1.02o Sketch reciprocal curves: y=a/x and y=a/x^2

10

1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{450c1c3a-9290-4afa-a051-112b60cf19c0-3_753_775_360_726} \captionsetup{labelformat=empty} \caption{Fig. 10}
   \end{figure} Fig. 10 shows a sketch of the graph of \(y = \frac { 1 } { x }\).
   Sketch the graph of \(y = \frac { 1 } { x - 2 }\), showing clearly the coordinates of any points where it crosses the axes.
2. Find the value of \(x\) for which \(\frac { 1 } { x - 2 } = 5\).
3. Find the \(x\)-coordinates of the points of intersection of the graphs of \(y = x\) and \(y = \frac { 1 } { x - 2 }\). Give your answers in the form \(a \pm \sqrt { b }\). Show the position of these points on your graph in part (i).

4. (i) Sketch on the same diagram the curves \(y = x ^ { 2 } - 4 x\) and \(y = - \frac { 1 } { x }\).
(ii) State, with a reason, the number of real solutions to the equation $$x ^ { 2 } - 4 x + \frac { 1 } { x } = 0 .$$

8. (i) Describe fully a single transformation that maps the graph of \(y = \frac { 1 } { x }\) onto the graph of \(y = \frac { 3 } { x }\).
(ii) Sketch the graph of \(y = \frac { 3 } { x }\) and write down the equations of any asymptotes.
(iii) Find the values of the constant \(c\) for which the straight line \(y = c - 3 x\) is a tangent to the curve \(y = \frac { 3 } { x }\).

3 \begin{figure}[h] \end{figure} Fig. 12 shows the graph of \(y = \frac { 4 } { x ^ { 2 } }\).

1. On the copy of Fig. 12, draw accurately the line \(y = 2 x + 5\) and hence find graphically the three roots of the equation \(\frac { 4 } { x ^ { 2 } } = 2 x + 5\).
   [0pt] [3]
2. Show that the equation you have solved in part (i) may be written as \(2 x ^ { 3 } + 5 x ^ { 2 } - 4 = 0\). Verify that \(x = - 2\) is a root of this equation and hence find, in exact form, the other two roots.
   [0pt] [6]
3. By drawing a suitable line on the copy of Fig. 12, find the number of real roots of the equation \(x ^ { 3 } + 2 x ^ { 2 } - 4 = 0\).
4. You are given that \(\mathrm { f } ( x ) = ( 2 x - 5 ) ( x - 1 ) ( x - 4 )\).
   (A) Sketch the graph of \(y = \mathrm { f } ( x )\).
   (B) Show that \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } + 33 x - 20\).
5. You are given that \(\mathrm { g } ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } + 33 x - 40\).
   (A) Show that \(\mathrm { g } ( 5 ) = 0\).
   (B) Express \(\mathrm { g } ( x )\) as the product of a linear and quadratic factor.
   (C) Hence show that the equation \(\mathrm { g } ( x ) = 0\) has only one real root.
6. Describe fully the transformation that maps \(y = \mathrm { f } ( x )\) onto \(y = \mathrm { g } ( x )\).

3 \begin{figure}[h] \end{figure} Fig. 12 shows the graph of \(y = \frac { 1 } { x - 2 }\).

1. Draw accurately the graph of \(y = 2 x + 3\) on the copy of Fig. 12 and use it to estimate the coordinates of the points of intersection of \(y = \frac { 1 } { x - 2 }\) and \(y = 2 x + 3\).
2. Show algebraically that the \(x\)-coordinates of the points of intersection of \(y = \frac { 1 } { x - 2 }\) and \(y = 2 x + 3\) satisfy the equation \(2 x ^ { 2 } - x - 7 = 0\). Hence find the exact values of the \(x\)-coordinates of the points of intersection.
3. Find the quadratic equation satisfied by the \(x\)-coordinates of the points of intersection of \(y = \frac { 1 } { x - 2 }\) and \(y = - x + k\). Hence find the exact values of \(k\) for which \(y = - x + k\) is a tangent to \(y = \frac { 1 } { x - 2 }\). [4]

1 Fig. 7 shows the curve \(y = _ { x - 1 }\). It has a minimum at the point P . The line \(l\) is an asymptote to the curve. \begin{figure}[h] \end{figure}

1. Write down the equation of the asymptote \(l\).
2. Find the coordinates of P .
3. Using the substitution \(u = x - 1\), show that the area of the region enclosed by the \(x\)-axis, the curve and the lines \(x = 2\) and \(x = 3\) is given by $$\int _ { 1 } ^ { 2 } \left( u + 2 + \frac { 4 } { u } \right) \mathrm { d } u$$ Evaluate this area exactly.
4. Another curve is defined by the equation \(\mathrm { e } ^ { y } = \frac { x ^ { 2 } + 3 } { x - 1 }\). Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\) by differentiating implicitly. Hence find the gradient of this curve at the point where \(x = 2\).

5.The function f is defined on the domain \([ - 2,2 ]\) by: $$f ( x ) = \left\{ \begin{array} { r l r } - k x ( 2 + x ) & \text { if } & - 2 \leq x < 0 , \\ k x ( 2 - x ) & \text { if } & 0 \leq x \leq 2 , \end{array} \right.$$ where \(k\) is a positive constant.
The function g is defined on the domain \([ - 2,2 ]\) by \(\mathrm { g } ( x ) = ( 2.5 ) ^ { 2 } - x ^ { 2 }\) .

1. Prove that there is a value of \(k\) such that the graph of f touches the graph of g .
2. For this value of \(k\) sketch the graphs of the functions f and g on the same axes,stating clearly where the graphs touch.
3. Find the exact area of the region bounded by the two graphs.

1. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = 1 + \frac { 4 } { x ( x - 3 ) }$$ The curve has a turning point at the point \(P\), and the lines with equations \(y = 1 , x = 0\) and \(x = a\) are asymptotes to the curve.

1. Write down the value of \(a\).
2. Find the coordinates of \(P\), justifying your answer.
3. Sketch the curve with equation \(y = \left| \mathrm { f } \left( x + \frac { 3 } { 2 } \right) \right| - 1\) On your sketch, you should show the coordinates of any points of intersection with the coordinate axes, the coordinates of any turning points and the equations of any asymptotes. \includegraphics[max width=\textwidth, alt={}, center]{4d5b914c-28b2-4485-a42e-627c95fa16e2-02_2255_50_311_1980}

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) with equation $$y = \frac { 1 } { x } + 1 , \quad x \neq 0$$ The curve \(C\) crosses the \(x\)-axis at the point \(A\).

1. State the \(x\) coordinate of the point \(A\). The curve \(D\) has equation \(y = x ^ { 2 } ( x - 2 )\), for all real values of \(x\).
2. A copy of Figure 1 is shown on page 7. On this copy, sketch a graph of curve \(D\).
   Show on the sketch the coordinates of each point where the curve \(D\) crosses the coordinate axes.
3. Using your sketch, state, giving a reason, the number of real solutions to the equation $$x ^ { 2 } ( x - 2 ) = \frac { 1 } { x } + 1$$ \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{64f015bf-29fb-4374-af34-3745ea49aced-06_942_1026_516_466} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure}

4

1. Sketch the curve \(y = \frac { 1 } { x ^ { 2 } }\).
2. The curve \(y = \frac { 1 } { x ^ { 2 } }\) is translated by 3 units in the negative \(x\)-direction. State the equation of the curve after it has been translated.
3. The curve \(y = \frac { 1 } { x ^ { 2 } }\) is stretched parallel to the \(y\)-axis with scale factor 4 and, as a result, the point \(P ( 1,1 )\) is transformed to the point \(Q\). State the coordinates of \(Q\).

2

1. Sketch the curve \(y = - \frac { 1 } { x ^ { 2 } }\).
2. Sketch the curve \(y = 3 - \frac { 1 } { x ^ { 2 } }\).
3. The curve \(y = - \frac { 1 } { x ^ { 2 } }\) is stretched parallel to the \(y\)-axis with scale factor 2 . State the equation of the transformed curve.

2

1. Sketch the curve \(y = \frac { 1 } { x }\).
2. Describe fully the single transformation that transforms the curve \(y = \frac { 1 } { x }\) to the curve \(y = \frac { 1 } { x } + 4\).

2

1. Sketch the curve \(y = - \frac { 1 } { x }\).
2. The curve \(y = - \frac { 1 } { x }\) is translated by 2 units parallel to the \(x\)-axis in the positive direction. State the equation of the transformed curve.
3. Describe a transformation that transforms the curve \(y = - \frac { 1 } { x }\) to the curve \(y = - \frac { 1 } { 3 x }\).

6 The curve \(C\) has equation \(y = \frac { x ^ { 2 } } { x - 2 }\). Find the equations of the asymptotes of \(C\). Find the coordinates of the turning points on \(C\). Draw a sketch of \(C\).

10 The curve \(C\) has equation \(y = \frac { 2 x ^ { 2 } - 3 x - 2 } { x ^ { 2 } - 2 x + 1 }\). State the equations of the asymptotes of \(C\). Show that \(y \leqslant \frac { 25 } { 12 }\) at all points of \(C\). Find the coordinates of any stationary points of \(C\). Sketch \(C\), stating the coordinates of any intersections of \(C\) with the coordinate axes and the asymptotes.

10 The curves \(C _ { 1 }\) and \(C _ { 2 }\) have equations $$y = \frac { a x } { x + 5 } \quad \text { and } \quad y = \frac { x ^ { 2 } + ( a + 10 ) x + 5 a + 26 } { x + 5 }$$ respectively, where \(a\) is a constant and \(a > 2\).

1. Find the equations of the asymptotes of \(C _ { 1 }\).
2. Find the equation of the oblique asymptote of \(C _ { 2 }\).
3. Show that \(C _ { 1 }\) and \(C _ { 2 }\) do not intersect.
4. Find the coordinates of the stationary points of \(C _ { 2 }\).
5. Sketch \(C _ { 1 }\) and \(C _ { 2 }\) on a single diagram. [You do not need to calculate the coordinates of any points where \(C _ { 2 }\) crosses the axes.]

6 The curve \(C\) has equation $$y = \frac { x ^ { 2 } } { k x - 1 }$$ where \(k\) is a positive constant.

1. Obtain the equations of the asymptotes of \(C\).
2. Find the coordinates of the stationary points of \(C\).
3. Sketch \(C\).

The curve \(C\) has equation $$y = \frac { ( x - 2 ) ( x - a ) } { ( x - 1 ) ( x - 3 ) } ,$$ where \(a\) is a constant not equal to 1,2 or 3 .

1. Write down the equations of the asymptotes of \(C\).
2. Show that \(C\) meets the asymptote parallel to the \(x\)-axis at the point where \(x = \frac { 2 a - 3 } { a - 2 }\).
3. Show that the \(x\)-coordinates of any stationary points on \(C\) satisfy $$( a - 2 ) x ^ { 2 } + ( 6 - 4 a ) x + ( 5 a - 6 ) = 0$$ and hence find the set of values of \(a\) for which \(C\) has stationary points.
4. Sketch the graph of \(C\) for
   1. \(a > 3\),
   2. \(2 < a < 3\).

3 The curve \(C\) has equation $$y = \frac { x ^ { 2 } - 5 x + 4 } { x + 1 }$$

1. Obtain the coordinates of the points of intersection of \(C\) with the axes.
2. Obtain the equation of each of the asymptotes of \(C\).
3. Draw a sketch of \(C\).

1. The curve \(C\) has equation

$$y = \frac { k ^ { 2 } } { x } + 1 \quad x \in \mathbb { R } , x \neq 0$$ where \(k\) is a constant.

1. Sketch \(C\) stating the equation of the horizontal asymptote. The line \(l\) has equation \(y = - 2 x + 5\)
2. Show that the \(x\) coordinate of any point of intersection of \(l\) with \(C\) is given by a solution of the equation $$2 x ^ { 2 } - 4 x + k ^ { 2 } = 0$$
3. Hence find the exact values of \(k\) for which \(l\) is a tangent to \(C\).

1. (a) Sketch the curve with equation

$$y = \frac { k } { x } \quad x \neq 0$$ where \(k\) is a positive constant.
(b) Hence or otherwise, solve $$\frac { 16 } { x } \leqslant 2$$

1. The curve \(C _ { 1 }\) has equation

$$y = \frac { 6 } { x } + 3$$

1. 1. Sketch \(C _ { 1 }\) stating the coordinates of any points where the curve cuts the coordinate axes.
   2. State the equations of any asymptotes to the curve \(C _ { 1 }\) The curve \(C _ { 2 }\) has equation $$y = 3 x ^ { 2 } - 4 x - 10$$
2. Show that \(C _ { 1 }\) and \(C _ { 2 }\) intersect when $$3 x ^ { 3 } - 4 x ^ { 2 } - 13 x - 6 = 0$$ Given that the \(x\) coordinate of one of the points of intersection is \(- \frac { 2 } { 3 }\)
3. use algebra to find the \(x\) coordinates of the other points of intersection between \(C _ { 1 }\) and \(C _ { 2 }\) (Solutions relying on calculator technology are not acceptable.)

3

1. Sketch the curve \(y = - \frac { 1 } { x ^ { 2 } }\).
2. The curve \(y = - \frac { 1 } { x ^ { 2 } }\) is translated by 2 units in the positive \(x\)-direction. State the equation of the curve after it has been translated.
3. The curve \(y = - \frac { 1 } { x ^ { 2 } }\) is stretched parallel to the \(y\)-axis with scale factor \(\frac { 1 } { 2 }\) and, as a result, the point \(\left( \frac { 1 } { 2 } , - 4 \right)\) on the curve is transformed to the point \(P\). State the coordinates of \(P\).

9

1. Sketch both of the following on the axes provided in the Printed Answer Booklet.
   1. The curve \(\mathrm { y } = \frac { 12 } { \mathrm { x } }\), stating the coordinates of at least one point on the curve.
   2. The line \(y = 2 x + 8\), stating the coordinates of the points at which the line crosses the axes.
2. In this question you must show detailed reasoning. Determine the exact coordinates of the points of intersection of the curve and the line.

10

1. Sketch the graph of \(y = \frac { 1 } { x } + a\), where \(a\) is a positive constant.