8 The diagram shows a sketch of the curve \(y = 2 ^ { 4 x }\).
\includegraphics[max width=\textwidth, alt={}, center]{f9a7a4dd-f7fd-4135-8872-2c1270d46a14-9_435_814_374_623}
The curve intersects the \(y\)-axis at the point \(A\).
- Find the value of the \(y\)-coordinate of \(A\).
- Use the trapezium rule with six ordinates (five strips) to find an approximate value for \(\int _ { 0 } ^ { 1 } 2 ^ { 4 x } \mathrm {~d} x\), giving your answer to two decimal places.
- Describe the geometrical transformation that maps the graph of \(y = 2 ^ { 4 x }\) onto the graph of \(y = 2 ^ { 4 x - 3 }\).
- The curve \(y = 2 ^ { 4 x }\) is translated by the vector \(\left[ \begin{array} { c } 1
- \frac { 1 } { 2 } \end{array} \right]\) to give the curve \(y = \mathrm { g } ( x )\).
The curve \(y = \mathrm { g } ( x )\) crosses the \(x\)-axis at the point \(Q\). Find the \(x\)-coordinate of \(Q\). - Given that
$$\log _ { a } k = 3 \log _ { a } 2 + \log _ { a } 5 - \log _ { a } 4$$
show that \(k = 10\).
- The line \(y = \frac { 5 } { 4 }\) crosses the curve \(y = 2 ^ { 4 x - 3 }\) at the point \(P\). Show that the \(x\)-coordinate of \(P\) is \(\frac { 1 } { 4 \log _ { 10 } 2 }\).