Sketch rational with linear numerator

5. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = \frac { 4 ( x - 1 ) } { x ( x - 3 ) }$$ The curve cuts the \(x\)-axis at \(( a , 0 )\). The lines \(y = 0 , x = 0\) and \(x = b\) are asymptotes to the curve.

1. Write down the value of \(a\) and the value of \(b\).
   (2)
2. On separate axes, sketch the curves with the following equations. On your sketches, you should mark the coordinates of any intersections with the coordinate axes and state the equations of any asymptotes.
   1. \(y = \mathrm { f } ( x + 2 ) - 4\)
   2. \(y = \mathrm { f } ( | x | ) - 3\)

7 A curve has equation $$y = \frac { 3 x - 1 } { x + 2 }$$

1. Write down the equations of the two asymptotes to the curve.
2. Sketch the curve, indicating the coordinates of the points where the curve intersects the coordinate axes.
3. Hence, or otherwise, solve the inequality $$0 < \frac { 3 x - 1 } { x + 2 } < 3$$

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

1. 1. Write down the equations of the asymptotes of the curve \(C\).
   2. Sketch the curve \(C\).
2. The line \(y = x - 3\) intersects the curve \(C\) at a point which has \(x\)-coordinate \(\alpha\).
   1. Show that \(\alpha\) lies within the interval \(3 < x < 4\).
   2. Starting from the interval \(3 < x < 4\), use interval bisection twice to obtain an interval of width 0.25 within which \(\alpha\) must lie.

13

1. Write down the equations of the asymptotes of curve \(C _ { 1 }\) 13 A curve \(C _ { 1 }\) has equation 13
2. On the axes below, sketch the graph of curve \(C _ { 1 }\) Indicate the values of the intercepts of the curve with the axes. \includegraphics[max width=\textwidth, alt={}, center]{fd9715c4-9ce1-4608-aed6-f3d4f71208b5-20_885_898_1192_571} 13
3. Hence, or otherwise, solve the inequality $$\frac { 2 x + 7 } { 3 x + 5 } \geq 0$$ 13
4. Curve \(C _ { 2 }\) is a reflection of curve \(C _ { 1 }\) in the line \(y = - x\) Find an equation for curve \(C _ { 2 }\) in the form \(y = \mathrm { f } ( x )\)

The function f is defined by $$f(x) = \frac{ax + 5}{x + b}$$ where \(a\) and \(b\) are constants. The graph of \(y = f(x)\) has asymptotes \(x = -2\) and \(y = 3\)

1. Write down the value of \(a\) and the value of \(b\) [2 marks]
2. The diagram shows the graph of \(y = f(x)\) and its asymptotes. The shaded region \(R\) is enclosed by the graph of \(y = f(x)\), the \(x\)-axis and the \(y\)-axis. \includegraphics{figure_16}
   1. The shaded region \(R\) is rotated through \(360°\) about the \(x\)-axis to form a solid. Find the volume of this solid. Give your answer to three significant figures. [3 marks]
   2. The shaded region \(R\) is rotated through \(360°\) about the \(y\)-axis to form a solid. Find the volume of this solid. Give your answer to three significant figures. [4 marks]