Identify transformation from equations

1. (a) On Diagram 1, sketch a graph of the curve \(C\) with equation

$$y = \frac { 6 } { x } \quad x \neq 0$$ The curve \(C\) is transformed onto the curve with equation \(y = \frac { 6 } { x - 2 } \quad x \neq 2\) (b) Fully describe this transformation. The curve with equation $$y = \frac { 6 } { x - 2 } \quad x \neq 2$$ and the line with equation $$y = k x + 7 \quad \text { where } k \text { is a constant }$$ intersect at exactly two points, \(P\) and \(Q\).
Given that the \(x\) coordinate of point \(P\) is - 4
(c) find the value of \(k\),
(d) find, using algebra, the coordinates of point \(Q\).
(Solutions relying entirely on calculator technology are not acceptable.) \section*{Diagram 1} Only use this copy of Diagram 1 if you need to redraw your graph. \includegraphics[max width=\textwidth, alt={}, center]{bb21001f-fe68-4776-992d-ede1aae233d7-19_709_739_1802_664} Copy of Diagram 1
(Total for Question 7 is 10 marks)

6

1. Sketch the graph of \(y = \frac { 1 } { x }\), where \(x \neq 0\), showing the parts of the graph corresponding to both positive and negative values of \(x\).
2. Describe fully the geometrical transformation that transforms the curve \(y = \frac { 1 } { x }\) to the curve \(y = \frac { 1 } { x + 2 }\). Hence sketch the curve \(y = \frac { 1 } { x + 2 }\).
3. Differentiate \(\frac { 1 } { x }\) with respect to \(x\).
4. Use parts (ii) and (iii) to find the gradient of the curve \(y = \frac { 1 } { x + 2 }\) at the point where it crosses the \(y\)-axis.

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 \(y = \frac { 3 } { x }\).

6

1. Sketch the curve \(y = - \sqrt { x }\).
2. Describe fully a transformation that transforms the curve \(y = - \sqrt { x }\) to the curve \(y = 5 - \sqrt { x }\).
3. The curve \(y = - \sqrt { x }\) is stretched by a scale factor of 2 parallel to the \(x\)-axis. State the equation of the curve after it has been stretched.

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 \(y = \frac { 1 } { x } + 4\).

5

1. Sketch the curve \(y = \sqrt { x }\).
2. Describe the transformation that transforms the curve \(y = \sqrt { x }\) to the curve \(y = \sqrt { x - 4 }\).
3. The curve \(y = \sqrt { x }\) is stretched by a scale factor of 5 parallel to the \(x\)-axis. State the equation of the transformed curve.

8 \includegraphics[max width=\textwidth, alt={}, center]{87012792-fa63-4003-875d-b8e7739037f1-4_489_697_274_667} The diagram shows the curves \(y = \log _ { 2 } x\) and \(y = \log _ { 2 } ( x - 3 )\).

1. Describe the geometrical transformation that transforms the curve \(y = \log _ { 2 } x\) to the curve \(y = \log _ { 2 } ( x - 3 )\).
2. The curve \(y = \log _ { 2 } x\) passes through the point ( \(a , 3\) ). State the value of \(a\).
3. The curve \(y = \log _ { 2 } ( x - 3 )\) passes through the point ( \(b , 1.8\) ). Find the value of \(b\), giving your answer correct to 3 significant figures.
4. The point \(P\) lies on \(y = \log _ { 2 } x\) and has an \(x\)-coordinate of \(c\). The point \(Q\) lies on \(y = \log _ { 2 } ( x - 3 )\) and also has an \(x\)-coordinate of \(c\). Given that the distance \(P Q\) is 4 units find the exact value of \(c\).

8 \includegraphics[max width=\textwidth, alt={}, center]{4d03f4e3-ae6c-4a0d-ae4d-d89258d2919a-4_417_931_255_607} The diagram shows the curve \(y = 2 ^ { x } - 3\).

1. Describe the geometrical transformation that transforms the curve \(y = 2 ^ { x }\) to the curve \(y = 2 ^ { x } - 3\).
2. State the \(y\)-coordinate of the point where the curve \(y = 2 ^ { x } - 3\) crosses the \(y\)-axis.
3. Find the \(x\)-coordinate of the point where the curve \(y = 2 ^ { x } - 3\) crosses the \(x\)-axis, giving your answer in the form \(\log _ { a } b\).
4. The curve \(y = 2 ^ { x } - 3\) passes through the point ( \(p , 62\) ). Use logarithms to find the value of \(p\), correct to 3 significant figures.
5. Use the trapezium rule, with 2 strips each of width 0.5 , to find an estimate for \(\int _ { 3 } ^ { 4 } \left( 2 ^ { x } - 3 \right) \mathrm { d } x\). Give your answer correct to 3 significant figures.

2

1. The curve \(y = \frac { 2 } { 3 + x }\) is translated by four units in the positive \(x\)-direction. State the equation of the curve after it has been translated.
2. Describe fully the single transformation that transforms the curve \(y = \frac { 2 } { 3 + x }\) to \(y = \frac { 5 } { 3 + x }\).

1. The curve \(C\) has the equation \(y = ( x - a ) ^ { 2 }\) where \(a\) is a constant.

Given that $$\frac { \mathrm { d } y } { \mathrm { dx } } = 2 x - 6 ,$$

1. find the value of \(a\),
2. describe fully a single transformation that would map \(C\) onto the graph of \(y = x ^ { 2 }\).

8

1. Sketch the graph of \(y = \frac { 1 } { x ^ { 2 } }\) \includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-12_1273_1083_404_482} 8
2. The graph of \(y = \frac { 1 } { x ^ { 2 } }\) can be transformed onto the graph of \(y = \frac { 9 } { x ^ { 2 } }\) using a stretch in one direction. Beth thinks the stretch should be in the \(y\)-direction.
   Paul thinks the stretch should be in the \(x\)-direction.
   State, giving reasons for your answer, whether Beth is correct, Paul is correct, both are correct or neither is correct.
   [0pt] [3 marks]