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

\includegraphics{figure_2} Fig. 12 shows the graph of \(y = \frac{1}{x-2}\).

1. Draw accurately the graph of \(y = 2x + 3\) 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 \(y = 2x + 3\). [3]
2. Show algebraically that the \(x\)-coordinates of the points of intersection of \(y = \frac{1}{x-2}\) and \(y = 2x + 3\) satisfy the equation \(2x^2 - x - 7 = 0\). Hence find the exact values of the \(x\)-coordinates of the points of intersection. [5]
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]

\includegraphics{figure_3} Fig. 12 shows the graph of \(y = \frac{1}{x-3}\).

1. Draw accurately, on the copy of Fig. 12, the graph of \(y = x^2 - 4x + 1\) for \(-1 < x < 5\). Use your graph to estimate the coordinates of the intersections of \(y = \frac{1}{x-3}\) and \(y = x^2 - 4x + 1\). [5]
2. Show algebraically that, where the curves intersect, \(x^3 - 7x^2 + 13x - 4 = 0\). [3]
3. Use the fact that \(x = 4\) is a root of \(x^3 - 7x^2 + 13x - 4 = 0\) to find a quadratic factor of \(x^3 - 7x^2 + 13x - 4\). Hence find the exact values of the other two roots of this equation. [5]

Answer the whole of this question on the insert provided. The insert shows the graph of \(y = \frac{1}{x}\), \(x \neq 0\).

1. Use the graph to find approximate roots of the equation \(\frac{1}{x} = 2x + 3\), showing your method clearly. [3]
2. Rearrange the equation \(\frac{1}{x} = 2x + 3\) to form a quadratic equation. Solve the resulting equation, leaving your answers in the form \(\frac{p \pm \sqrt{q}}{r}\). [5]
3. Draw the graph of \(y = \frac{1}{x} + 2\), \(x \neq 0\), on the grid used for part (i). [2]
4. Write down the values of \(x\) which satisfy the equation \(\frac{1}{x} + 2 = 2x + 3\). [2]

Answer part (i) of this question on the insert provided. The insert shows the graph of \(y = \frac{1}{x}\).

1. On the insert, on the same axes, plot the graph of \(y = x^2 - 5x + 5\) for \(0 \leq x \leq 5\). [4]
2. Show algebraically that the \(x\)-coordinates of the points of intersection of the curves \(y = \frac{1}{x}\) and \(y = x^2 - 5x + 5\) satisfy the equation \(x^3 - 5x^2 + 5x - 1 = 0\). [2]
3. Given that \(x = 1\) at one of the points of intersection of the curves, factorise \(x^3 - 5x^2 + 5x - 1\) into a linear and a quadratic factor. Show that only one of the three roots of \(x^3 - 5x^2 + 5x - 1 = 0\) is rational. [5]

The graph of the equation \(y = \frac{1}{x}\) is translated by the vector \(\begin{bmatrix}3\\0\end{bmatrix}\)

1. Write down the equation of the transformed graph. [1 mark]
2. State the equations of the asymptotes of the transformed graph. [2 marks]

Curve \(C\) has equation \(y = \frac{1}{(x-1)^2}\) State the equations of the asymptotes to curve \(C\) Tick (\(\checkmark\)) one box. [1 mark] \(x = 0\) and \(y = 0\) \qquad \(\square\) \(x = 0\) and \(y = 1\) \qquad \(\square\) \(x = 1\) and \(y = 0\) \qquad \(\square\) \(x = 1\) and \(y = 1\) \qquad \(\square\)

Sketch the following curves.

1. \(y = \frac{2}{x}\) [2]
2. \(y = x^3 - 6x^2 + 9x\) [5]

The function \(f\) is defined by \(f(x) = \frac{8}{x^2}\).

1. Sketch the graph of \(y = f(x)\). [2]
2. On a separate set of axes, sketch the graph of \(y = f(x - 2)\). Indicate the vertical asymptote and the point where the curve crosses the \(y\)-axis. [3]
3. Sketch the graphs of \(y = \frac{8}{x}\) and \(y = \frac{8}{(x-2)^2}\) on the same set of axes. Hence state the number of roots of the equation \(\frac{8}{(x-2)^2} = \frac{8}{x}\). [2]

1. Sketch the curve with equation $$y = k - \frac{1}{2x}$$ where \(k\) is a positive constant State, in terms of \(k\), the coordinates of any points of intersection with the coordinate axes and the equation of the horizontal asymptote. [3]

The straight line \(l\) has equation \(y = 2x + 3\) Given that \(l\) cuts the curve in two distinct places,

1. find the range of values of \(k\), writing your answer in set notation. [6]

Sketch the curve with equation \(y = \frac{x^2 + 4x}{2x - 1}\), justifying all significant features. [11]

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

1. Find the equations of the asymptotes of \(C\). [4]
2. Draw a sketch of \(C\), which should include the asymptotes, and state the coordinates of the points of intersection of \(C\) with the \(x\)-axis. [5]