

8 The curve \(C _ { 1 }\) has equation \(y = \frac { \mathrm { p } ( x ) } { \mathrm { q } ( x ) }\), where \(\mathrm { p } ( x )\) and \(\mathrm { q } ( x )\) are polynomials of degree 2 and 1 respectively. The asymptotes of the curve are \(x = - 2\) and \(y = \frac { 1 } { 2 } x + 1\), and the curve passes through the point \(\left( - 1 , \frac { 17 } { 2 } \right)\).

1. Express the equation of \(C _ { 1 }\) in the form \(y = \frac { \mathrm { p } ( x ) } { \mathrm { q } ( x ) }\).
2. For the curve \(C _ { 1 }\), find the range of values that \(y\) can take.
3. For the curve \(C _ { 1 }\), find the range of values that \(y\) can take.
   Another curve, \(C _ { 2 }\), has equation \(y ^ { 2 } = \frac { \mathrm { p } ( x ) } { \mathrm { q } ( x ) }\), where \(\mathrm { p } ( x )\) and \(\mathrm { q } ( x )\) are the polynomials found in part (i).
4. It is given that \(C _ { 2 }\) intersects the line \(y = \frac { 1 } { 2 } x + 1\) exactly once. Find the coordinates of the point of intersection. Another curve, \(C _ { 2 }\), has equation \(y ^ { 2 } = \frac { \mathrm { p } ( x ) } { \mathrm { q } ( x ) }\), where \(\mathrm { p } ( x )\) and \(\mathrm { q } ( x )\) are the polynomials found in part (i).
5. It is given that \(C _ { 2 }\) intersects the line \(y = \frac { 1 } { 2 } x + 1\) exactly once. Find the coordinates of the point of
   intersection. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}

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.

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

1. Find the equations of the asymptotes of the curve.
2. Prove that there are no points on the curve for which \(4 < y < 8\).
3. Sketch the curve. Indicate the asymptotes in your sketch.

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 \(y ^ { 2 } = \mathrm { f } ( x )\).