Solutions from graphical analysis

5 \includegraphics[max width=\textwidth, alt={}, center]{af7aeda9-2ded-4db4-9ff3-ed6adc67859f-07_778_878_255_630} The diagram shows part of the graph of \(y = a \cos ( b x ) + c\).

1. Find the values of the positive integers \(a , b\) and \(c\).
2. For these values of \(a\), \(b\) and \(c\), use the given diagram to determine the number of solutions in the interval \(0 \leqslant x \leqslant 2 \pi\) for each of the following equations.
   1. \(a \cos ( b x ) + c = \frac { 6 } { \pi } x\)
   2. \(a \cos ( b x ) + c = 6 - \frac { 6 } { \pi } x\) The diagram shows a metal plate \(A B C\) in which the sides are the straight line \(A B\) and the arcs \(A C\) and \(B C\). The line \(A B\) has length 6 cm . The arc \(A C\) is part of a circle with centre \(B\) and radius 6 cm , and the arc \(B C\) is part of a circle with centre \(A\) and radius 6 cm .

7. \begin{figure}[h] \end{figure} Figure 1 shows the curve with equation \(y = \mathrm { f } ( x )\).
The points \(P ( - 4,6 ) , Q ( - 1,6 ) , R ( 2,6 )\) and \(S ( 3,6 )\) lie on the curve.

1. Using Figure 1, find the range of values of \(x\) for which $$\mathrm { f } ( x ) < 6$$
2. State the largest solution of the equation $$f ( 2 x ) = 6$$
3. 1. Sketch the curve with equation \(y = \mathrm { f } ( - x )\). On your sketch, state the coordinates of the points to which \(P , Q , R\) and \(S\) are transformed.
   2. Hence find the set of values of \(x\) for which $$f ( - x ) \geqslant 6 \text { and } x < 0$$

6. \includegraphics[max width=\textwidth, alt={}, center]{00364339-8108-4031-8e67-6100810e8297-2_549_885_251_370} The diagram shows the graph of \(y = \mathrm { f } ( x )\).

1. Write down the number of solutions that exist for the equation
   1. \(\mathrm { f } ( x ) = 1\),
   2. \(\mathrm { f } ( x ) = - x\).
2. Labelling the axes in a similar way, sketch on separate diagrams the graphs of
   1. \(\quad y = \mathrm { f } ( x - 2 )\),
   2. \(y = \mathrm { f } ( 2 x )\).

5.The function \(f\) is given by $$f ( x ) = \frac { 1 } { \lambda } \left( x ^ { 2 } - 4 \right) \left( x ^ { 2 } - 25 \right)$$ where \(x\) is real and \(\lambda\) is a positive integer.

1. Sketch the graph of \(y = \mathrm { f } ( x )\) showing clearly where the graph crosses the coordinate axes.
2. Find,in terms of \(\lambda\) ,the range of f .
3. Find the sets of positive integers \(k\) and \(\lambda\) such that the equation $$k = | \mathrm { f } ( x ) |$$ has exactly \(k\) distinct real roots.

8. \begin{figure}[h] \end{figure} Figure 1 shows the graph of \(y = \mathrm { f } ( x )\).

1. Write down the number of solutions that exist for the equation
   1. \(\mathrm { f } ( x ) = 1\),
   2. \(\mathrm { f } ( x ) = - x\).
2. Labelling the axes in a similar way, sketch on separate diagrams the graphs of
   1. \(\quad y = \mathrm { f } ( x - 2 )\),
   2. \(y = \mathrm { f } ( 2 x )\).

\includegraphics{figure_1} Figure 1 shows a sketch of a curve C with equation \(y = \text{f}(x)\), where f(x) is a quartic expression in \(x\). The curve • has maximum turning points at \((-1, 0)\) and \((5, 0)\) • crosses the \(y\)-axis at \((0, -75)\) • has a minimum turning point at \(x = 2\)

1. Find the set of values of \(x\) for which $$\text{f}'(x) \geq 0$$ writing your answer in set notation. [2]
2. Find the equation of C. You may leave your answer in factorised form. [3]

The curve \(C_1\) has equation \(y = \text{f}(x) + k\), where \(k\) is a constant. Given that the graph of \(C_1\) intersects the \(x\)-axis at exactly four places,

1. find the range of possible values for \(k\). [2]