Sketch rational function from transformation

1 Sketch the graph of \(y = \mathrm { e } ^ { a x } - 1\) where \(a\) is a positive constant.

6. (a) Sketch the curve with equation $$y = - \frac { k } { x } \quad k > 0 \quad x \neq 0$$ (b) On a separate diagram, sketch the curve with equation $$y = - \frac { k } { x } + k \quad k > 0 \quad x \neq 0$$ stating the coordinates of the point of intersection with the \(x\)-axis and, in terms of \(k\), the equation of the horizontal asymptote.
(c) Find the range of possible values of \(k\) for which the curve with equation $$y = - \frac { k } { x } + k \quad k > 0 \quad x \neq 0$$ does not touch or intersect the line with equation \(y = 3 x + 4\) \includegraphics[max width=\textwidth, alt={}, center]{6a5d0ffc-a725-404b-842a-f3b6000e6fed-21_72_47_2615_1886}

6. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \frac { 2 } { x } , x \neq 0\) The curve \(C\) has equation \(y = \frac { 2 } { x } - 5 , x \neq 0\), and the line \(l\) has equation \(y = 4 x + 2\)

1. Sketch and clearly label the graphs of \(C\) and \(l\) on a single diagram. On your diagram, show clearly the coordinates of the points where \(C\) and \(l\) cross the coordinate axes.
2. Write down the equations of the asymptotes of the curve \(C\).
3. Find the coordinates of the points of intersection of \(y = \frac { 2 } { x } - 5\) and \(y = 4 x + 2\)

\begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \frac { 3 } { x } , x \neq 0\).

1. On a separate diagram, sketch the curve with equation \(y = \frac { 3 } { x + 2 } , x \neq - 2\), showing the coordinates of any point at which the curve crosses a coordinate axis.
2. Write down the equations of the asymptotes of the curve in part (a).

4

1. Sketch the curve \(y = \frac { 1 } { x ^ { 2 } }\).
2. Hence sketch the curve \(y = \frac { 1 } { ( x - 3 ) ^ { 2 } }\).
3. Describe fully a transformation that transforms the curve \(y = \frac { 1 } { x ^ { 2 } }\) to the curve \(y = \frac { 2 } { x ^ { 2 } }\).

7. (i) Describe fully the single transformation that maps the graph of \(y = \mathrm { f } ( x )\) onto the graph of \(y = \mathrm { f } ( x - 1 )\).
(ii) Showing the coordinates of any points of intersection with the coordinate axes and the equations of any asymptotes, sketch the graph of \(y = \frac { 1 } { x - 1 }\).
(iii) Find the \(x\)-coordinates of any points where the graph of \(y = \frac { 1 } { x - 1 }\) intersects the graph of \(y = 2 + \frac { 1 } { x }\). Give your answers in the form \(a + b \sqrt { 3 }\), where \(a\) and \(b\) are rational.

1. (a) Describe fully the single transformation that maps the graph of \(y = \mathrm { f } ( x )\) onto the graph of \(y = \mathrm { f } ( x - 1 )\).
   (b) Showing the coordinates of any points of intersection with the coordinate axes and the equations of any asymptotes, sketch the graph of \(y = \frac { 1 } { x - 1 }\).
   (c) Find the \(x\)-coordinates of any points where the graph of \(y = \frac { 1 } { x - 1 }\) intersects the graph of \(y = 2 + \frac { 1 } { x }\). Give your answers in the form \(a + b \sqrt { 3 }\), where \(a\) and \(b\) are rational.
2. A store begins to stock a new range of DVD players and achieves sales of \(\pounds 1500\) of these products during the first month.

In a model it is assumed that sales will decrease by \(\pounds x\) in each subsequent month, so that sales of \(\pounds ( 1500 - x )\) and \(\pounds ( 1500 - 2 x )\) will be achieved in the second and third months respectively. Given that sales total \(\pounds 8100\) during the first six months, use the model to

7 A curve \(C\) has equation $$y = 7 + \frac { 1 } { x + 1 }$$

1. Define the translation which transforms the curve with equation \(y = \frac { 1 } { x }\) onto the curve \(C\).
2. 1. Write down the equations of the two asymptotes of \(C\).
   2. Find the coordinates of the points where the curve \(C\) intersects the coordinate axes.
3. Sketch the curve \(C\) and its two asymptotes.